الفصل الاول مبادى الاحصاء والاحتمالات



????? ????? ????? ??????? ??????????? ??????? ??? ????????? ????????? ? ????? ???????? ? ????? ??????? ? ?????? ???????? ? ????????????? ???????? ?????? ???????? ? ??????????????? ?????? ? ??????? ???????? ?????? . ??????? ???????? ( ??????? ???????? ? ??????? ???????? ? ???? ??????? ? ??? ?????? ????????? ????????? ? ??????? ??????? ? ????? ?????? ? ????? ??????? ) .?????? ?????? ???????? ???????? ??? ??????? ???????? ?????? ( ????? ??????? ?????? ???????? ????????? ??????? ???????? ????????? ?????????? ? ????? ???????? ????? ???????? ? ????? ???????? ? ?????? ? ????? ??????? ?????? ? ??????? ?????? ? ??????? ).?????? ?????? ???????? ??? ??????? ???????? ?????? ( ??????? ? ???????? ???????? ???????? ??????? ???????? ????????? ?????????? ? ??????????????? ( ??????? ) ??????????? ? ?????? ???????? ? ?????? ??????? ? ????? ???????? )? ????? ????????? ???????? ????? ???????? ???????? .????? ??????? ??????? ????? .????? ????? ??????? ????? .???? ????? ??????? ? ?????? ???????-???? ? ?????? ??????? ?????????? ???????? ??????? (??? ????????? ???????? ( ????? ????? ?????? ????? ?????? ? ????? ?????? ) ? ??? ????????? ???????? ( ??????? ??????? ? ??????? ??????? ).??????? :- ??????? ?????? ?? ????? ???? ???? ??? ??? ???? ??????? ?????? ???? ??? ???? ?????? ????? ????? ?????? ????? ??????? ???? ???? ??????? ?????????? ??? ???? ??????? ?????????? 'Statista' ??? ???? ?????? ???? ??????? ?????????? ' Statistik ' ???? ???? ???? ??? ???????? ??? ??? ?????? ???????? ('political state') . ????? ????? ????? ?? ?????? ?????? ??????? ????????? ? ????????? ????? ?????? ????? ????? ????? ??? ????? ???????? ???????? ?????? ? ????????? ????? ?????? ??????? ?????? ???? ????? ??? ??????? ???????? ?????? ?. ????? ??????? ??? ??? ???????? ???????? ?????? ????? ? ???? ????? ???????? ?????????? ???????? ???????? ????????????? ? ???? ?????? ?? ???? ??????? ?????? ??????? ?????? ????? ????? ???? ????????? ? ??? ??? ??????? ? ???????? ?? ??????? ???????? ???????? ?? ??????? ???? ?? ?????? ????? 3050 ??? ??????? ?????? ???????? ????? ??????? ??? ???? ????? ?????? ?? ??? ?? 2030 ??? ??????? ???? ??? ?????? ?????? 3,800,000 ? ???? ????? ?????? ?????? ???? ?????? ?????? ?? 435 ??? ??????? . ???? ????? ?????? ????? ?????????? ???? ???????? ?????????? ??? ???????? ?????? ????????? ??? ???????? ???????? ??????? ???? ???????? ??? ????? (Graunt John) ?? ????? ?????? ????? ????? ????? ??? ?? ???? ????? ??????? ????????? ????? ??? ?????? " Bills of Mortality' Observation on London " ????? ?????? ?? ??? 1662?????? ??????? ????? ? ??????? ??????? ?????? ??????????? ??????????? ????? ?????? ???? ???????? ??????? ???? ????? ?? ??????? ??????????? ?? ?????? ??????? ???? ???? ????? ??? ?????? ' ????? ???????? ? ?????? ?? ?? ?? ????? ??????? ??? ??????? ??????? ???? ???? ?? ???????? (A.L. Bowley) ?????????? ????? ?? ?????? ?????? ?? ??? ??? ????? ????? ?????? ????? . ??? ????????? ????????? ???????? ?????? (i) ?????????? (??? ????????) ??????? ??? ???????? ???????? .(ii) ????? ??? ????? ????? ?????????? ????????? (iii) ??????? ??????? ?????? ???? ??????? ????? ??? ???? ??????? ???? ????? ?????? ????? ? ??? ???? ????????? ?????? ???????? ???? ??????? ???? ????????? ?? ??? ???????? ???????? ???? ???? ??????? ???????? ???????? ??????????? ?????? ????? ???????? ??? ?? ???????? ?????? ???? ???? ??? ????? ????????? . - ????? ?? ?????? (Yule & Kendal ) ???????????? ?????? ??????? ?????? ???????? ???? ????? ?? ???????? . ?????? ??? ???????? ????? ???? ???????? ??????? ? ??????? ??? ???? ??? ???? ?? ???????? .- ????? ?????? (Webster) ?????????? ??????? ???????? ???? ??????? ????? ?????? ??? ???????? ???? ???????? ???? ???????? ?? ???? ?????? ??? ?? ???? ???????? ??? ?? ??? ????? ????? . - ????? ?????? (Achenwall) ?? ?????? ?? ???????? ???????? ???? ???????? ???????? ? ????? ??????? ? ?????. - ????? ???? (L.R. Connor) ?????????? ????? ?? ??????? ?? ??????? ??? ???????? ??????? ??? ????????? ???????? ?????? ???? ????? ? ???? ?????? ???????? ?????????? . - ????? ????? (Secrist H.) ??????????? ?????? ?????? ???????? ???????? ????? ?????? ?? ???????? ???? ? ???? ???? ? ????? ??? ???????? ??????? ??? ????? ????? ???? ?????? ? ????? ????? ??????? ?????? ?????? ?????? ?????? ????? ?????? ???? ????? ????. ??? ???????? ???????? ???? ???????? ?????? ???? ???? ??????????? ??? ?????????? ??????????? ???? ??????? .??? ????? ??????????? ??????? ???? ???? ????????? ?????? ???????? ???? ??????? ???????? ???????? ??? : ?????????? ???? ?????? ?????? .??? ?????? ?? ???????? ???????? ???? ???? ???????? ??? ?????? ? ?????? ???? ??????? ????? ??? ???? ??????? ???? ???? ??? ???? ???? ? ???? ??? ??????? ???????? ???? ??????? ??? ?????: (a) ????? ????? ?????? ????????? ???? ??????? ?????????? ???????? ? ?????? ??? ??????? ?? ??????? ??? ???????? ?? ???? ? ???? ????? ??????? ???? ?????? ?? ????? ?????? ? ?????? ? ????? ????? ?????? ??????? ????? ??????? .(b) ??????? ???? ?????? ????? ??????? ???????? ???????? ???? ?????? ????? ???????? . ????? ??????? ? ??? ??????? ??? ??? ??????? ? ?????? ? ??? ???? ?????? ??????? ? ???? ??????? ...?????? ? ?? ?????????? ??? ???????? ???????? ???? ????? ????? ?? ??? ??? ???????? ?????? ??????? ???? ???????? ?? ???????? ????????? . ???? ??? ???? ???? ????? ????? ???????? ???? ??????? ??? ??? ???? (??????) ??? ????? ????? ??????? ?????? ??? ??? ? ???????? ??????? ???????? ? ?????? ? ??????????? .?????????? ???? ?????? ???? ???? ???????? .?? ?? ??? ???? ???? ?? ?????? ??? ???????????????? ? ???? ??????? , ?????? ???????? ???????? ????? ??????? ?? ?????? ??? ??????? ????????? . ????? ?????? ??????????? ??????? ??? ?????? ???????? ???????? ???? ???????? ?????? ?????? ??? ????????? ??????? ?? ????? ?????? ?? ?????? ???? ????????????? ?? ??? ????? ???? ????????? ??????? ?? ????????? ???????????? ?? ????? ?????? ????? ????????? ????? ?????? ?? ?????? ???????? . ???? ???????? ???????? ???? ???????? ????????? ???? ??????? ?? ???????? ???? ??? ??????? ?????? ? ????????? ??? ???? ????? ????? ????? ?????? ? ??????? ????? ???? ??????? ??? ???? ???????? ????????? ???????? ???? ??? ???????? ???????? ??? ???????? ??? ?????? ??????? ?????? ???? ??????? ?????? ???? ??????? ?? ??? ?????? ?????? ??? ????? ???????? ?????? ????.?????????? ??????? ??? ???? ????? ???????? ???????? . ??????? ?????? ????????? ??? ?????? ???????? ?? ?????? ???????? ? ????? ??? ??? ? ?? ???? ??????? ??? ??? ????? ??? ??????? ???? ?????? ???? ????? ?? ???? ??? ???? ???? ???????? ???? ?????? ???? ??? ?? ????? ??? ???? ?????? ???? ?????? ???? ??? ???????? ???? ???? ???????? ???????? ? ??? ???????? ?????? ???? ??????? ???????????? ?? ???? ????? ?? ?? ???? ??????? ??? ????? ???? ?????????? ? ??? ????????? ? ????? ?????? ?????? ? ??? ?????? ?????? ????????? ? ??? ???????? ???? ??? ??????? ???????? ??? ?????? ???????? ???????? ???? ??????? ??? ???? ?????? . ???? ??????? ???????? ????? ?? ???? ????? ?????????? ?????? ??????? ? ???? ??? ???????? ???? ?????? ??? ??? ??? ??????? ?????? ??????????? ??? ???????? ??????????.?????????? ???? ?????? ??? ??????? (????) ??????? ?????? ?? ????? ??? ???????? ????? ???????? ????????? . ????????? ???????? ???? ?????? ??? ??????? ???????? ??? ???????? ??? ???? ?????? ? ??? ???????? ?? ???? ??? ????? ??? ?????? ???? ?????? ????? ? ????? ???? ???????? ??? ??? ???? ?????? ????? ?????? ??? ????? ???????? ?????? . ??? ?????????? ?????????? ????? ??????? ?????? ??? (????) ?? ??????? ?????? ???? ?????????? ???? ???? ????? ??????? ???????? ? ????? ?? ???????? ?????? ???? ?????? ??? ????? ?????? ?? ????????? ??? ?? ???? ?????? ?????? ??? ?????? ???????? ?????? ????????? ???????? ???? ???? ?????? ? ?? ??? ?????? ? ??? ???????????? ???? ?????? ?????? ????? ???? ??????? ???? ???????? ???? ?? ?????? ???? ???? ???????? ??? ???? ???? ???? ?????? ???????? ???? ??????? ?????? ??? ????? ???????? ????? ?????? ??????? ? ???????? ??? ??? ??? ???? ?????? ??????? ????? ??? ????? ?????? ????????? ???? ??? ????? ???????? ?????? ???????? ???? ??????? ??????? ?????? ??????? ??? ??????? ??? ??????? ????????? ?????? ?? ??????????? ????? ??? ???????? ??????? (???????? ???????? ) ????? ?? ???? ??????? ??? ???? ????????? ???????? (???????? ?????????). ??? ???????? ?????? ??? ??????? ?????? ???????? ?? ???? ????????? ??? ????? ?? ???????? ???? ???? ??????? ?????? ????? ???? ????? ?????? ??? ?? ????? ?? ???? ???? ?????? ???????? ??????? .?????????? ????? ??????? ?????? ????? ???? ????? .??? ?????? ??? ???????? ???? ??????????? ????? ?? ???????? ???? ?? ??????? ??????? ??????? ???????? ??????? ?????? ???????? ???????? ???? ?????? ?????? ?????? ??? ??? ???? ???? ?????? ??? ?????????? ?????? ??? ??????? ????? ??? ??? ?????????? ?????? ???? ???????? ????? ?????? ????? ? ?? ??? ??? ???? ? ???? ??????? ?? ??????????? ????? ??????? ??????? ????? ?????????? ???????? ??????? . ????? ??? ??????? ???????? ???????? ??????? ?? ?????? ??? ???????? ????????? ??? ?? ??????? ??????? ??????? ????????? ?????? ???????? .?????????? ?????? ???? ??????? ????? ??? ???? ???????? ?????? ???? ????? .??? ??????? ???????? ???? ??????????? ????????? ?????? ???? ??????? ????? ???????? ????? ???? ??? ????? ???? ??? ??? ??? ??? ? ??? ????? ?????????? ???? ?? ???????? ?? ???? ??????? ???????? ????? ???? ? ???? ???????? ???? ??????? ???? ????????? ?? ?????? ?????? ??????? ??????? ??????? ??? ???????? ????????? ??? ?? ????? ??????? . ????? ??? ?????? ?? ???????? ???????? ???? ??????? ???????? ???????????? ?????? ???????? ???? ?????? ????????? ??? ?????? .1-2 ??? ????????? ????????? ??????? (population) :- ????? ?? ?????? ??? ??????? ?? ??????? ??? ????????? ???? ???? ??? ??? ??????? ?????? .???????? (sample) :- ?????? ????? ??? ????????? ??????? ???? ??????? ? ???? ???? ??????? (size) ???? ??? ???????? ?? ?????? .????(1-1 ):-?????????? ???????? ??????? ????????? ???? ???????? ? ????? ??????? ????????? ??????? ????? ? ?? ???????? ??? ????? ???????? ?? ???????? ? ?? ????????? ???????? ??????? ????????? ???? ?????? ???? ?????? ? ????? ??????? ????????? ??????? ????? ?????? ???????? ?? ???????? ? ??? ?? ??????? ???? ?????? ???????? ?? ??? ?????? ??????? ???????? ???? ??????? ??? ?? ?????? ????? ????? ?? (4) ???? ?????? ??????? ????? ?????? ???? ?????? ???? ??? ?????????? ????????? ??????? ? ???????? ?????? ????????? ( ??????????? ? ???? ?????? ? ???? ??? ???? ????? ????? ? ??? ??????? ? ??? ??????? .....)????? ??????? ???????? ????? ?????? ??? ????? ??? ??? ?????? ???? (84) ????? ??? ???? ?????? ???????? ?????? ?? ??? ?????? ??????? ??????? . ????????? ???????? (statistical inference) :- ???? ?????? , ???? , ????? ??? ????? ??? ??????? ????? ??? ????? ?????????? ???????? ?? ??????? .????? ??????? (statistical classification)??????? ?????? ????? ???????? ???? ??????? ??? ??????? ?????????:(a) ????? ???? (Theoretical Statistics ) ?(b) ????? ??????(Applied Statistics) ????? ??? ??????? ???? ??????? .(a) ??????? ?????? (Theoretical Statistics) ????? ?? ???? ??? ??? ???? ???? ????? ?????? ???? ???? ???:i)) ????? ???? (Descriptive Statistics) :- ????? ???? ??? ?????? ???? ?????? ??????? ?? ???????? ? ?????? ????????? ????? ??????? ????????? ??????? ?????? ????? ??????? ??????? ????????? ? ??????? ???????? ? ??? ? ??? ????? ???????? ?? ??????? ?????? .(ii) ????? ???????? (Inductive Statistics):- ???????? ???? ??? ?????? ???? ????????? ???? ??????? ??? ????? ??????? ??? ???? ??????? ? ????? ???? ?????? ?????? ??????? ??????? ????? ????? ??????? ??????????.(iii) ????? ???????? (Inferential Statistics):- ???????? ???? ??? ?????? ????????? ???? ????????? ????????????? ????????? ???? ????? ??????? ??????? .(b) ????? ?????? (Applied Statistics) :- ??????? ????? ?????? ?????? ?????????? ??????? ???????? ? ?????? ???????? ??????? ?????????? ? ?????? ?????? ??????? ? ?????? ???????? ?? ???? ??????? ? ...???. ? ???????? ?? ??????? ???????? .1-3 ????? ???????? (Type of data)?? ???????? ????? ???????? ??? ??? ????? ???? ???????? ??????? (Quantitative data) ????? ?? ??????? ????? ??? ?????? ???? . ??? ???????? ??? ??????? ???? ???????? ???? ???? ??? ??? ??? ?????????? ?? ???????? ???? ????? ??????? ????? (Qualitative ) ??? ????? ????????? ??????? (Categorical Data) ?????? ???? ??????? ???? ?? ??? ????? ??????? ????? (Nominal Data) ??????? ??????? (Ordinal Data) .?? ????? ?????? ????????? ???????? (binary response) ????? ?????? ??? ??? ???? ?? ??? ????? ? ????? ??? ???? ????? ?????? ??? ???? ?? ???? ??? ????? ? ????? ?????? ???????? ??? ???? ?? ??? ? ??? ???? ????????? ???????? ????? ???? ????? ??? ??????? ?????? . ????(1-2) :- ???? ???? ???? ?? ????? ????? ??????? ????????? ????? ???? ?? ????????? ??????? ???? ?????? ????????? ???? ????? ?????? ?????? ???? ?? ??????? ????? ??????? ??? ???? ????? ???? ?? ??? ??????? ???? ?????? ?????? ?? ??? ?? ?? ????? ????? ??????? ?????? ? ???????? ??? ?????? ??? ( ??? ?? ??? )??? ????? ?? ?? ???? ?? 100???? ????? ? ???????? ?????? ??? (??? ????? ??? ?? ???? ???? ?? 100 ?? ??? ????? ??? ?? ???? ??? ?? 100)?? ?? ????? ????? .?? ???? ????? .?? ?? ????? ???? ??????? ??????? ??? ???? ????? ????? ??? ??????? ??? (??? ?? ???) ?????(0?1) .??? ????????? ????????? (ordinal response) ??? ????? ?? ??????? ????? (categories) ????? ????? ???? ?? ???? ?? ??????? ????????? ???????? ????? ?????? ??? ??????? ????? ????????? ?? ???? ????? ? ?? ????? ????? ? ???? ? ???? ????? ?????(1?2?3?4?5) ???????? (-2? -1? 0? 1? 2) ??? ????? ????? ??????? ?? ????? ????? ????? ??? ??? ???? ??? (??? ? ????? ? ???? ) ????? (1?2?3) ?? (-1?0?1) . ??? ???????? ????????? ????? ????? ??? ??????? ????? ?????? ???? ?????? ??? ???? ???????? ? ???????? ??? ??????? ? ??? ???????? ?????? ???? ??????? ?????? ?????? ??? 1, 2, 3 ??? ????? ???? ??? ????? ? ??? ? ??? ????? ????? .??? ???????? ??????? ???? ????????? ??????? (nominal response) ????? ?? ??????? ????? ????? ????? ???? ?? ???? ?? ??????? ????????? ????? ??????? ????? ?????? ??? ?????????? ????? ????????? ???? ?????? ?? ???? ???? ??????? ??????? ?? ??? ????? ????? ????? ???? ??? ?????? ???? ???? ??????????? ??? ?????? ???? ??????? ?????? ?????? ? ??? ????? ???????? ? ??? ????? ?????? ???????? ??? ???????? . ???????? ???????? (Cross-sectional data) ??????? ?????? ??? ?????? ??????? ??? ??????????? ????????? ?? ??? ?????? ??????? ??? ???? ?????? ???????? . ?????? ??????? ??????? (Time series data):- ????? ?? ??????? ?????? ??? ??? ??????? ??? ????????? ?? ???? ?????? ????? ????? ??? ??????? ??????? ???? ?????? ? ???? ??? ????? ????????? ??????? ????? ?????? ? ????? ???????? ??????? ???? ??????? ? ????? ???? ????? ?????? ??????? ?? ?????? . 1-4 ????? ??????? ( samples types) ?? ??????? ???? ?????????? ??????? ?? ????? ????? ???????? ??? ?? ??????? ?????? ?????? ???? ???? ???? ??????? ????????? ??????? ???? ??????? ???????? ???? ???? ??????? ?????? ??????? ???? ???? ??? ??????? ????? ??????? . ??? ????? ???? ???? ????? ??????? ???? ??????? ????? ??????? ( representative ) ? ??? ?? ?? ????? ???? ?????? ????? ??????? ??????? ??????? ???? ??????? ( biased ) ??? ??? ????? ??? ????? ????????????? ????????? ??????? ??????? ??? ????? ?????? ???????? ???? ???? ???? ??????? ??????? ? ???? ?? ???? ?????? ???? ?????? ????????? ???? ??????? ??? ???????? ????????? ?? ??? ???? ??? ??????? ??????? ??? ?????? ?? ??????? ???? . ????? ???????? ??? ??????? . ??? ???? ??????? ??????? ??????? ( ????? ???? ???????(???????) ?? ??????? ). ????????? ??? ??? ???? ??????? ??????? . ??????? ??????? ?????? . ??????? ???? ???? ???? ??? ???????? ??? ???????????? ??? ?????????? .??? ??? ?????? ??????? ?????? ?? ??? ?????? ????????? ??????? (simple random sample) ?? ????? ?????? ?????? ???? ??? ???? ???? ?? ??????? ???? ????? ??????? ???????? ? ??????? ???? ??????? ????? ? ????? ????? ???? ????? ????? ????? n ???? ???? ??????? ????????? ??? ????? ??????? ?????? ????? ????????? ???????? ??????? ??? ?????? ? ? ???????? ????????? ????? ?????? ? ????? ???????? ???????? ???????? ???? ???????? ??? ?????? ???? ???????? ? ???? ???? ?? ??????? ????????? ???????? ?????? ???? ????????? ????? ??????? ????? ???????? .??????? ????????? (systematic sample) ??????? ???? ???? ????? ?????? K th ?? ????? ??????? ??? ????? ??????? ?????? ?????? ????? ??? ?????? ?????? ??????? ?? ????? ????? (???? ????? ??? ?????? ???????? ) ???????? ????? ??????? ?? ??????? ? ??? ?????????? ???? ?????? ????? ????? ??????? ?? 1 ??? N .????? ???? ???????? n ???? ??????? ???????? .??? ??????? K = N/n .???? ??????? ??? ????? ???? ?????? ??? 1 ??? K .??? ??????? ???? K th???? ??????? .???? (1-3 ):- ???? ?? ??? ??????? (N=200) ??? ????? ???????????? ?? 1 ??? 200 ???? ????? ???? ??????? ????? ?? ???? ??? ?????? ????? (n=10) ????? 200/10 = 20 =K=N/n ? ???????? ?? 1 ??? 20 ????? ??????? ????? 12????? ????? ??????? ??????? ??????? 12,32,52,72,92,112,……………………,192 ????????? ???? ????? ?????? .?????? ???????? (strati?ed sample ) ??????? ??????? ???? ????? ??? ?????? ?????????? ???????? ??????? ???????? ???????? ???????? ??? ????? ???? ????? ????? ???????? ? ??? ????? ??????? ???? ?????? ??????? ?? ??????? ??????? ?????????? ??????? ? ??????? ???????? ??????? ?? ???? ???????? ??? ?? ?????? ??????? ????????? ????? ( ????? ??????? ??? ???????? ???? ??????????) ????? ??????? ????? ??? ???? ?? ?????? ?? ? ???????? ???? ??? ??? ???????? ????? ???????? ???? ???? ???? ???? ???????? ????? ????? ?????? ? ???????? ??????? ?? N ?? ??????? ???? ??? ??????? ????? ??? ??? ????? ???? ???? ????? ???? ????? ?????? ??? ?? ??? ???? ????? ???? ????? ???????? ?? ???? ????? ???? ???? ???????? ?????? ???? ????? .??????? ??????? ????????? ?????? ??????? ????? ????? ??????? ????????? ??? (???? ????? ???? ???. ) ????? ???? ??????? ??? ???????? ??? ????? ?? ??????? ????? (sub populations) ????????? ??? ??????? ????????? ????????? ?????? ??????? ???? ???????? ???? ????? .????? ?????? ???????? ?? ???????? ???? ???????? ????? ???????? ?????? ??? ????? ???????? ???? ???? ????? ???? ??? ????? ??????? ???? ??????? ??? ????? ??????? ??????? ?.?????? ?????????? (cluster sampling) ?????? ????? ???????? ??? ???????? ?? ???????? ???? ????????? ????? ??? ???????? ???????? ???? ??????? ? ? ??? ?? ????? ??????? ????? ????? ?????? ? ??? ???? ?????? ???????? ??? ????? ??????? ????????? (?????) ??? ???? ??????? ?????????? ??? ????????? ????????? ??? ???????? ?????? ???? ?????? ????? ??????? ??? ???? ??????? ???????? ????????? ??? ?? ?????? ????????? ???? ?? ??? ??????? ?????? ?????????? (area sampling. ) ????(1-4):- ?????? ???? ????? ?? ???????? ????? ??? 10 % ???? ???? ??????? ?? ??? 6 ????? ?? ????? ?????? ? ??? ???? ????? ?????? ????? 10 % ?? ??????? ??????????? ? ??????? ??????? ??????? ???? ??? ???? ?????? ????6 ????? ?? ??? ??????? ????? ???? ??????????? ??????? ???? 6 ????? ???? ??? ?????????? ??? ????? ???? ??????? ?????????? ? ???? ????? ???? ? ????? ????????? ?? ???????? .??????? ??????? ??????? (Multiphase sampling) ????? ??? ??????? ????? ?????????? ???? ???? ???????? ?????? ???????? ?????? ??? ?????? ????? ???? ?? ??? ????? ??? ?????? ???? ??????? ??????? (sub samples) ???????? ?????? ?. ?? ?????? ???????? ??????? ?????? ????? ?? ?????? ?? ?????? ????????? ??????? ??????? ?????? .????(1-5 ):- ???? ?? ????? ?????? ???? ??????? ?????? ??????? ??? ??????? ??????? ??? ?????? ??????? ??? ??????? ? ??? ??? 10 % ?? ??????? ?? ???? ?? ??????? ????????? ??? ??? ?? ??? ??????? ?????? ?? ???? ???????? ???? ????????? .1-5 ?????? ???????? (frequency table):- ?? ?? ??? ??????? ?????? ?????? ???????? ?? ????? ????? ???????? ???? ?? ???? ????? ??????? ???? ??????? ???????? ?????? ?? ???? ????? ???????? ????? ??????? ?? ????? ???? ???????? ????? ??? ????????? ??????? ??? ??????? ???? ?????? ??? ????? ??????? ???????? ?? ?? ??? ????????? ??? ?? ???? ???? ???????? ??? ??????? ??? ?? ??????? ???????? ????? ??? ??? ?? ?????? ????? ????????? ????? ????? ????? ?? ?????? ? ??????? ???? ?????? ???? ??? ?????? ???????? ??? ??? ????? ???? ?????? (???????) ? ?? ????????? ????? ??????? ????????? ???? ????? ?????? ??? ??? ?????? ???????? ?????? ??????? ??? ???? ??? ????????? ???? ???? ?? ??? ????? ???? ????? ??????? ???? ???????? ???? ????? ???????? ???? ??????? .????? ???? ???? ?????? ????? ?????? (FREQUENCY TABLE AND HISTOGRAM ) ??? ???? ?????? ??????? ???? ????????? ? ?? ???? ????? ?? ???? ??????? R* = maximum value ? minimum value.???? ?????? ????? ????? ??????? R = maximum value ? minimum value + 1????? ???? ?? ???????? ??? ???? ?????? ???????? . ????? ??? ?????? (k) ????? ????? ??????? ???? ??? ?? ?????? ???? ?? (5) ??? (20) ???? ???? ???? ???????? ????? ?????????? ???? ????? ????????????? ??? ???? ?????? .????????? ???? ???(yule) ??? ????? k=2.5 4n????????? ???? ????? (sturges) ????? ?????? k=1+3.322 log10(n) ????? ???? ????? ??? ???? .????? ??? ????? (????? ) ????? ???? ????????? ???? ??????? ??????? l=Rk????? ?????? ?????? ???? ??????? ????? ??? ?????? ?? ?????? ?????? l1 ???? ???????? ?? ?????? ??????? ?????? ???? ?????? ????? ???? ?????? ?? ?????? ?????? uk ????? ???????? . ??? ?????? ?????? (class intervals) ????? ?? ??? ???? (class boundaries )?. ??? ?? ?? ??? ???? ?? ?????? ????????? ??? ????? ?????? .???? ???? (???? ??????) ????? ??? ?????? (class intervals)? ????????? ??? ??? fi? (?/ ??) ????????????? fi/n??? ?????? ??? n ???? ??? ????????? ????? ????? ?????? ??? ??? ??????? (grouped frequency distribution) ??? ?????? ????? ?? ???? ????? ? ??????? ?? ?????????????? .Frequency?????????Class intervals ??????f1 l1-u1f2 l2-u2f3 l3-u3--------------------------fklk-ukn=i=1kfi??????? ???????? ?? ???? ????? ?? ????? ??? ?? ???? ????? ????? ?? ????? ??? ???? ????? ????? ??????? ?? ??????? ??????? ?????? ???? ??? ?????? ????? (ungrouped frequency distribution).???? ????? ??????? ???? ???? ???? ???? ????? ???????? ???? ????????? ??????????? ??????? .????? ????? ????? ???? ?? ?????? ??? ???? ???????? ???? ??????? ???????mi=(li+ui)/2????????????? (relative frequency)???? ?? fi ????? ?? ??????? ????? (?????) i ? ????? ?? n ???? ????? ?? ????????? .n=fi ?? ????????????? (relative frequency) ????? (?????) i ????? ?? ?????? fi/n ? ??? ?? ??????? ?????? ???????? (cumulative relative frequency) ????? (?????) i ????? ?? ?????? ??????? ??????? j=1ifj/n .??????????????? ?????? :- ????? ???????? j=1ifj , i=1,2,3,….k ? ??? ?? k ???? ??? ?????? ????? ????? ????? ????????? ????????? ??? ?????? ?????? ?????? .??????????????? ?????? :- ????n ????? ????? ?? ????????? (????? ????????? ?????) ??? ????? ?????? ????? ????? ?????? ???????? n- j=1ifj , i=1,2,3,….k ? .????(1-6 ):- ???????? ??????? ???? ??????? ???? ??? ??? ?? ?????? ?????? ???? ???? ????? ??????? ????? ???? ????? ?????? .615585824898776574409981854479675649778655494374707154999173 ????:- ??? ????? ????? ???? ?????? ???? ????? ??? ?????R = maximum value ? minimum value + 1R = 99 ? 40 + 1 = 60k=2.5 4n=2.5 430=5.6≈6l=Rk=606=10frequencyclasses640-450-360-870-580-490-10030j=16fj ????? ????? ??????? ???????? ??????? ?????? ??????? ???? ?????? ?????? ??????? ??????? ?????? ??????? ??????? ??????ficlasses306640-2410450-2013360-1721870-926580-430490-10030i=16fi ????(1-7 ):- ??? ??? ??? ??????? ??????? ???? ????? (30) ?????? ?? ????? ??????? ??? ?? ?????? ?????? ??? ???? ?????? ?????? ?????? ?????? ???? ?????????? ?????? ???????? ??????? ???? ?????????? ?????344354265535446751236655423445????:- ?? ???? ???? (MINITAB) ???? ???????? ?? ??? ??????? ????? c1 ?? ????? ???????????? > > ???????? ???? ????? ??? ?? ??? ?????????? ?? ???? ??? ??????? ??????? ??? ???? ?? ???? ???? (SPSS) ???? ?? ??? ?????? ???? ?????? ? ??? ????? ???????? ?? ???? ????? ???????? ????? ?????? ?????? ?? ????? ??????? ???????? ???? ???????? ????? ???????????? ?? ??????? ?????? ???? ?????? ??? ????? ?? ?????? ???????? ????? ????? ????? R=7-1+1=7 ??? ?????? ??????? ???????? ???? ?? ??? ?????? (7) ???? ??? ??? ????? l=Rk=77=1 ?? ?????? ????? ???? ?????? ?????? (7) ???? ??? (1) ? ??????? ?????? ???? ????????? fi ??? (1) ???? ???? ???? ????? ??? ??????? ???? ? ?(3) ???? ???? ???? ??? ??? ??????? (2) ???????? ?????? ??????? ?????? ????????????? ???? 130=3.33 , 330=10,530=16.67,….. ??????? ?????? ??????????????? ????? ???? ????????? ??? ??? ?????? ?????? 1 , 1+3=4 , 4+5=9, 9+8=17,……… ??? ?????? ?????? ??????? ?????? ???????? ??? ??? ??? ?????? ?????????? ??? ??? ????????????? ?? ?????? ?????? ?????? 3.33 , 3.33+10=13.33 , 13.33+16.67=30.00,….. , .??? ?????? ???????? ?????? ??????? ?? ????? ????? ???? ????? ?????? ?????? ?????? ????????? (?????????) ???? ?????? ????(1- 8):- ???? ??? ???? ????? ?????? ?? ??? ????????? ???????? ???? ????????????? ???????? ??????? ??????????? ?? ?????? ?????? ?? ?????? ? ???? ?? n=fi=1669 ???????? ???? ???? ??? ?? ???.1-6 ??????? ???????? ?? ????? ??????? ???? ?? ??????? ?? ??? ??????? ?????????? ??????? ???????? ??? ???? ??????? ??? ????? ????????? ???? ????? ??? ?? ????? ????? ????? ??????? ???? ?????? ??????? ?? ?????? ????? ??? ??? ??? ??????? ? ?????? ????? ???????? ???? ?????? ??? ????? ??????? ?? ??? ??????? ???????? .1-6-1 ??????? ???????? ( bar graph):- ??? ????? ?????? ????? ??????? ?????? ?? ????? ???????? ???? ????????? ( ?? ????????? ??????? ) ????? ???? . ????(1-9 ):- ???? ??????? ???????? ???????? ??????? ????:- ????? ?????? Graph > ?Bar Chart 13252450????? ?? ????? ???????? ??????? ????????????? ????????? ????? ???? ???? ????? ?????? ????????????? ?????? ???? ??? ?? ?????? ????? ?????? ??????? .???? ??? ???? ?????? ?????? ??????? ?? ??????? (??????? ? ??????? ???? ?????? ? ?????? ?????? ???? ???? ?????? ?????? ????????? ??????? ??????? ?????????? ??? ??? ??????? ???? ?????? ?? ???? . 1-6-2 ??????? ???????? (pie chart) :- ????? ?? ????? ????? ??? ????? ????? ??? ?????? ??? ??????? ?? ?????? ???? ???? ??? ????? ?????? ? ??? ???? ???? ???????? ??????? pci=nin×3600???? (1-10 ):- ?????? ?????? ???? ????? ???????? ??? ????????? ???????? ? ??????? ??? ??????? ???????? .???? :- ???? ???? ???????? ?????? ?????????? ??? ??????? ????? ??????? ?? ????????? ????? ???? (1669) ? ??? ?????? ???????? ???? ??? ?????? ?????? ?? ?????? ?????? ???? ???????????? ??? ?????? ????? ??????? ??????? 4141669×360=89.299 , 1321669×360=28.472,……??????? ?? ??????? ???????? ???? ??? ????? ?????? ????? ???? ?? ?????? ?????? ????? ???? ????? ?? ??????? ???? ????? ??? ?????? ????? ? ??? ???? ??????? ???? ?????? ???????? ??? ?? ???? ?????? ????? ???? ????????? ?????????? ?? ??????????? Graph > Pie Chart.?? ???? ?????? ???? ????????? ??? ?????????? ??????? ???? ??? ?? ???? ????? ??????? ????? ????? ?? ???? ??? ????? ???? ?? ?????? ?? ???? ?? . ???? ??? ?? ???? ????? ?? ???? ?? ???? ????? ??????? ??? ?? ????? ??????? ??????? .1-6-3 ???? ???????(line chart) ????(1-11 ):- ???? ?? ????? ??????? ???? ??????? ???????? ??????? ?????? ?? (2014-2021) ?? ??????? ??? ??????? .202120202019201820172016201520143965332515974???? :- ?????? ?????? ???? ??????? ??????? ???? ???????? ?????? ??? ???? ?? ???? ????? ?????? ??? ?????? ??? ?? ????? ????? ??? ???? ????? ??? ?????? ??????? ?????? ????? ??? (2020) .1-6-4 ??? ?????? ????????? ????????? (Empirical CDF graphs)???? ????? ????? ????? ?? ???? ?????? ?????????? (1-12 ):- ???? ?????? ????????? ????????? ???????? ??????? (0.5, 1.0, 2.1, 2.1, and 7.3.) ?? ????? ??????? .???? :- ???????? (5) ? ????? ?? ????? ?????? ??? ?????? (2.1) ????? ????? ???? ????? ???? ???????? ????? ?? ???????? (0.5) ?????? ???? ??????? ?? ????? ????? ????????? ???? (20%) ???? ??? ???? ?????? (0.5,20) ??????????? ?? ????? ??????? ?????? ???? ?? ???? ?? ????? ????? ?????? ?????? ?????? ????? ?????? ?? ?????? ? ????? ?? (20%) ?? ????? ???????? ? ???? ??? ?? (0.5). ?????? ?????? ??????? (1.0) ????? ???????? ?????? ?????? ?? (40%) ??????? ???? ?????? ???????? ??? ???? ??? ????? ??? (1.0). ?????? (2.1) ??????? ?????? ? ???? ????? ?????? ????? ?? (80%) ???? ?????? ???????? ??? ???? ??? ????? ??? (2.1) . ?????? ?????? (7.3); ???? ????? ?????? ????? ?? (100%) ???? ?????? ???????? ??? ???? ??? ????? ??? (7.3) .??? ???? ???????? ???? ??? ????? ?????? ???????? ??????? ?????? ??? ?? ??????? ?? ????? ????? ??? ?????? ????????? ?? ???? ???????? ???? ??? ??????? ? ?????? ??????? ?????? (2.6) ???????? ?????? (2.718) ???????? ??????? .1-6-5 ??????? ???????(boxplot )??? ????? n ?? ????????? ??? ???? ???? ??? ???? ???? ?? ???? ?????? ???????? ??? ????? ?????? ?????? ???? ??? ????? ??????? ?? ??? ????? ??????? ? ?? ??? ???? ?????? ???? ????? ?????? ????? ????? ??????lower fourth xi ? ????? ???? ?????? ???? ????? ?????? ????? ????? ?????? upper fourth xi ?? ???? ???? ?????? ??? ???????? ????? ?????? ?????? ? ?? ???? ????? (fourth spread ) ???????? ??????? fs=upper fourth xi-lower fourth xi??????? ????? ??? ??????? ?????? ???? ???????? ????????? smallest xi, lower fourth xi, median xi, upperfourth xi, largest xi???? ??? ??????? ??????? (poxplot) ? ?? ?? ?????? ?? ????? ???? ???? (outlier ) ?? ????? ?????? ?????? ??? ????? ?????? (extreme) ??? ?? ???? ?????? ????? ??? ?? ???????? ???? ??? ????? ?????? (mild) ????? ??? ??? ??? ???? ????? ???????? ????? ???? ???? ?????? ?????? lower fourth xi-1.5fslower fourth xi+1.5fs??? ???????? ????? ???? ((outlier ??? ????? ??????? (mild) ??? ???? ???? ?????? ? ???????? ?????? ??? ?? ?????? ??? ???? ?????? [36.5,108.5] ????????? ??? ????? ??????? ??? ????????? (115,125,125).??? ??????? ???? ???????? ????? ???? ?????? ?????? upper fourth xi-3fsupper fourth xi+3fs??? ???????? ????? ???? ((outlier ??? ????? ??????? (extreme) ??? ???? ???? ?????? ? ???????? ?????? ??? ?? ?????? ??? ???? ?????? [24.5,162.5] ????????? ??? ????? ??????? ??? ???? ?? ?????? ????? ???? ?????? .?????? ?????? ???? ??????? ?? ??? ????? ?????? ??????? ??????? ????(1-13 ):- ???? ??????? ??????? ???????? ??????? ? ???? ???????? ????? ?????? ???? ???? ????????? ??? ????? ??? ???????? ???????? ??? ???????? ??????? ??????? ?? ??? ??????? smallest xi=40lower fourth xi=70+752=72.5median xi=90 upper fourth xi=95+982=96.5largest xi=125fs=96.5-72.5=24????? ???? ??????? ??????? ?????? ?????? ?? ?????? ????? ?????? ???? ??? ?????? ?? ?????? ????? ?????? ? ???? ??? ??? ???? ?????? ?? ???????? ? ??? ?? ??? ??????? ???? ????????? ?? ??? ???????? ( ????? ????? ???? ?????? ?????? ??????? ???????).??? ???? ????? ???????? ????? ???? ?????? ?????? lower fourth xi-1.5fs=72.5-1.5*24=36.5lower fourth xi+1.5fs=72.5+1.5*24=108.5??? ???????? ????? ???? ((outlier ??? ????? ??????? (mild) ??? ???? ???? ?????? ???????? ?????? ??? ?? ?????? ??? ???? ?????? [36.5,108.5] ????????? ??? ????? ??????? ??? ????????? (115,125,125).??? ??????? ???? ???????? ????? ???? ?????? ?????? upper fourth xi-3fs=96.5-3*24=24.5upper fourth xi+3fs=96.5-3*24=162.5??? ???????? ????? ???? ((outlier ??? ????? ??????? (extreme) ??? ???? ???? ?????? ???????? ?????? ??? ?? ?????? ??? ???? ?????? [24.5,162.5] ????????? ??? ????? ??????? ??? ???? ?? ?????? ????? ???? ?????? .1-6-6 ????? ?????? (dotplot)?? ????? ?????? ????? ????? ???? ?????? ???? ??? ???????? ??????? ?????? ???? ?????? ???????? ????? ?? ????? ?? ????? ???? ??? ??? ???? . ?? ?? ?????? ??? ????? ????? ??? ???? ???? ?????? .?????? ???? ???????? ???? ?? ??? ? ??? ??? ??? ?? ?????? ???? ??? ??? ????????? ???????? ??? ??? ??????? ?????? ? ????? ??? ????? ??????? (stem-and-leaf) ??? ????? ?????? ?????? ???????? ??? ??????? ??????? ?? ???????? ??????? .????(1-14 ) :- ???????? ??????? ??????? ???? ?????? (33) ???? ?? ????? ??????? ????? ????? ? ??????? ?????? ????? ??????? ?????? ???? ???????? ? ?? ??? ??????? .???? :- ??? ???? ???? ??????? ???? ?????? ?? ???????? ?? ??? ???? ??? ???? ?? ??? ????????? ???????????? ???? ??? ?????? ????? ??? ??? ???? ? ?? ????? ???????? ??? ????? (101) ??? ??????? ???????? ????? ?? ??????? .????? ?? ??? ??????? ???? (Minitab) ??? ??? ???????? ?? ??? ??????? ????? c1 ???????? ?????? ?????? Graph > Dotplot1-6-7 ?????? ??????? (stem-and-leaf ) ??????? ?????? ???? ????? ??????? ???????? ? ???????? ???? ???? ? ???? ? ??????? ?????? ?????? ???????? ? ?????? ???? ??????? ? ??? ??? ????? ???? ?????? ????????? ???????? ?????? ??? ????? ?? ??? ??? ?????? ? ????????????? ???? ??????? ???? ????????? ???????? ?????? ???? ??? ???? ??? ???????? ???????? . ?????? ???? ?????? ???????? ????? ?????? ??? ???? ???? ??????? ??????? ??? "???" ? “ ???? ? "??????" ???? ??? ????? ?? ????? ??? ??????? ?????? ??? ???? "???????" ???? “ ??? ” . ?????? ??????? ????? ??? ??????? ????????? ???? ???? ????? ??? ???????? ???? ??????? ???? ??????? ????????? ??? ???????? ????? ??? ??????? ??? ????? ???? ?? ?? ? ??? ???????? ????? ????? ??????? ??????? ??? ????? ???? ?? ??????? ??????? ??????????? . ?????? ????????? ?????? ??? ??? ????? ? ??????? ?? ?? ??????? ?????? ????? ??????? ????????? ??? ??? ????? ?? ???? ????????? ???? ????????? ??? ?? ???????? ???????? ? ?????? ???? ???????? ??????? ???????? . ???????? ???? ????? ??????? ??? ????? ????? ?????? ??? ?????? ? ????? ???? ??????? ?? ?? ??? ????? ???? ?????? ????????? ???????? ?????? ??? ????? ???? (???). ??? ??????? ??????? .???? (1-15 ):- ?????? ??????? ????????? ???????? ?????: 3,4,8,8,10)) .???? :- ?? ???? ???? (MINITAB) ???? ?????? ??????.??????? (?????? ??????? ) ???? ???? ?? ????? ??????? ??????? ??? ???? ?????? ????? . ?? ??????? ? ???? ??????? ?? ????? (1.0). ????? ??????? ???????? 8 ?? 8 ????? ????? ??????? ???????? 10 ?? . 0 ????? (???? ???????) ???? ?????? ??????? ?????? ????? ??? ???? ???? ???????. ?? ???????? ???? ?????? 0 ?????? ???? ???????? ?? ??? ?????? ???? ????????? ????????? ???? ???? ??? ?????? ??? ???? ??? ??? ???? 10 . ???? ???? 1 ?????? ??? ?? ???????? ???? ???? ??? ??????? ??? 10? ??? ??? ???? 20.???? ????? (???? ??????) ??? ???? ????? ??????? ???????? ????????? ????????? ? ???? ???? ?????? ?????????? ??????? ?????? ( ) ???? ???? ?????? . ??? ???????? ??????? ??? ???? ??????? ?? ??? ??????? . ???? ???? ??? ?????? ? ??????? ?????? ??????? ???? ?????? ???????? ???? ???. ??????? ???? ??? ??????? ??????? ????? ??????? ???? ?????? ???????? ???? ? ???? ?????? ??? ????? ??? ?????? (2) ? ???? ?????? (1) .?? ???????? ?????? ???????? 8 ? ???? ????? ?? ?????? ??????? ??????? ?? ?????? ?????? . ????? ?? ?????? ??????? ??????? ?????? ??????? ????????? ?? ??????? ???????? .????(1-16 ):-???????? ??????? ???? ????? (20) ???? ?? ???? ??????? ??????? ( 78,74,82,66,94,71,67,88,69,80,91,74,82,75,96,78,84,77,71,83) ??????? ??? ????? ??????? ?????? ????? ??????? ?? ??????? ??? ??????? .????:-?????? ?????? :- ?????? ????? ?? ??????? (???? ??????) ????? ??????? ???? ????? (6,7,8,9) ?????? (6) ???? ??? ????? ?? ? ?? ????? ????? ????? (6) ???? ?? (65) ? ?????? (7) ???? ????? ??? ??? ????? ????? ????? (7) ??? ?? (75) ???????? ???? ???? ??????? (75) ????? ??????? ?????? ????? (8) ?(9) .??????????? (?????) :- ????? (6) ????? ?? ??????? ?????? (65) ??(66,67,69) ???? ???? ????? ?????? ??? ????? (679) ? ??? ????? (7) ?????? ????? ?? ??????? ??? ?? (75) ?? (71,71,74,74) ???? ???? ????? ?????? ??? ????? (1144) ? ????? (7) ??????? ????? ?? ??????? ???? ?? ??????? (75) ?? (75,77,78,78) ???? ???? ????? ?????? ??? ????? (5788) ??????? ????? ????? ?? ?????? .????????????(?????):-??? ??????? ????????? ?? ?????? ????? ??? (3) ? (3+4=7) ?(4) ?? ???? ??? ??????? ??? ??? ???? ?????? ?????? (78) ???? (9) ????? ??? ??? ??????? ??? ?? ???? ??? ???? ??? ??? ???? ?? ?????? ? ?????? (4) ??? ??????? ???? ????? ????? ???? ??????? ??????????? (80) ??????? ????? ????? ???????? .1-7 ?????? ?????? ???????? (Measures of central tendency)??? ??????? ?? ???????? ????? ???? ??????? ?? ????? ??????? ???? ???? ?????? ?? ???????? ???? ?????? ?? ??? ???????? ???????? ??? ??????? ???? ??????? ? ????? ??? ??????? ???? ???????? 1-7-1 ????? ??????? (arithmetic mean) :-?? ?????? ?????? ????? ??????? ???? ????? 01??? ??????? ??????? ???? ????????? ?? ??? ???????? :- ?? ?????? ???? ????? ?????? ??????? ?? ??? ???? ???? ???????? ???????? ? ???? ?????? ????? ??????? (?????? )? ??????? ??? ??? ????????? ??? ??? ???? ???? ??????? ????? ???? ????? ??????? ????????? .2. ??????? ????????? :- ?? ????????? ???????? ????????? ???????? ???? ???????? ??????????? ????????. ??? ???? ?????? ? ????? ????? ????????? ?? ?????? ?????? ???? ?????? ???????? ??????? (??? ????) ???? ????????? ?? ???? ???? .3. ???????????? ?? ?????? ????????? :- ???? ????????? ????? ???????? ?? ?????? ? ???????? ? ???. ? ?????? ??? ???? ??????? ?????? ??????????? . ??? ???? ?????? ? ??? ???? ????????? ???????? ????????? ????? ?????? ? ???? ???? ???????? ???? ?????? ????? ????????? ??????? ????? ???????? .??? ???? n ?? ????????? ???????Xi ??? ????? ??????? ??( ??????? ) ???? ????? ??? ??????? ????? ??? ????? ????? ?????? ??????? X=i=1nxin???? (1-17 ):- ??????? ???? ???????? ??????? ?? ??????? Xi ????? ???? ???? ???????? ??????? ????? ???????? ??????????? ?? ??????? ?????????? Xi =45,34,66,45,47,87,90??????? ???? ???????????? ?????? ??????? .?? ???? ??????? ???????? ???? ?????? X=i=1n Xin=45+34+66+45+47+87+907=59.14 ?????? ???? ??? ????????? ????? ???????????????? ??? ???? ??????? ???????? ??? ???????? (??????? ????????) ?? ???? ?????? ??????? X=A+d=A+dik???? ??? ??? ??? ???? ?????A ?? ?????? ??????? X ???????? di=Xi-A ???? ?????? ????? ??????? ???? ???????? ???????? ? ??? ??? ?? ???? ???? ????????? ??? ?????? ???????? ??????? ??????? ???????? . ??????? ?????? ( 1-17 ) ???? ?? ???? ????? ?????? A=34 ??? ?????????? ???? ??????90874745663445Xdi=1765653131132011di=Xi-A ???? ??? ??????? ??? ??????? ???????? ???? ?????? X=34+1767=59.14????? ??????? ???????? ??????? ?? ???? (ungrouped frequency distribution)?????? ???????? ???? ??????? ??? ???? ???? ???? ??? ????????? ?? ??? ????? ??????? ??????? ? ???? ???? n ?? ??? ??????? X1 ,X2 ,…Xk ??? ????? ???????? ????? ??????? ??? ??? ????????? ??? ???? ?? ??? ??????? X ? ???? ???X1 ????f1 ?? ?????? ? ?X2 ????f2 ?? ?????? ? 0000 ?Xk ????fk ?? ?????? ???????? ?? n ????? ?? ????? ????? ?? ????????? .n= i=1kfi????? ?? ?????? ?? ???? ?? ???? ?????? ?????? XkX2…….X1??????? Xnfkf2…….f1??????? f ??? ????? ??????? ??? ???? ??????? ???????? ???????? k ?? ??????? ???? ???????? ??????? X=f1X1+f2X2+…+fKXKf1+f2+…+fK=i=1KfiXii=1Kfi=i=1KfiXin?? ??? ???????? ??????? ??????? ???????? (short cut method) ??? ?????? ??? ???? ?? ??? ??????? X ????? A ????? ???? ???????? ????????? ??????? ??????? di=Xi-A ? ??? ???? ?????? ???? ?????? X=A+fidin???? (1-18):- ???????? ??????? ???? ??????? ???????? ????? ????? ?????? ??????? ( ????????? ) ????? ????? ??????? ?? ??? ?????? ??????? ??????? ????? ?????? ??? ? ??????? ???? ????? ??????? ?????? ??????? 25201510???????? X312114??????? f????:-????? ??????? ???????? ???????? X=i=1KfiXin=10×4+15×11+20×12+25×330=52030=17.333??? ???? ?? ???? ??????? ???????? ???????? ??? ???? ???? ??????? X (???????? ) ????? ??? 15 ??? ???? ??????? ???? ?????? 25201510???????? Xfi=n=30312114??????? fdi = 101050-5di=Xi-Afidi=7030600-20fidi??? ?? ????? ??????? ????? X=15+7030=17.333????? ??????? ???????? ??????? ?? ???? (grouped frequency distribution)??????? ?? ????????? ???????? ???????? ???????? ???????? ???? ???? ???? (?????) ????? ????????? . ???? ?? li ????? ?????? ?????? ????? ?i ? ui ?????? ?????? ????? i th . ????? ?? ??? ?????? k ? ??? ?????? i = 1, 2,. . . . . n. ????? ?? fi ????? ??????? ????? i th . ??? ??????? ???????? ???? ??????? ??? ??? ???? ????? ??? ?? ????? ?????? .Frequency?????????Class intervals ??????f1 l1-u1f2 l2-u2f3 l3-u3--------------------------fklk-ukn=i=1kfi??????? ??????? ??????? ???? ?? ??????? ??? ??? ????????? ??????? ?? ????? (????) ????????? ??????? ????????? ???????? ????????? ? ??? ?? ??????? ???????? ?? ?? ????????? ???? ?? ??? ????? ????? ????? ?????? ??????? ??? ????? ????? ????? ????? ???????? ???? ????? ????? ????? i ????? ???????? ??????? mi=(li+ui)/2????? ????? ???? ?? ????? ?????? ??????? ????????? . ???? ?? k ???? ??? ?????? ??? mi ???? ????? ????? i ? fi ?????????? i ??? ?? i=1,2,3,…k ????? ?? n=i=1kfi ??? ????? ??????? ???? ?????? ??????? ??????? ???????? X=mififi??? ?? mi ???? ????? ????? i .??????? ???????? X=A+fidifi??? ?? (di=mi-A) ???? ?????? ????? ????? mi ?? ????? ?????? A .????(1- 19 ):- ??? ??? ??? ????? (??????? ?????????) ??????? ?? ??????? ??????? ??????? ?? ??? ?????? ??????? ??????? ???? ??? ?? ??????? ???????? ??????28-32 24 >20 >16 >12 >8 >4 >class3818271152fi??????? :- ???? ????? ??????? ???????? ???????? ????????? .????:- ???? ???? ???????? ????? ????????? fimifididi=mi-18mificlass12-24-12624 >50-40-81058 >154-44-4141112 >48600182716 >396724221820 >20864826824 >90361230328-32139664074???? ???????? ???????? X=mififi= 139674=18.8648 ???? ???? mi ?? ??? ??? ???? (18 = A) ???????? ???????? ????????X=A+fidifi=18+6474=18.86481-7-2 ????? ??????? (geometric mean)?? ???? ??????? ???????? ??????? ????????? ?????? ???? ????? ?????? ???????? ?? ???????? ????? ????????? ?????? ??? ?? ???? ??? ????? ??????? ???? ???? n ?? ????????? ??????? X1,X2,X3,…….Xn ???? ?? Xi>0 ??? ??? ??? ????? ??????? ???????? ??? ??????? ???? ??????? ??????? ge= X1X2X3…..Xn1/n=( i=1nXi)1/n????? ????????? ??? ???? ?????? ??? ?? log10(ge)= i=1nlog10(X)n??? ???? ??????? ????? ???? ????????? ???? ???? ????? ??????? ??? ??? ge=antilog[ i=1nlog10(X)n]????( 1-20 ):- ???? ????? ??????? ???????? ??????? (2,12,26,33,67,99) .???? :- ???? ?????? ?????? ????????? ??? ?????? ??????? .i=16log10(xi)99673326122xi8.1351.995641.826071.518511.414971.079180.30103log10(xi)????????? ????? ?????? ??????????? ??????? ge=antilog i=16log10(xi)n=antilog8.1356=22.6986??? ????? ??????? ???????? ??????? ??? ???? (Ungrouped Frequency Distribution)??? ???? k ?? ????????? X1,X2,X3,…….Xk ??????????? f1,f2,f3,…fk ??? ????? ??????? ???????? ???? ???????? ???????log10(ge)= i=1k filog10(X)i=1k fi??? ???? ??????? ????? ???? ????????? ???? ???? ????? ??????? ??? ??? ge=antilog[ i=1k filog10(X)i=1k fi ]??? ????? ??????? ???????? ??????? ??? ???? (grouped Frequency Distribution)log10(ge)= i=1k filog10(mi)i=1k fi??? ?? mi ???? ????? ????? i ???? k ?? ?????? ??? ???? li ????? ui .??? ???? ??????? ????? ???? ????????? ???? ???? ????? ??????? ??? ??? ge=antilog[ i=1k filog10(mi)i=1k fi ]????(1-21 ):- ???? ????? ??????? ??????? ???????? ?????? i=16fi 90-10080-70-60-50-40-classes31458743fi??? ???? ????? ?????? ???? ?????? ??????fi Log(mi)Log(mi)mificlasses4.9591.65345340-6.9611.74055450-12.6901.81365760-15.0001.87575870-9.6471.92985580-7.9111.97795490-10057.16831i=16fi ge=antilog i=1k filog10mii=1k fi =antilog57.16831=69.82321-7-3 ????? ???????? (harmonic mean)??? ???? n ?? ????????? X1,X2,X3,…….Xn ??? ????? ???????? ???????? ?????????? ???? ???????? ??????? hm=n1X1+1X2+…+1Xn=ni=1n1Xi????(1-22 ) :- ???? ????? ???????? ???????? ??????? (11,21,27,31,33) .hm=5111+121+127+131+133=50.2381=20.9995??? ????? ???????? ???????? ??????? ??? ???? (Ungrouped Frequency Distribution)??? ???? k ?? ????????? X1,X2,X3,…….Xk ??????????? f1,f2,f3,…fk??? ????? ???????? ???????? ???? ???????? ???????hm=ni=1kfiXi????(1-23 ):- ???? ????? ???????? ?????? ???????? ?? ???? ??????? ????????? ????? ???? ??? ??????? ???????? ?????? .i=16fi 2017141185classes23258521fi????:- ???? ?????? ?????? i=16fi 2017141185Classes(Xi)23258521fi1.860.10.290.570.450.250.2fi/Xihm=ni=1kfiXi=231.86=12.365????? ???????? ???????? ??????? ??? ???? (grouped Frequency Distribution) ????????? ???????? ???? ???????? ???????hm=ni=1kfimi??? ?? mi ???? ????? ????? i ???? k ?? ?????? ??? ???? ????? .????(1-24 ):- ???? ????? ???????? ??????? ???????? ?????? i=16fi 60-7050-40-30-20-10-classes22128731fi????:- ???? ?????? ?????? i=16fi 60-7050-40-30-20-10-Classes(Xi)22128731fi655545352515mi0.6140.0150.0360.1770.20.120.066fi/mihm=ni=1kfimi= 220.614=35.8301-7-4 ????? ???????? (quadratic mean)????? ???????? ????? ?? ????? ???????? ????? ??????? ??????? ????????? ? X1,X2,X3,…….Xn??? n ?? ????????? ????? ??????? qm=X12+X22+……+Xn2n=Xi2n????(1-25):- ???? ????? ???????? ???????? ??????? (10,24,34,46,50) .qm=102+242+342+462+5025=64485=35.9111-7-5 ?????? (Median)?? ?????? ????? ?? ???? ??????? ???? ???? ???????? ??? ????? ???????? ? ??????? ????????? ??? ??????? ??? ????? ???????? ??? ????? ??????? ?? ??????? ???? ???? n ?? ????????? ??? ?????? md ???? ????? ??? ????????? ???? ???? n ????? ??? ?????? ???????? ??????? ??? ??????? n+1 /2 ?????? ?????? ? ??? ??? ???? n ????? ??? ???? ??????? ??? ???????? ??????? ??? ??????? n/2 ? (n /2)+1 ????????? ?????? ? ????? ???????? ?????? ?????? . ???? ??? ???????? ???? md= n+12th????????????? ???? ??? ???????? ???? md= average of n2thand n 2 +1thordered values ????( 1-26 ):- ???? ?????? ???????? ???????? ??????? (22,12,34,96,33) . ????:- ???? ????????? ????? ??????? ????? (12,22,33,34,96) ???? ?? ??? ????????? ???? ????? (5) ??? ?????? ???? ???? ???????? ??? ??????? (3) ???? ??? ?=33 md ??? ??????? X=1975=39.4 ? ?????? ?? ??????? ?? ????????? ?? ???? ?????? ???? ??? ?? ?????? ??? ???? ??? ??? ????? ???????? ??????? ???????? ?? ?????? ??? ????? ??? ????? ??????? ????? ?????? ??????? ???????? ?? ?????? ????? ????? ??? ???? ??? ??????? ??????? ??? ?????? .????(1-27):- ???? ?????? ???????? ???????? ??????? (10,18,8,22,12,31,44,29) ? ???? ????????? ????? ??????? ????? (8,10,12,18,22,29,31,44) ???? ?? ??? ????????? ???? ????? (8) ??? ?????? ???? ??????? ????? ???????? ??? ??????? (4) ?(5) ???? ??? ?????? ???? ??????? ??????? md=18+222=20??? ??????? ( X=21.75) ? ?????? ?? ????? ???? ?? ?????? ????? .?????? ???????? ??????? ?????? ?????? ?????? ???? ???? (ungrouped frequency distribution) :- ??? ????????? ???? ?? ????? ???? ??? ???? ?????? ?? ????? ?? ????? ?????? ???????? ?????????? ???? ??? ???????? ??? ????????? ????????? ???????? ??? ?? ?????? ?????? ???? (1-28 ):- ???? ?????? ???????? ??????? 3229262320???????? X2510144??????? f????:- ???? ???? ???????? ?????????? ??? ?????? ????????? ( cdf or F) ??????? ? ???? ?? (fi=n=35) ?????? ???? ? ??? ?????? ?? ?????? ??? ???????35+12th ????? ???? ???????? (18th) ????? ????? 23 ? ???? ??? ?????? md=23 . ????? ???? ??? ???? (18) ?????? ??? ??????? ?????? .3229262320???????? X2510144??????? ffi=n=35353328184cdf or F?????? ?????? ?????? ????? (grouped frequency distribution)?????? ??????:- ?????? ????????? ??? ??????? ???????? ?????? .??? ?? lm ???? ?????? ???? ?????? ? fm ???????? ?????? ? h ??? ????? ??? F ??????? ???????? ????? ??????? ???? ?????? .md=lm+n2-Ffmh??? ?????? ?????? ?????? ???1- ?? ??????????? ??????? ???????? ????? ? ????? ?? ??????? ?? ????? ???????? ? ??? ??????? ????? ??????? ?????? ????????? n2 ? ???? ??? ???????? ??? ?????? ????????? ????? n ?????? .2- ?????? ????? ????? ???????? ????? ??? ?????? ??? ?????? ??? ??????? .3- ?????? ???????? ???? ??????? ??? ?????? ?????? ?????? ????? ????? ??? ?????? ????????? ????? ??????? ?? ??????? ???? ?????? .?????? ??????? :- ?????? ????????? ??? ??????? ???????? ?????? . md=um-n2-Ffmh??? ?? um ???? ?????? ???? ?????? ? fm ???????? ?????? ? h ??? ????? ??? F ??????? ???????? ????? ??????? ???? ?????? .???? (1-29 ):- ???????? ??????? ???? ??????? ???????? ?????? ???? ????? ??????? ?????? ??????? ???? ?????? ?????? ??????? .Frequency?????????Class intervals ?????? 120 30-40 14540-50 20050-6025060-7018570-80900??????? ????????? :- ??? ?????????????? ?????? ????? ?? ??? ?????? ????? ?? ????? ????????? ????????? ??? ???? ??? ?????? ?????? . less than cdf?????????? ???? Fgrather than cdf ?????????? ????FFrequency?????????(fi)Class intervals ??????120900 120 30-40265780 14540-50465635 20050-6071543525060-7090018518570-80900??????? ?????????????????:- ??????????? ???? ?? n2= 9002=450 ? ????? ??? ??? ?????? 50-60 ? lm=50 ?fm=200 F=265 ??????? ???????? ????? ??????? ???? ?????? . ???? ????? h=10 . md=lm+n2-Ffmh=50+450-26520010=59.25????????????? :- ??????????? ?????? n2= 9002=450 ? ????? ??? ??? ?????? 50-60 ? um=60 ?fm=200 F=435 ??????? ???????? ????? ??????? ???? ?????? .???? ????? h=10 .md=um-n2-Ffmh=60-450-43520010=59.25??????? ???????? ??????? ?????? :- ???? ?????????????? ?????? ???????? ?????? ?? ??? ?????? n2 ????? ?????? ??? ?????? ?????? ???? ?? ?????? ????? ?? ?????? ???? ???? ????? ??????? ???? ???? ??????? ???? ?? ?????? ????? ??? ?????? ?????? ? ????? ??????? ?? ?????? ?????? ???? ?????? (md).??????? ??????? :- ???? ????? ??????? ??????? ?????? ??????? ??? ????? ????????? ???? ?? ????? ??? ?????? ?????? ??? ???? ?????? ?? ?????? ???? ???? ?????? . 1-7-6 ????? ??????? ?????? ( WEIGHTED MEAN )?? ????? ??????? ??????? ??????? ? ???? ????? ?????? ??? ???? ????????? ?? ??????? ??? ??? ?????? ???? ?? ?????? ????? ????????. ???? ????? ?? ????????? ?????? ??????? ?? ??????? ??? ??????? ??????? ?????? ???? ??????? ???? ??? ????????? ? ???? ?? ?? ?? ????? ??? ????????? ????????? ???? ??????? ??????????????? ??????? ??? ??? ???? ? ???? ???? ??????? ?????? ????? ?????? ??????? ?? ???? ????????? ???? ????? ???? ??? ????? ??? ????? ??????? ??????? ??????? ??? ?? ??? ???? ???? ?????? ( ???????? ???????????? ) ?????? ??????? ???? ????? ???? ????? ??????? ????????? ?? ??????? ???????? ???? ???????. ?? ??????? ??????? ?????? ? ?? ??? ??? ???????? ??????? ???? ???? ?? ????? ????? ?? ?????? ??? ???????? ???? ?????? ???????? ???????? ?? ?????? . ??? ???? ????? ????? ????? ?????? ??? ???? ???????? ?????? ???? ????? ????? ??? ?? ??????. ??? ???????? ??????? ??? ??? ???????? ??????? ????? ??????? ????????.??? ??????? ??? ??? ??????? ????????? ???? ?????? ??????? ??????? ???????? ?????????.... ???.????? ????? ?? ???? ???????? ?????????? ???? ?????? .??????? ???????? Xw=wiXiwi??? ?? wi ???? ??? (??????) ???????? Xi .??????? ???????? Xw=A+widiwi??? ?? (di=Xi-A) ???? ?????? ?????????????? Xi ?? ????? ?????? A .????(1-30 ):-????? ??????? ?????? ???????? ??????? ????? (ungrouped frequency distribution)Xw=i=1kwifiXii=1kwifi????(1-31 ):- ???? ????? ??????? ?????? ???????? ??????? 10864Xi3562fi2431wi???? :- ???? ???? ???????? ?????? ??????? 10864Xi3562fi2431wi336601601088wifiXi46620182wifiXw=i=1kwifiXii=1kwifi=33646=7.3041-7-7 ????? ????????? :- ??? ???? k ?? ????????? ?????????? Xi ???? ??? ????????? ??? ?????Ni ???? ?? ??? ????? ????????? ??????? ????? ???? ?? ??? ?? ????????? ???? ??????? ??????? X=N1X1+N2X2+…+NKXKN1+N2+…+NK=i=1KNiXii=1KNi????( 1-32):- ?? ??? ????? ?????? ?? ???? ?????? ????? ????? ?? ???? XiNi???????? 94??????11112??????2157??????365??????4??????? ???? ????? ????????? .???? :- X=4*9+12*11+7*15+5*64+12+7+5=30328 = 10.82141-7-8 ??????? ?????? (compsite average):-?? ??????? ?????? ????? ?? ????? ????????? ?????? ? ???? ??? X1,X2,…,XK ????? ?? ??????? ??? k ?? ????????? ??? ca???? ???????? ??????? . ca =X1+X2+…+XKK ????(1-33 ):- ???? ??????? ?????? ??????? ?????? (1-32 ) .????:-ca =X1+X2+X3+X44=9+11+15+64=414=10.25???? (1-34 ) :- ???? ??????? ?????? ??????? ???? ?????? ?? ???? ????? ??????? ??????? ???? ?? ???????? ???????? ???? ?? ?????? ?????? .45513612?????? ??????16433?????? ??????2225644?????? ??????3 ???? :- ???? ????? ?? ???? ?? ???????? ???? ???? ??????xn36445513612?????? ??????147.5026433?????? ??????240.663225644?????? ??????3ca =X1+X2+X33=36+47.50+40.663=413=41.386??????? (mode) ??? ????? ?? ?????? ???? ????? ???? ?? ????? ?? ???????? ?????????? ? ???? ???? ????? ???? ?? ???????? ???? ????? (modal) ??? ??? ???? ???? ?? ?????? ??? ?????? ??? ???????? ???? ????? (bimodal ) ???? ??? ???? ??????? ???????? ?????? ?????? .????(1-35):- ???? ??????? ???????? ??????? (11,12,12,23,32,12,27) ???? ?? ???????? ????? ?? ?????? (12) ??? ??????? ????? (mo=12)??? ??????? ?????? ?????? ???? ???? (ungrouped frequency distribution) ???? ???? ??????? ??? ??? ?????? ???? ????? ??????????? ??????? .????(1-36 ):- ???? ??????? ???????? ??????? 30252015105 X5121319174??????? f????:- ?? ????????? ???? (19) ???? ??? ??????? (mo=15) .??? ????? ????????? ????? ??????? ?? ???? (grouped frequency distribution) ?????? ?? ???? ???????? ??? ???? ?????? ???? ??????? (lm)mo=lm+f2f1+f2×h?????? ?? ???? ???????? ??? ???? ?????? ???? ??????? (um)mo=um-f2f1+f2×h ?? f1 ???? ???? ????? ????? ??? ????????????? ?? ? ??? f2 ???? ???? ????? ????? ??? ????????????? ?? ? h ??? ??? ??????? ???? ?? ??????? ?? ???? ??????? . mo=3 md- 2 x????(1-37 ):- 20-2317-2014-1711-148-115-8 X9122225173??????? f????????? ??? ???? ?????? ???? ??????? mo=lm+f2f1+f2×h=11+38+3×3=11.8181 ???????????? ?????????? ???? ??????? mo=um-f2f1+f2×h=14-38+3×3=13.1818??? ????? ????? ?????? ??? ??? ????????? ??????? ????? ???????.1-8 ????? ?????? (Measures of dispersion) 1-8-1 ??????? (variance)?? ????? ????? ?????? ?????????? ????????? ?? ??????? ???? ???????? (variance) ? ??????? ???? ??? ???? ???????? ??? ??????? ?????? ??????? ??????? ???? ??? ??? ????? ?? ??????? ???????? ???? ???????? ??????? sx2=1n-1i=1nxi-x2??? ???? ????? ?????? ??????? ??????? sx2=1n-1i=1nxi2-1ni=1nxi2??? ???????? ???????? (sx) ???? ??? ??????? ? ??????sx= sx2???? (1-38 ):- ???? ??????? ????????? ???????? ???????? ??????? ?????? ??????? ??????? ???? ???? ????? ????? ( 1,4,5,6,3,7).????:- ?? ?????? ???? ???? ???? ??? ????? ??????????? ????????26736541xi1364993625161xi2?? ???? ?????? ??????? sx2=1n-1i=1nxi2-1ni=1nxi2=15136-16(26)2=4.666???????? ?????? sx= sx2= 4.666=2.160 ???????? ????? ??? ???? ?? ??? :- ?? ???? ??????? ???? ??????? ??? (var(x)≥0) ? ?????? ????? ??????? ??? (varx=0) ??? ??? ??? pX=μ=1 ??? ?? μ=E(X).??? ????? a ,b ??? ??????? ??????? ?????varax+b=a2varx.?????? ???? ??? ?? ???????? ???????? ???? sax+b=a sx????? ?????? ????? ??? ????? ??????? ?????? ???????( ??????? ????????) ?????? ? ?????? ?????? ??? ??????? ????? ???????.??? ??????? ?? ???????????? ??????? ????? ??? ????????????? ?????? ???????? ???????? grouped frequency distribution)) .sx2=1n-1i=1kfimi2-1ni=1kfimi2??? n=i=1kfi ????? ????????? ? k ???? ??? ?????? ?? ??????? ? ??? ???? ??? ??????? ?? ??? (ungrouped frequency distribution))) ????? ?????? ?? ??????? ??????? ???? ??????? ???? ?? ????? mi ???????? xi ????? ???? ???? ??????? ??????? ????? i .????(1-39 ):- ???? ??????? ????????? ???????? ???????? ??????? ??????? ?????? ?????? ??? ?? ???????????? ??????? ?? ????? ????? ?????? ????? (25).ficlasses212-522-932-642-352-62???? :- ???? ?????? ?????? ??? ???????????? ?????????? ??? ????? ?????? fimi2fimimificlasses5783417212-364513527522-1232133337932-1325428247642-974717157352-623954595525???????sx2=1n-1i=1kfimi2-1ni=1kfimi2=12439545-125(955)2sx2=127.666??? ???????? ???????? ????? ??? ??????? ??? ?????? sx= sx2= 127.666=11.298????(1-40 ):- ???? ??????? ????????? ???????? ???????? ??????? ??????? ?????? ???? ??????? ????????? ???? ?????? ????? ?? ????????? ?????? ?????? ????? (13) ???? .5432classes1741fi???? :- ???? ?????? ?????? ??? ???????????? ?????????? ??? ??? xi .fixi2fixifiClasses(xi)42125761243548828742551560934713??????? sx2=1n-1i=1kfixi2-1ni=1kfixi2=1126093-113(47)2sx2=493.589???????? ?????? sx= sx2= 493.589=22.21681-8-2 ????? (range ) :- ????? ???? ??? ???????? ???? ??? ??? ????? ??????? ????? ????? ??? ????? ???????? ?????? ?? ??? ???? ???? ??????? ???????? ??????? ra=X(n)-X(1)????(1-41 ):- ???? ????? ??????? ?????? (34,12,61,54,78,23 ).???? :- ??? ?? ???????? ???????? ??? ????? ????? ra=X(n)-X(1)=78-12=66??? ?? ???? ???????? ??????? ?????? (grouped frequency distribution) ????????? ???????? ??? ????? ???? ????? ??? ???? ?????? ????? ??????? ????? ?????? ????? ?????? .????(1-42 ):- ???? ????? ??????? ?????? (1- 39 ).ra=u5-l1=62-12=501-8-2 ??????? ???????? (???????) (z-score )???????? ????? ???????? ?????????? ???????? ?? ???? ????? ??? ???????? ?? ???? ??? ???????? ?????? ??? ?????? ??????? X ???? ??? ??? ????? ???? ???????? ??????? .zi=Xi-Xs??? ??? ???? ?????? ??? Xi ?? ??????? X ?????? ??? ???????? ???????? ??? ??? ???? ??????? ???????? ???? zi ????????? ??? ?? ????? ??? zi ????? ?????????? ??????? ?????? ????? ?????? ??? ?? zi ????? ?? ???? ?????? ??????? ???? ?????? ??? ???????? ??????? ?? ???? . ?? ??????(z-test) ???? ??? ???? ????? ????????? ???????? ???? ???? ??? zi ???????? ??? ??????? ??????? ?????? (3) ????? ?? (-3) ??? ???????? ??????? ??????????? ???? (outlier) ? ????? ?? ????? ?????? ????????? ???????? ?????? ? ?????? ?????? ??? ???? ?? ????? ??????(z-test) ??? ??? ??????? ? ????? ???? ??? ???? ????? ???????? ???? ((modified z-score test ????? ?? ?????? ???? ?????? ????????? ????? ??????? ??? ???? ????? ??????? (md) ??????? (MAD) ?????? ??????? ??????? MAD=median(Xi-md)???? ??? ??? ???? ????? ????????? ??????? zmi ??? ???????zmi=(Xi-X )MAD??? ??????? ????????????? ???? ??? ???? ???? zmi ?????? 3.5 ? ???????? ??????? ??? ?????? ???? ?????? ?? ?????? ?????? ??????? Xi-mdMAD > 5????? ??? ????? ???????? .????(1-43) :- ????? ???? ???????? ??????? ?? ?????? ???? ??????? ?? ????? ??????? ??????? ???? ?????? ( 5,21,27,31,26,98) ? ???? ????? ????????? ????????????? ?????? ?? ???? ????? ???????? .???? :- ??? ??????? ????????? ???????? (X=34.7) ?(s=32.3) ??? ??????? ???????? ???? ???? zi ???? ??? ?????? ?????? . ?? ???? ????? ?????? ????????? ??? ?? ?? ????? zi??? ?????? ????? ???????????? ??? ?? ?? ????????? ?????? ????? ???????? ????? ???????? ??????? ? ??? ?? ???? ?????? ??????? ??? ???? ?????? ???????? ??????? ??? ??????? ?????? md=26.5 ? ?? ???? ?????? ????????? ??????? ????? ???????? ?? ???? ????? ??????? ????? (MAD=5) ?????? ?????? ?????? zmi ???????? ???? ??????? ?????? ??? ??????? ??????? ??????? ???? .?? ???????? ??????? ???????? (98) ???? ???? ?? ???? ????? ??????? ? ????? ??????? ??????? ???????? ????? ????? ??? 1-8-3 ????? ???????? (Skewness?)????? ????? ????? ?? ???? ??????? ??????? α1=3(X-md)3s????(1-44 ):- ???? ????? ???????? ( ????? ????? ) ???????? ??????? ?? ????? ??????? 178156124xi???? :- ?? ??????? ???? (X= 7.57) ??????? (md=7) ???????? (=22.29 s2) ???????? ???????? ???? ??? ??????? ?????? (s=s2=(22.29)=4.72) ??? ????? ????? ????? ???????? ??????sk1=3(X-md)3s=3(7.57-7)34.72= 0.1177?? ???????? ??? ?????? ???? ??? ??? ??????? ???? ??????? ????? ????? ?????? .????? ?????? ????? ?? ???? ??????? ??????? α2=m3s3??? ?? s ???????? ???????? ?m3 ????? ?????? ??? ????? ??????? ????? ???????? ??????? m3 = i=1nfi (Xi-X)i=1nfi????? ?????? ??????? ??? ??????? ???? ????? ???????? ?????? ??????? ???????? X ????? α3=E(X-μ)3σ3= μ3μ23/2????? ??? ??????? (symmtric) ????? ??????? ?????????? ??????? ???? ???? ????? ?????? ????? ??? ???? ??????? ?????????? ????? ??????? ? ??? ?????? ??????? ???? ??? ?????? ??? ??? ?????? ??? ?????? ??????? ????? ??? ?????? ??? ??? ?????? ????? ?? ???? ?????? ?????? ??????? ??????? ?????? ???????? ??? ??????? sk=nn-1(n-2)[ Xi-X s ]3 ????(1-45 ):- ???? ????? ???????? ???????? (10,24,34,46,50) .???? :-X=i=1n Xin=10+24+34+46+505=32.8 sx2=1n-1i=1nxi2-1ni=1nxi2=146448-1462=267.322sx= sx2=267.322=16.35, n=5??? ??????? ???? ?????? ?????? ?????? ??? ??????? ??? ??? ??? ????? ???????? ???? ?????? sk=nn-1(n-2)[ Xi-X s ]3 =54×3 -1.17685=-0.49??? ???? ???????? ??????? ???? ???????? ??????? sk=i=1nmi-X 3fii=1nfi1-8-5 ????? ??????? ( (kurtosis?? ????? ?????? ??????? ??? ??????? ???? ????? ??????? ????? ??? ?????? ??? ??????? ???????? ?? ??????? ??????? ??? ?? ?????? ??????? ??????? ???? sharper peak ??????? ??????? ??????? ??????? ??????? ???? ?? ????? ???????? ????? flatter peak ?? ??????? ??????? ?????? α4=E(X-μ)4σ4- 3????? ??? ???? ????? ???????? ? ????? ?? ?????? ?????? ??? ????? n ??????? ??????? ku=ni=1nXi-X 4(i=1nXi-X 2)2- 3=ni=1nXi-X 4(n-1)2 s4-3????????? ??????? ??????? kum=n(n+1)n-1n-2(n-3)[ Xi-X s ]4 -3(n-1)2n-2(n-3) ????(1-46 ):- ??????? ?????? (1-45) ???? ????? ??????? .????:- ??? ?? ??????? ???? ?? ???? ?????? ?????? X =32.8 , sx=16.35?????????? ????? ???? ???? ?????? ??????? ???? ?????? ?????? ?? ???? ??????? ??????? kum=n(n+1)n-1n-2(n-3)[ Xi-X s ]4 -3(n-1)2n-2(n-3)kum=5×64×3×25.515-3×423×2=-1.010??? ??????? ??? ?????? ?????????? ???? (ku=-1.621) ??????? ??????? ??????? ???? ?? ????? ???????? ????? ??? (flatter peak ) ????????? ?? ??????? ??????? .????????? ??????? (grouped frequency distribution) ???? ???????? ku=i=1nmi-X 4 fii=1nfi??? ?? mi ???? ????? i .1-8-6 ????? ???????? ( coefficient of variation) ????? ????? ????? ?????? ???? ???????? ???????? ????? ?? ???? ???? ???????? ???????? ??? ??????? ??? ?? ????? ?????? ??????? ??????? ?? ???? ?????? ???? ?????? ???????? ?? ??? ????????? ?????? ???? ?????? ?? ?????? ??? ?????? ??????? .cv=sx×100%??? ?? s ???????? ???????? ???????? ??????? ? x ????? ???????????(1-47 ):- ??????? ?????? (1- 45) ???? ????? ???????? ????:- ??? ?? ??????? ???????? ????????? ???????? ????? X=i=1n Xin=10+24+34+46+505=32.8 sx2=267.322 , sx= 267.322=16.35 ???? ??? ????? ???????? ????? cv=sx×100%=16.3532.8×100%=49.84????????? ????????? ??? ??? ???????? (Symmetrical or non-skewed distributions)???? ???? ???????? ???????? ????? ? ??????? ????? ????????? ?? ?????? ????? ??????? ?????? ??????? ????? ? ??????? ??????? ??????? ??? ???? ??????? ????? ????? ?? ???????? ??? ???? ????? ?? ??? ???? ?????? ?????? ???????? ???? ??????? ??????? ?? ??????? ????? ?? ????????? ?????? ???? ????? ??? ?? ??????? ??????? . ??????? ?????? ??? ??????? ????? ?????? ?????? ???????x =md=mo????????? ???????? ?? ????????? (Positive or Right skewed distributions)???????? ???????? ??? ????????? ?????? ???? ??? "????" ???????? ?????? ??? ?????? ? ???? ????? ????????? ???????? ???????? ??? (??? ????? ) ??? ?????? ??????? ?????? ?? ???? ??????? ?? ??? ???????? ??? ?????? ???? ?????????? ?? ???? ?????? ?????? ?????? ?????? ?? ????????? ?? ??? ?? ?????? ???????? ??????? ?????? ????. ??????? ???? ??? ?????? ( ??????? ?? ?????? ???? ) ??? ?????? ???? ?? ??????? ???? ?? ???????mo<md < x ????????? ???????? ?????? ?? ??????? (Negative or Left skewed distributions)??? ????? ????????? ???????? ????? ??? ???????? ??? ( ?????? ???? ???????? ??? ???????? ??? ?????? ?? ???? ??????? ??????) . ??????? ????? ?????? ???????? ?????? ???? ?? ???? ???????.???? ? ????????? ???? ???? ???? ????????? ??????? ??? ????????? ? ????????? ??????? ???? ??????? ????? ????? ???? . ??????? ???? ??? ?????? ( ??????? ?? ?????? ???? ) ??? ?????? ???? ?? ??????? ???? ?? ???????x <md<mo????(1-48 ) ????? ????? (SPSS)??? ???? ????? ?????? ??????? ?????? ?????? ???? ??? ??????? ?????344354265535446751236655423445??????? ????? ???? ?????? ?? ??????? ????????? ?????? ?????? ???????? ??????? ???????? ? ?????? ????? ??????? ??????? . ????:- ?? ???? ???? (SPSS) ???? ?? ??? ?????? ???? ?????? ? ??? ????? ???????? ?? ???? ????? ???????? ????? ?????? ?????? ?? ????? ??????? ???????? ???? ???????? ????? ???????????? ?? ??????? ?????? ???? ?????? ???? ??? ?? ???? ????? ?????? ????????? ?????? ???????? ???????? ????? (??????? ??????? ???????? ????????) ? ?(????? ??????? ?????? ????????) ????? ??????? ????????? ???????? ?????? ....??????) ????? ??????? ??????? ?????? ????? ?? ???? ?? ???? ????? ?????? ???????? ??????? ???????? ???? ?? ???? ???????? ?????? ?? ????? ????? ????? ????? ???????? ???? ???? ???? ???????????? ??????? ??? ?? ??????????? ???? ??? ???? ??????? ??????? ???? ??????? ?????? ??????? ??????? ???? ??? ?? 1-9 :- ???????? (correlation) ???????? ???? ????? ??? ??????? ?? ???? ?????? ??? ???? ????? ??????? ??? ????????? ?????? ???? ???????? ??? ????? ????????? ????? ( ??????? ?????? ?? ????? ) ?? ????????? ??????? ??????? ?????? ?? ??????? ????? ??? ??? ??? ???? ???????? . 1-9-1 ????? ???????? (correlation coefficient)??? ??? X ? Y ????? ?? ??????? ???????? ??? ????? ??????? ?????? ?? ???? ???????? (ρxy) ????? ???? ???????? ??????? corr(X,Y)=cov(X,Y)varX var(Y)??? ?????? ?????? ??????? ???????? ???? ???????? ?? ??? ??????? ?????? rxy=sxysxsy ;rxy≤1??? ?? ??? ??? ??????? ???????? ????????? ???? ?????? sxy=1n-1i=1nxi-x(yi-y)???? ?? ??????? ????????? x,y ???? ?????? ??????? sx2=1n-1i=1nxi-x2sy2=1n-1i=1nyi-y2 ????????? ???? ??????? ?????? ??? ???????? ???????? ???? ?? ???? ???????? ??????? rxy=sxy sxxsyyrxy= i=1n(xi-x)( yi-y)i=1n(xi-x)2 i=1n(yi-y)2 ????????? ????? ???? ????? ???????? ?? ??????? ????????? ??????? ??????? rxy= ni=1nxi yi-i=1nxi i=1n yini=1nxi2 –(i=1nxi )2[ni=1nyi2 –(i=1nyi )2]??? ????? ???????? ???? ?? ??? ?? ???? ????? ???????? ???? ????? -1 ? 1 .??? ??? rxy???? ?? 1 ??? X ? Y ?????? ?????? ???? ???? (positive relation) ???? ???? ??? ?? ??????? ??? ???????? ??????? X ?????? (??????) ?? ??????? ??????? ???? ???????? ??????? Y ?????? (??????) ?? ??????? .??? ??? rxy ???? ?? -1 ??? X ? Y ?????? ?????? ???? ???? (negative relation ) ????? ??? ?? ??????? ??? ???????? ??????? X ?????? (??????) ?? ??????? ??????? ???? ???????? ??????? Y ?????? (??????) ?? ??????? .??? ??? rxy ????? ????? ???? ??? ??? ???? ????? ?????? ?? ??? X ?Y.?? ????? ????? ???????? r ?? ??????? ?????? ?????????? ??? ???????? ???????? ???????? ?? ?? ????????? (X,Y) ?????? ??????? ??????? ??????? ? ??????? ??????? ???? ??????? ????? ???(fisher z~transform) ????? ???? ????????z=0.5 ln?[(1+r)/(1-r)]????? ????? ?????? ??????? ????? ????? ??????? ??????? z ~N( 0.5ln1+ρ1-ρ, 1n-3)???? ???????? ??? ?????? ??????? ????? ?????????????? ??? ρ ????? ????? ???????? ?? ?????? ??????? H0: ρ = 0 ?versus ?H1 : ρ ≠ 0.0?????? ?????? ???????? ??????? z=0.5ln1+r1-r-0.5 ln1+ρ1-ρ1/n-3??? ??? ????? ρ ? α ( ????? ?? ????? ????? ) ???? ??????????? ?? ????? ????? ???? ??? ??? z ≥ zα/2??????? ???? ??? ?? ???? z ?? ????? ????? ? ???? ????? ????? ???? ????? ?????? α ? ??? ???????? ?????? ????? ???????? ???? ????????? (X,Y) ?????? ??????? ??????? ??????? .????(1-49 ):- ???? ????? ???????? ?? ??? ????????? ??????? ???? ?? ???????? ???? :- ???? ???? ???????? ????????? ??????? ?????? ???? ?? ?????? ???????? ?? ??? ????????? ???? rxy= ni=1nxi yi-i=1nxi i=1n yini=1nxi2 –(i=1nxi )2[ni=1nyi2 –(i=1nyi )2]??????? ????? ????????? ?? ???? ???????? ???? ??? rxy= 6*355-47*16 6*719-( 47)2[6*210-( 16)2]=0.948??? ?? ???????? ( rxy=0.948) ??? ???? ?? 1 ??? X ? Y ?????? ?????? ???? ???? (positive relation) (???? ???? ) ???? ???? ??? ?? ??????? ??? ???????? ??????? X ???? ???? ?????? (????????) ?? ??????? ??????? ???? ???????? ??????? Y ?????? (?? ??????) ?? ??????? ? ?????? ??? ?? ????? ??? X ?????? ???? ??? ????? ??? Y ??? ?????? ? ???? ?????? ?? ???? ???????? .????? ????????????? ?? ?????? ??? ????? ???????? (ρ?) ???? ??????? ?????? ??? H0 ???? ???? H1??????? H0: ρ? = 0 ?versus ?H1 : ρ? ≠ 0.0????????? ????????? ????????z=0.5ln1+r1-r-0.5ln1+ρ1-ρ1/n-3z=0.5ln1+0.9481-0.948-0.5ln1+01-01/6-3=3.1389 ???? ?? ?????????? ?????? ???( 1.96 = zα/2=z0.025) ? ??? ????? ????? ?? ( zα/2≥1.96) ???? ?? ?????? ???????? ???????? ??? ?? ????? ????? (3.1389>1.96) ??? ??? ???? ????? ????? ??? ????? ?????? (α=0.05 ) ??????? ?? ??? ?? ??????? X,Y ?????? ?????? ??? ?????( ?????????? ) .? ??? ?????? ?????? ?????????? (p-value) ?? ???????? ???????? ????????? ???????????? ?? ????? ?????? ???? ??? ?????? ??? ??? ????? ????? ????? ? ??? ?? (p-value) ???? ???? ???????? ?????? ??? ?????? ??????? ????? ???? ??? ????? ??????? (???????? ??????) ?? ??????? ???????????? ???????? ? ??? ???? ?????? ????? ??????? ? ?? ?????? ??? ??????? ???? ??? p-value ?????? ??? (0.05) ? ???? ???? ???? (p-value) ???????? ???????? ????????? ??? ???? (0.05)? ??????? ?????? ?????? ???????? ??? ?? ?????? ( 0.004= p-value) ???? ???? ??? ?? (0.05) ????? ???? ????? ????? ????? ???? ??? ?? ????? ???????? ????? ??? . 1-9-2 ????? ?????? ??????? ????? (spearman rank correlation coefficient)??? ???? n ?? ??? ??????? ??????? X ??? ???? ???? ??? ???? ???? ????? ????? ????????? ???????? ?? Ri ???? ???????? Xi ????? i=1,2,3,…n ?????? ?????? ??? ???? n ?? ??? ??????? ???????Y ??? ???? ???? ??? ???? ???? ????? ????? ????????? ???????? ?? Si ???? ???????? Yi ????? i=1,2,3,…n .?? ????? ?????? ??????? ????? ????rs= 12 i=1n [Ri - n+12][Si - n+12]n(n2-1) ?? ??????? ???????? ??????? rs=1 - 6i=1ndi2 n(n2-1)??? di ???? ???????? ?????? di=Si - Ri, i=1,2,3,…n????? ?????? ????? ???????? ???? ?? ??? ??? ??? ????? ?????? Ho:FX,Yx,y≡FXx FYy , for all x,y pairs?????? ?? ????? ????? ?? ??? ??????? ???????? ?????? ??? ???? ??????? ? ??? ????? ?????? ????? ??? ????? ??????? ?????????? ?? ??? X ?Y . ?????? ?? ???? ???? ????? ?????? (one sided upper tail test)??? ??? ????? ?????? H1:X and Y are positively associated????? ??? ??? ?? X ?Y ??????? ????? ? ???? ????? ?????? α ????? ?????? ??? ??? ????? ??? ???? ?? ??? Reject H0 if rs≥rs,α???? ??? ?? ??? ????? . ?? ?????? rs,α ( ???? ?????? ) ????? ???? ???? ?????? ????? ?? ????? ????? ????? ??? α .??? ??? ??? ?????? ?????? ??????? ????? ??????? ??????? ??????? ???????? ??????? rs*=rs-E0(rs)var0(rs)1/2=(n-1)1/2 rs?????? H0 ???? ????? ??????? ??? rs* ??????? ??? ?????? n ????????? ?? ?? ??????? ????? ?? ??????? N(0,1) ??? ??? ??? ????? ????? ?????????? ?????? ??????? ???? ???? ??????? ???? ??? ??????? ??? ????????? ?? ???? ???? ????? ?????? ??????? ??? ??? rs*≥zα ???? ????? ????? H0 .1-9-3 ??? ???? ????? ???????? ????? rk= 2i=1nDi n(n-1)??? ?? Di ???? ????? ???????? ????? ????? ??????? i ??????? ??????? ?????? ???????? ??????? ????? .????(1-50 ):- ???????? ??????? ???? ???????????? (Y) ??????? ?????? (X) ????? ?? ???????? ??????? ???? ????? ?????? ????? ???????? ?????? ?????? ????? ?????? ?? ??????? ??? ??????? .???????????????????????? ?????????? ???X995560784266Y????:- ???? ???? ????? ?????? ?????? ??? ??????? ???? ???????? .Didi2di???(Y) ???????????(Y) (Y)???(X) (X)5-0=50011421??? ???????3-0=31-132552?????0-0=09-363603???1-1=00044664??????0-1=-10055785?????0-4=-416426996????????326??????? ??? ???? ??????? rs=1 - 6i=1ndi2 n(n2-1)=1 – 6(26)6(62-1)=0.2571 ??? ???? ????? rk= 2i=1nDi n(n-1)= 2(3 )6(6-1)=0.20 ?????? ??????? ??? ??? ??? ?????? ???? ?? ???? ???????????? ?????? ?? ?????? ????? ???? ??? ???? ???? ????? ????????? ???????? ??? ????? (????? ? ????? ? ?????? ? ???????? ??????? ? .... ?????? ) ??? ??? ??????? ????? ??? ???????(???? ?????? ?????? ) ??? ??? ??????? X ???? ???? ???? ??????? ? ???????? ?? ????? ?????? ???? ?????? ????????????????????? ?????????? ?? ????? ??? ??????? ??? ???? ??? ???? ?? ?? Y ??? ???? ?????? ??????? ??? ???????? ???? ???? ???? ?????? ????? ???? ??????? ?????? ??????? ?????????? ???? ????? ????????? ?????? ????? ?????? ??? ?? ????? ??? ??? ?? ??? ??????? ?? ????? ?????? ???? ??? ?? ???? ????? ?????? ????????? ???? ?? ?? ??????? ??????? ?????? ???? ??? ????? ??????? ????? ????????? ???????? ??????? ?????????????? ???? ???????? ????? ????????? ?????? ??? ???? ????? ? ?????? ???? ???????? ???????? ??????? ???? ????? ??????? ????? ?????? ????????????????? ???? ???? ?? ?????? ???????????? ?????? ?????????? ??? ???? ?????? ???? ????????????? ??? ?? ???? ???? ??????? ?????? ????? ?? ??????? ???????? ??? ?????? ????? ?????? ???? ?????? .????(1- 51 ):- ??? ?????? ?? ????? ????? ?? ????? ?????? ????? ?? ?????? ?? ???? ???? ????? ???? ?????? ???????? ??? ????? ????? ??????? ????? ???? ?????? ?????? ??????? ?????????????????????????????????? 12345?? ??? ???? ???????? ???? ??????? ?? ?????? ????????? ???? ???? ??? ?? ?????? ???????????? ?????????????? (Y)?????? ????????????? (X)????????????????????????????????????????????????????????????????????????? :- ???? ????? ?????? ????? ???????? .???? :- ???? ???? ????? ?????? ?????? ??? ??????? ???? ???????? . di2di???(Y)???(X)?????? ?????????????? (Y)?????? ????????????? (X)9336??????????0.250.5-5.55?????????????2.251.523.5??????????9-341????????????1112???????????4-25.53.5??????????25.5??????? rs=1 - 6i=1ndi2 n(n2-1)=1 – 6(25.5)6(62-1)=0.271 ??? ????? ????? ??????? ????? ???????? (tie) .????? ??? ???? ????? ???????? ?? ??? ??? ?????? ???????? ????? (?????? ?? ????? ????? ) ??????? ( ?????? ?? ????? ?????? )? ?? ?? ????? ?????? ?? ????? ?????? ????? ???? ??? ????????? ??????? ?? ?????? .10 : ????? ??????? ?????? ???? ( one-way analysis of variance) ???? ?????? ???????? ?? ??? ?????? ?? ????????? ??????? ?? ???? ???????? ???????? ?? ????? ????? ?? ?????? ?????? ??????? ?????????? ??? ????? ????? ??????? ??? ???? ?????? ?? ???????? ??? ?????? ??????? ?????? ??????? ?????? ??????? ????????? ?????? (?????? ???????).????????? ???? ???????? ?????????? ???????? ? ???? ?????? ????? ???? ??? ?????? ?????????.??? ??????? ??????? ??????? .. ??? ?????????? ?????? ? ??? ???? ????? ??? ????? ?? ??????? ????? ????????? ?? ????.?? ?????? ???? ??????? ????? ???? ??????? ????? ????? ?? ??????? ??? ? ??????? ??????? ???? ????????? ??????? ?? ???????? .???? ????? ?? ??? ???? ???? ?? ????? ??? ?????? ???? ??? ????? ????? ??????? (ANOVA models) ????? ??? ????? ??????? ????????? ?? ????? ??????? ?????? ???? . ??? ??????? ?????? ????? ???? ????? i = 1…r ??? i ????? ?????? ??? ?? j ???? ??? ???????? ???????? ?????? ????? ????? j = 1…ni ???? ??? ni ??? ??? ??? ??????? ??????? ith ? ??? yij ????? ????????? ???????? jth ??????? ?? ??????? ith ?? ??????? ?????? ??? ?? nT ??? ????????? ????? ??? ??i ???? ??????? ?? ??????? ??????? ????????? ??? ??????? ith ? ??? yi. ????? ????????? ????????? ??? ????? ?????? ith ????? ????? ??? ??????? ????? ????????? ??? ??????? ith ????? ???? ?????? yi. . ?? ????? ????? ??????? ?????? ???? (The one-way analysis of variance model) ???? ?? ???? ???? ??? ??????? ????? ????? ??????? ?????? (cell means model ) ?????? yij = ??i ? ??ij??? ?? ??i ???? ??????? ?? ??????? ??????? ????????? ??? ??????? ith ?? ?????? ??? ??ij ??????????????? ????? ????? ????? ????? ?????? 0 ?????? ???? ?2 .??? ?? ?? ????????? ???? ??????? ?????? ????? ??? ?????? ?i?? ??? ?i ???? ? ?? ????????? ????? ??? ??????? ?????? ?????? . ?? ????? ??????? ?????? ?????? ???????? ?????? ???? ????? ??????? ??????? ??????? ??????? ?i? .??? ????? ????? ?????? ??????? ?????? ???? ?????? ?????? ???? ??????? ??????? ?????????? ??? ??????? ??????? ?????? ?? ??????? . Ho: ??1 = ??2 = … = ??r Ha: At least one mean is not equal to the others??? ?????? ??????? (fitted value) ?? ??????? ????????? ???? ??? ????? ????? ?????? ??? ??????? ???????yij = yi .???????? (residuals) ??????? ?? ????? ??????? ????? ???? ????? ????? ???????? ??????? ???????? ??? ???????? ?????? ????? ?????? ? ??? ????? ??????? ????? ??? ??? ????? ?? ??????? ?????? ????? ???????? eij = yij ? yij= ?ij ? yi.??? ??????? ??? ????? ???? ?? ??????? ?????? ???? ???????? ??????? yi. = j=1niyij / ni??? ?? ni ??? ??????? ??????? i ?? ?????? .??? ???????? ???????? (standard deviation ) ???????? ?? ???? ????? ???????? ????? ?? s ??????? ???????? ???????? ???????? (Pooled standard deviation?) ??? ??????? ??????? ??? ??????? ??????? ?????? ????? s2 ??s = j=1ni( yij -yi. )2 / (ni - 1 )s2= i=1r j=1ni( yij -yi. )2 / i=1r(ni- 1 )???? ???? ?????? ??? s2 ??? ??????? s2= i=1r(( ni -1 )2 si2) / (nT - r )???? ???? ???? ????? (Individual confidence intervals? ) ??? ????? ?? ??????? ?????? ?? ??????? ??? ???? ??????? ???????? ???????? ???????? ??? ?????? ??????? yi. ? ?t (1 ????/ 2 ; nT ? r) (S) / ni?? S ???? ???????? ???????? ????????(pooled standard deviation ) ? ??? ???? ????? ??????? ???? ????? ??? ?????? ?? ???????? ???????? ????? ????? ?????? (Degrees of freedom (df )?) ????? ???? ??? ??????? ???????? ?? ????? ???????? ????? ????? ?????? ??? ????? ?? ??????? ?????? ??????dffactor=r-1dferror=nT-rdft=nT-1?? nT ??? ????????? ????? ? r ??? ??????? ?????? ?? ??????? .??? ?? ????? ???????? ?? ??????? ???? ????? ???????? ????? SS Total ????? ???? ?? ?????????? ????????? ?? ???????? ? SS (Factor) ????? ???????? ?????? ????? ?????? ?????? ?? ??????? ????? ???? ??? ???? ???? ????? ???????? ??? ????????? ?. SS Error ????? ?????? ???????? ????????? ?? ????? ????? ?????? ????? ???? ???? ????? ???? ????????? ????????? ???? ???????? ???? ?????? ?????????? ????? ??????? ???? ?? ????????? ?????? ??????? ?????? ??? ??? ????????? ?????? cf=(i=1rj=1niyij)2 nT??????? ???????? ??? ??????? (????????? ) SSFactor.SSFactor=i=1rni(yi.-y..)2=(i=1ryi.2ni )-cf ??? ????? ???????? ????? SSTotal .SSTotal=(yij-y..)2=i=1rj=1niyij2-cf??????? ?????? ???????? ???? ????????? ??????? SSError.SSError=i=1rj=1ni(yij -?y.. )2=SSTotal-SSFactor??? ?? ????? ???????? (Mean square (MS)?) ?????? ?????? ???? ???????? ???????MS(Facter)=SSFacter/dfFacterMS(Error)=SSError/dferror???? ???????? ???????? ???? ??? ???? ????? ??????? ?????? ??????? ?? ?? ???? ?? ???? ?????? ??????? Fcal= MS(Facter)/MS(Error)?????? ???? ????? ?????? ??? r-1 ??????? nT-r ??? ?????? ??????? ??? Fcal ??? ??? ??? ???? ????? ????? ???? ???? ??? ?? ????????? ??????? .?? ?? ??? ????? ?????? ????????? ??? (p-value) ?? ??? ?? ??? ??? ????? ????? ? ???? ?? ???? (p-value) ???? ???????? ?????? ??? ?????? ?????? ???? ?????? ??? ????? ?? ?????? ???????? ??? ?? ????? ???? ????? ?? ?????? ??? ???? (p-value) ???? ????? ??? 0.05 ???? ???? ?????? ???????? ??? (p-value) ?? ???? ??????? ??? ?? 0.05 ???? ????? ????? .? ??????? ????????? ?????? ???? ??? ???? ???? ???? ????? ??????? ??? ??????????? ????? ? ????? ???? ???????? ???????? ???? ?????? ? ????? ????? ?? ?? ???? ????? ??????? ??? S2 = MS Error ????? ????? ???????? ???????? ???????? .??? ????? ??????? (Coefficient of determination) ???? ??? ????? ???????? ??????????? ?? ????? ????????? ????? ???? ?? ??? ??????? ?????? ? ??? ?? ?????? ??????? ??? ??? ????? ??????? ?????? ???????? ?????? R-sq = 1 ??? (SS Error / SS Total)????? ???? ?????? ??????? ????? ?????? R-sq = SS Regression / SS Total ??? ????? ??????? ?????? (Adjusted ?R2 ) ????? ???? ???? ??????????? ????????? ????????? ?? ??????? ????? ???? ????? ??? ???????? ?? ??? ??? ????? ???????? ??????? ?????? ?????? R-sq Adj = 1 ? MS Error / (SS Total / DF Total)????(1-52):- ???? ?????? ??????? ???? ?? ????? ????? ?????? ??????? ????? ????? ?? ??????? ??? ?? ????? ???? ?? ?? ??? ?? ??????? ??????? ?? ????? ???????? ??? ???? ???????? ??? 60 ??? . ?? ??????? ?????? ???? ?? ?????? ????? ????????? ?? ????? ???????? ??? ????????? ?? ??????? ?????? ????? ??????? ??????? ???? one-way ANOVA .????? ???? ??????? ???????? ??????? cf=(i=1rj=1niyij)2 nT=(18.95+12.62+…………+18.10+22.50)216=(220.56)216=3040.4??? ????????????????Fcal=MSFacterMSError=48.813.6=3.58 ????? ??? ??????? ??? ?? ???? ???? ???? ????? ??????? ANOVA .One-way ANOVA: Durability versus CarpetSource DF SS MS F PCarpet 3 146.4 48.8 3.58 0.047Error 12 163.5 13.6Total 15 309.9S = 3.691 R-Sq = 47.24% R-Sq(adj) = 34.05%?? ???? Fcal ???????? ?? ???? ????? ??????? ????3.58 ??? ?????? ?????? ???????? ?????? ?????? (3,12) ??????? ?????? (0.05) ????? ???? (3.49) ??????? ?? ????? ??? ?????? ?????? ????? ?????? ?? ??? ??? ???? ?????? ???????? ?? ??????? ?? ??? ????? ?????? ???????? ? ???? ???? ???????? ?? ???? ???? (0.047) ???? ???? ??? ?? (0.05) ??? ??? ??? ??? ???? ?????? ???????? ??? ????? ?????? ? ???? ?????? ???? ???? ??? ????? ????? ?????? ??????? ?????? ?????? ???? (1-53 ) :- ????? ????? ????? ????? ???? ????? ???? ????????? ???? ?? ???????? ?? ????????????? ?? ???? ??????? ??????? ?? ????? ????????? ?? ??? ?????? ????? ??????? ???? ????? ????? ??? ?? ?????? ( 1 ) ? ??? ?????? (1 - 2- 3- 4- 5 ) ???? ??? ??????? ???????? ??????? ( ?????? ? ???? ????? ? ????? ? ???? ? ?????? ).??????? :- ??? ???? ????? ??????? ??????? ?????? ????? ??? ????????? ( ???????? ) .????? ?????????? ( ???? ? ??? ? ???? ?????? ?????? ) ??????? ?????? ( 0.05 ) . ??? ??????? ???????? ????????? ( ????? ???? ???????? ) ? ?????? ??? ??????? ?????? ????????? ??? ??????? .????? ????? ??????? ?? ????? ????????? ????????? ?? ??????? ???????? ??? .??????? ???????? ????? ?????? ??????? ?? ????? ???????? ?????? ?????????? ?? ????? ??????? . ???? ( 1 ) ??????? ????? ????? ???? ???????? ???????? ?????? ???????????? ???????? ?????? ????? ??????? ?????? ???? ??? ?????? (1) ?? ????? ???????????? ?????????????? ( ???????? ) ???????? ??????? ?? 1-5 ? ?? ??? ???????? ???? ?????? ?????? ???? ??????? .?????? ?????? ?????? ??? ???? ???? ???? ?????? ????? ??????? ?????? ???? ???? ??? ????? ????????? ??????? ????? ??? ????? ???? ????? ????????? ?????? ?????? .?????? ???? ???? ????????? ( ??????? ) ????? ????? ?????? ???? ??????? .????? ???? ????? ??????? ?? ??????? ??? ??? ????? ?? ????? ????? ????? ?? ???? ???????? ? ????? ???? ???? ?????? ?????? ?????? .????? ????? ????????? ????? ????????? ???? ??? ?? ????????? ???? ?????? ???????? ?? ???? ??? ?????? ? ??????? ?????? ???? ?? ????? ???? ????? ????? ???? ?????? ?????? ????? ??????? ????? ??? ??????? ??? ?????? ??? ???? ???????? ???? ?? ????? ??? ?????? ??????? ? ???? ?? ?????? ???????? ??????? ( ????? ????? ) ????? ???????? ? ???? ?????????? ??????? ?????? ?? ?????? ?????? ?? ?????? ?????? ???? ?? ?????? ????????? ????????? ? ????? ???? ???? ?????? ??? ????? ?????? ???? ??? ?? ?????? ??? ???? ?????? ?????? ??????? , ????? ???? ?????? ????? ????? ?? ??????? ???????? ??????? ???? ???? ???? ???? ??????? ?????? ????? ?? ????? ????? ????? ????? ???? ??? ????????? ( ??????? ) ??? ???????? ? ????? ????? ?????? ????? ??????? ??????? ??? ???? ???????? ??? ????? ???????? ????? ???????? ?????? ?? ?? ?????? .????? ?????? ?????? ???? ????? ?????? ?? ????? ??????? ??????? ?????? (?????? ??????? ? ????? ?????) ?????? ?? ??????? ????? ???? ?????? ?????? ?????? ???? ??? ?? ?????? ??? ???? ?????? ?????? ??????? ? ?????? ????? ???? ?? ??????? ???????? ???? ??? ??????? ??????? .One-way ANOVA: DEGREE versus COLEG Source DF SS MS F PCOLEG 4 1391.4 347.9 3.81 0.018Error 20 1825.6 91.3Total 24 3217.0S = 9.554 R-Sq = 43.25% R-Sq(adj) = 31.90% Individual 95% CIs For Mean Based on Pooled StDevLevel N Mean StDev -----+---------+---------+---------+----1 5 63.800 6.573 (--------*--------)2 5 67.400 6.148 (--------*--------)3 5 83.600 10.431 (--------*--------)4 5 64.400 4.879 (--------*--------)5 5 74.400 15.582 (--------*--------) -----+---------+---------+---------+---- 60 70 80 90Pooled StDev = 9.554Grouping Information Using Dunnett MethodLevel N Mean Grouping3 (control) 5 83.600 A5 5 74.400 A2 5 67.4004 5 64.4001 5 63.800Means not labeled with letter A are significantly different from control level mean.Dunnett's comparisons with a controlFamily error rate = 0.05Individual error rate = 0.0153Critical value = 2.65Control = level (3) of COLEGIntervals for treatment mean minus control meanLevel Lower Center Upper +---------+---------+---------+---------1 -35.819 -19.800 -3.781 (-------------*------------)2 -32.219 -16.200 -0.181 (-------------*------------)4 -35.219 -19.200 -3.181 (------------*------------)5 -25.219 -9.200 6.819 (------------*-------------) +---------+---------+---------+--------- -36 -24 -12 0Hsu's MCB (Multiple Comparisons with the Best)Family error rate = 0.05Critical value = 2.30Intervals for level mean minus largest of other level meansLevel Lower Center Upper --+---------+---------+---------+-------1 -33.725 -19.800 0.000 (--------*------------)2 -30.125 -16.200 0.000 (--------*----------)3 -4.725 9.200 23.125 (--------*--------)4 -33.125 -19.200 0.000 (--------*------------)5 -23.125 -9.200 4.725 (--------*--------) --+---------+---------+---------+------- -30 -15 0 15Grouping Information Using Tukey MethodCOLEG N Mean Grouping3 5 83.600 A5 5 74.400 A B2 5 67.400 A B4 5 64.400 B1 5 63.800 BMeans that do not share a letter are significantly different.Tukey 95% Simultaneous Confidence IntervalsAll Pairwise Comparisons among Levels of COLEGIndividual confidence level = 99.28%COLEG = 1 subtracted from:COLEG Lower Center Upper ---------+---------+---------+---------+2 -14.474 3.600 21.674 (--------*--------)3 1.726 19.800 37.874 (--------*--------)4 -17.474 0.600 18.674 (--------*--------)5 -7.474 10.600 28.674 (--------*--------) ---------+---------+---------+---------+ -20 0 20 40COLEG = 2 subtracted from:COLEG Lower Center Upper ---------+---------+---------+---------+3 -1.874 16.200 34.274 (--------*--------)4 -21.074 -3.000 15.074 (---------*--------)5 -11.074 7.000 25.074 (---------*--------) ---------+---------+---------+---------+ -20 0 20 40COLEG = 3 subtracted from:COLEG Lower Center Upper ---------+---------+---------+---------+4 -37.274 -19.200 -1.126 (--------*--------)5 -27.274 -9.200 8.874 (--------*--------) ---------+---------+---------+---------+ -20 0 20 40COLEG = 4 subtracted from:COLEG Lower Center Upper ---------+---------+---------+---------+5 -8.074 10.000 28.074 (--------*--------) ---------+---------+---------+---------+ -20 0 20 40Grouping Information Using Fisher MethodCOLEG N Mean Grouping3 5 83.600 A5 5 74.400 A B2 5 67.400 B4 5 64.400 B1 5 63.800 BMeans that do not share a letter are significantly different.Fisher 95% Individual Confidence IntervalsAll Pairwise Comparisons among Levels of COLEGSimultaneous confidence level = 73.57%COLEG = 1 subtracted from:COLEG Lower Center Upper ------+---------+---------+---------+---2 -9.004 3.600 16.204 (------*-----)3 7.196 19.800 32.404 (-----*-----)4 -12.004 0.600 13.204 (-----*------)5 -2.004 10.600 23.204 (-----*------) ------+---------+---------+---------+--- -20 0 20 40COLEG = 2 subtracted from:COLEG Lower Center Upper ------+---------+---------+---------+---3 3.596 16.200 28.804 (-----*-----)4 -15.604 -3.000 9.604 (------*-----)5 -5.604 7.000 19.604 (------*-----) ------+---------+---------+---------+--- -20 0 20 40COLEG = 3 subtracted from:COLEG Lower Center Upper ------+---------+---------+---------+---4 -31.804 -19.200 -6.596 (-----*------)5 -21.804 -9.200 3.404 (-----*------) ------+---------+---------+---------+--- -20 0 20 40COLEG = 4 subtracted from:COLEG Lower Center Upper ------+---------+---------+---------+---5 -2.604 10.000 22.604 (-----*-----) ------+---------+---------+---------+--- -20 0 20 40 ??????? ???????? ????? ?? ???? ????? ??????? ??? ?? ?????? ?????????? ???????? ???? (0.018)??? ??? ???? ????? ??????? ????? ??? ???? ?????? ???????? ??? ????? ??????? ???? ???? ??? ????? ??????? ?????? ?????? ?? ??????? ??? ???????? ??? ?? ???? ???????? ???? (0.018) ??? ??? ?? ????? ???????? ?????? ???????? ??????? (0.050) ?????? ????? ?? ???????? ??? ????????? ?????? ???????? ????????? ??? ??? ??????? ?? ?????? ????? ?? ????? ?? ?????? (???? ???? ??????? ) ??? ??? ?????? ????? ?????? ??????? ( ??????(1) ? ???? ????? (2) ? ???? (4) ? ?????? (5) ) ?? ????? ????? ???? ????? ????? (3) ??? ????? ????? ???? ????? ????? ?????? ???? ?? ???? ????? ??? ????? ??? ???? ????? ????? ?? ?????? ????????? ?? ???? ???????? ? ??? ?? ????? ????? ?????? ??????? ( ??????(1) ? ???? ????? (2) ? ???? (4) ) ?? ?????? ???? ????? ??? ?????? ??????? ??? ??? ??? ??? ??????? ) ??? ????? ?????? ??????? ????? ?????? ??????? ????? ????? ??? ???? ??? ????? ????? ?? ??? ?????? ? ????????? ????? ???? ?? ?????? ?????? ???????? ?????? ?? ?????? ??????? ??? ?? ???? ????? ?????? .???? ???????? ??? ???? ?????? ????? ????? ???????? ???? ?? ???????? ( ??????(1) ? ???? ????? (2) ? ???? (4) ) ???? (0.000) ??? ?????? ????? ??? ???? ?????? ????? ????? () ??? ?????? ?????? ????? ?????? ?? ???? ???????? ????? ??? ????? ?????? ??? ??? ???? ?????? ?????? ?????? () .?????? ??? () ??? ?????? ???????? ?? ????? ???? ????? ?????(3) ??? ??? ???????? ????? ????? ??????? ???? ??? () ?????? ????? ???????? ??? ???? ????? ??? ???? ???? ?????? ???????? ?? ??? ????????? ?????? ????? ????? ??????? ( ????? ????? ) ???? ????????? ???????? ??? ?????? ?????? ???? () ????? ?????? ??????? (??????(1) ? ???? ????? (2) ? ???? (4) ) ???? ??? ?????? ????? ??????? ?????? ??????? ??? ?????? ???????? ???????? ???????? ????? ????? ?? ????? ?????? ??? ???? ?????? () ????? ????? ?? ?????? ?????? .?????? ???? () ??? ?????? ????? ??????? ????? ????? ??? ????? ????? ??????? ?????? ????? ????? ????? ??????? ??? ????? ?????? ?? ???? ?????? ?????? ?????????? ??? ????? ?????? ????? ????? ??? ??? ??????? ?????????? ???? ???? ?????? () ??????? ???? ???????? ??? ?? ???????? ???? ?? ????? ??? ???? ?????? ???? ??? ???? ?????? ???????? ??? ???????? ???? ????? .????? ????? ???? ????? ?????? (1) ??? ???? ?? ????? ???????? ?????? ????? ????????? ?? ????? ????? (3) ??? ?? ???? ????? ???? ?????? ?????? ??? ?? ????? ??? ?????? ??? ??? ??? ??? ???? ???????? ???????? ?? ??? ????? ?????? ?????? ????? ? ??? ???? ???????? ??? ?? ???? ????? ??????? ???????? ????? ??? ?????? ??????? ??? ??? ??? ??? ???? ???????? ???????? ???? ??? ???? ???????? ?????? ?????? .????? ????? ???? ????? ???? ????? (2) ?????? ???? ?????? ????? ???? ????? ???? ?????? ?????? ??????? ?????? ??? ?????? ??????? ? ??? ?? ???????? ?? ????? ????? ?????? ?? ?? ???? ?????? ??? ???? ????? ???????? ???? .????? ????? ???? ????? ????? (3) ????????? ?? ????? ???? ??????? ???? ???? ?????? ????????? ?? ????? ???? ???? ?????? ?????? ??? ?????? ??????? ??? ???? ? ???? ???? ???? ?????? ?? ??? ????? ????? ??????? ??? ?????? ???????? ????? ??? ?????? ??????? ????? .??? ?????? ???? ?? ?????? ??? ???????? ???? ??? ???? ?????? ???????? ??????? ?????? ??? ????? ????? ???? .??? ?????? ??? () ??? ??? ???????? ????? ??????? ?????? ????? ???? ?????? ????? () ???? ?????? ????? ????? ????? ??? ????? ?????? ???? ?????? ????? () ???? ????? ??? ????? ????? () ??? ????? ?????? .????? ????? ???? ????? ?????? (1) ??? ???? ?? ????? ???????? ?????? ????? ????????? ?? ????? ?????(3) ??? ?? ???? ????? ???? ?????? ?????? ??? ?? ????? ??? ?????? ??? ??? ??? ??? ???? ???????? ???????? ?? ??? ????? ?????? ?????? ????? ? ??? ???? ???????? ??? ?? ???? ????? ??????? ???????? ????? ??? ?????? ??????? ??? ??? ??? ??? ???? ???????? ???????? ???? ??? ???? ???????? ?????? ?????? .????? ????? ???? ????? ???? ????? (2) ?????? ???? ?????? ????? ???? ????? ???? ?????? ?????? ??????? ?????? ??? ?????? ??????? ? ??? ?? ???????? ?? ????? ????? ?????? ??? ???? ????? ???????? ???? ??? ?? ????? ??? ?????? ??????? .????? ????? ???? ????? ????? (3) ????????? ?? ????? ???? ??????? ???? ???? ?????? ????????? ?? ????? ???? ???? ?????? ?????? ??? ?????? ??????? ??? ???? ? ???? ???? ???? ?????? ?? ??? ????? ????? ??????? ??? ?????? ???????? ????? ??? ?????? ??????? ????? .??? ?????? ???? ?? ?????? ??? ???????? ???? ??? ???? ?????? ???????? ??????? ?????? ??? ????? ????? ???? .??? ????? ?????????? ??????? ??????? ??? ???? ???? ??? ?????? ????? ???? ????? ????? ??????? ????????? ?? ???? ??? ???????? ( ?????? ????? ????? ?????) ?? ???? ??????? ?????? ??? ?? ??? ? ?? ??? ?????????? ??????? ?? ??? ??? ???? ????????? . ??? ????? ??????? ?????? ?????? ????? ???????? ( ?????????) ????? ???? ?????? ????? ????? ???? ????? ????? ?? ???? ???????? ??????? ?????? ??? ??? ???? ??? ??????? ?? ??? ???? ?? ????????? ??????? ? ??? ?????? ?????? ????? ????? ????? ????? ????????? ????????? ????? ????? ?? ??? ????????? .??? ???? ???????? ?????? ?????? ???? ????????? ?? ???? ???????? ??? ???? ????? ???? ????? ?????? ?? ???????? ??? ?????? ?????? ???? . ??? ??????? ??????? ??????? ???? ???? ????? ????? ??????? ??? ????? ?????? ????? ???????? ?? ??? ??? ??? ??? ?????? ?????? ?????? ????? ?????? ???? ???? ??????? ???? ????? ???????? ?????? ?????? ???????? ?? ???? ?????? ????? ???????? ???????? ?? ???? ???????? . ??? ????? ????? ??????? ????? ?? ???? ?????? ?? ??????? ???? ????? ??????? ??????? ????? ???????? ??? ????? () ??????? ???????? ???? ???????? ??? ??? ??? ???? ????? ????? ??????? ??? ?? ???? ??? ????? ?? ???? ?????? ?????? ??? ??? ?????? ??????? ??? ??? ????? ?????? . ??? ??? () ????? ??????? ????? ??????? ?? ??? ????? ?????? ????? ?????? ????? ????? ????? ??????????? ???????? ??? ????? ??????? ???? ?????? ?????? ?? ????? ??? ???? ??? ????? ?????? ?? ???????? ?? ?????? ??????? ?????? ?????????? .?????? ???????? () ?????? ??? ????? ???????? ??????? ?? ????? ??? ????? ???? ????? ?????? ?? ?????? ?????? .??? ??? () ?????? ??????? ?? ??????? ??? ??? ?????? ??????? ??? ????? ???? ???? ????? ??? ?????? ?? ????? ???????? ??? ??? ??? ????????? ??????? ?? ????? ??? ??? ???? ????? ???? ????? ?????? ?? ???????? . ?? ????? ??????? ?????? ??????? ????? ???????? ??????? ?????? ??? ??? ??????? 1-11:- ?????? ??????? - ???? (Kruskal-wallis)??? ???? ?????? ????? ??????? ????? ?????? ????? ??? ????? ?????? ?? ?? ???? ?????? ????? ??????? ???? ????? ?? ??? ?????????? ???? ????? ??????? ????? ?????????? ???? ?????? ????? ? ?????? ?? ??????? ?? ????? ??????? ????? ??? ????? ??? ?? ???????? ?????? ????? ??? ??? ???? ???????? ??? ????????? . ???? ?????????????- ????????? ??? ?????? ??????- ???- ????. ??? ?????? ??? ????? ?????? ???????? ?? ??? ???? ?? ???? ?? ????????? ( ????????) ??????? ??????? ????? ????? ???? ??? ??? ?? ?? ??????? ??????? ??? ???? ????? ??????? ????? ????? ?????? ???? ???? ??? ????? ????? ?? ????? ????????? ????? ??? ?? ?????? ( ?????? ) . ?? ???????? ????? ??? ?????? ???? ????? ??????? ?????? ???? ANOVA ????? ????? ??? ????? ?? ?? ??????? ????????? ??? ???????? ???? ?????? ?????? ????? ????? .????? ?????? ??????? - ???? ?? ????? ??? ??? ?????? ??? ????????? .??????? ???????? ???? ??????? ??????? ??????? ??? ????????? ??? ????????? ?? ????? ??????? ( ????? ?????? ) ? ??? ??? ??????? ??????? ??? ????? ???? ?? ????? ???????? . ??? ???????? ????????? ??????? ???? ??????? ? ????? ?? ????? ??????? ??? ??????? ????????? ?????? . ??? ?????????? ???? ?????? . ???? ????????? ????? ?? ????? ???????? ?????? ?????? ?? ??????? ??? ???? ????? ?? ??????? ??? ? ??????? ??????? ???? ????????? ??????? ?? ?????? .??????? ?? ???????? ?????? ?? k ?? ??????? ???????? ?????? n1,n2,n3,…nk ???????? ?? n=n1+n2+n3…+nk ??? ???? ?????????? ???? ???????? ??????? ?????????? (????????)??? ?????? ???????? ??????? Ri ????? ????? ????? ????? ???????? ??????? Ri-ERiVAR(Ri) ~ N(0,1)ERi=ni(n+1)/2???? ?? ???? ????? ??????? ??????? ??????? Ri ????? ????? ???? ??? ?????? ???? ???? ??? ??? ???? k ?? ??????? ??? i=1k(Ri-ERi)2VAR(Ri)~ Xk-12?? ????????????? - ???? (Kruskal-wallis) ????? ?????? ???????? ??????? H*=1S2i=1kRi2ni-n(n+1)24??? ?? s2=1(n-1)i=1kj=1nir(Xij)2-n(n+1)24????? ???????? ?? ???? ??? ?????? ??????? ???????? ?? ???? ??? ???? ??? (ties) ??? ????????? H=12n(n+1)i=1k1niRi-ni(n+1)22????? ????? ??? ???????? ??????? H= 12nn+1 i=1kRi2ni – 3(n+1)?? ???????? H ????? ????? ???? ??? X2 ?????? ???? k-1 ??? ?????? ??????? - ???? ?????? ???? ???? ?????????? ???? ???? ????? ??????? ???? ???? ???? ????????? .????(1-54) :- ???? ????? ?????? ???????? ????? 100? ?????? ????? ?????? ?? ????????? ?? ???? ??????? ????? ??? ????? ?????? ?? ??? ??????? (A,B,C) ????? ?????????? ??? ?? ?????? ?? ???? ???? ?? ?????? ??????? ??????? ??? ???????? ????????? ??????? ?????? (0.05) .CBA554333154432???? ?????? :- ???? ????????? ????? ?????? x(i) ????????????? ??????? ???? ???????? ???????? ?? ??????? ?? ??? ????? ??? ?? ???? ???? ?????? 555444333321x(i)121110987654321???????1111118884.54.54.54.521?????CCACAACBBAAB??????????? ????? ?? ????? 3 ???? ???? ???? ????? ???? 4 = 4.5/(4+5+6+7) ????? ??????? ????? ????? ? ??? ???? ????? ????? ??? ?????? ?????? ?????? ??? ??? ?? ?????? ?? ??????? ??????? A (2,3,4,4,5) R1= 2+4.5+8+8+11 = 33.5 B (1,3,3) R2= 1+4.5+4.5 = 10C (3,4,5,5) R3= 4.5+8+11+11 = 34.5????? ??? ?????? ????????H=12n(n+1)i=1k1niRi-ni(n+1)22 H= 1212(13) 15(33.5- 5(13)2)2+13(10- 3(13)2)2+14(34.5- 4(13)2)2 =3.72????????? ???????? ??????? H=1212(13) 33.525+1023+34.524- 3(12+1) =3.72??? ????? ????? ??? ????? H ? Hadj ??????? ????? ???? ??? ????? ???? df=k-1 ??? ???? ???????? ???? ???????? ??????? p_value=1-CDF( XH,df2)?? ??????? ??????? ???????? p_value=1-CDF( XHadj,df2)???????? ??????? ????? ??????? ??????? ???????? ???????? . ????? ????? ?????? ??????? ????????????? (tie) ??? ????????? ??????Hadj= H1-ti3-tin3- n???? ??????? ??? ????? ???????? ??? ???? (MINITAB ) ?? ???? ????? ??????? ?????? ?????? ??? ?? ??? ????? ? ???? ?? ????? ????? ???? ??? ????? ????? ????????? H0: ?1=?2=?3????? ????? ?????? ???? ???? ??? ??? ????? ????? . H0:not all ?, s are equal ??? ?? ? ???? ????? ????????? population medians)).????? ?????? ?????? ?????? ??????????? ?????????? ????? ???????? ?????????? ???? ????? ???? ????? ????????? ?? ?????? ????? ?????? ?????? ????? ????? ??????? ?????? ??? ??? ???? ???? ?????? ?????? ?????? ???? ??? ?? ???? ??????? ?????? ??????? ??????? Kruskal-Wallis Test: TR versus GR Kruskal-Wallis Test on TR AveGR N Median Rank Z1 5 4.000 6.7 0.162 3 3.000 3.3 -1.763 4 4.500 8.6 1.44Overall 12 6.5H = 3.72 DF = 2 P = 0.156H = 3.97 DF = 2 P = 0.137 (adjusted for ties)??????? ??? ??????? :- ?? ??? (P = 0.156) ?? ?????? ??? ??????? ??? ???? ?? (0.05) ???? ?????? ??????? ??? ???? (P = 0.137) ???? ????? ??? ??? ???? ???? ?????? ?? ??? ????? ????????? ??? ??????? ??????? . 1-12 :- ????? ???? ??? ??????? ??????? ?? ???? ???????? ????? ??? ?? ????????? ? ??? ?? cij ???? ???????? ??????? ?? ?????? ith ????????? jth ???? ??? i=1,2,….,n ? j=1,2,……,k ? ????? ???????? ?? ???? ????? ???????? ?????? .???? ???? ?????? ????? ??????? ?????????????????? ???? ??? ????? ?? ???????? ???? ????? ?? ??????? ????? N ?? ????????? ????????? ???????? . i=1,2,….n and j=1,2,…k, ??? ?? (Xij1,…..,Xijcij) ?? ??? ???? ????? (i,j) ???????? i∈{1,2,…n} ? j∈{1,2,…k} ??? cij ?? ????????? ????????? ???? ???? ??????? ?? ????? ????? ?????? ????? Fij .?? ???? ??????? F11,….,F1k,……,Fn1……..Fnk ?????? ???????? ??????? Fiju= Fu-βi -τj, -∞<u< ∞ ???? ??? i=1,2,….,n ? j=1,2,……,k ??? ?? F ???? ???? ??????? ????????? ??????? ??????? ????? ??? ????? θ ? ??? βi ????? ??????? ???????? ??? ??????? ??????? ??????? i ? ? τj ????? ???????? ???????? ??? ??????? ??????? ?? ???????? jth . ?? ???????? ??????? ????? ???? ?????? ?????? ??? ????? ?? ??????? ????? ????? ????? ??????? ??????? ???? ?? ????? ?? ????? ???????? ?????? ????????? ??????? ???????? ????? ???????? ??????? ??? ??? ?? ??????? ???? ?? ???? ??????? ??????? Xijt=θ+βi+τj+eijt??? ??.i=1,2,….,n , j=1,2,……,k , t=1,2,……cij ??? θ ?????? ????? ? τj ????? ???????? j ? βi ????? ?????? i ? ?? ???? ?? ??????? ????????? ??? ????? ????? ????? ????? ????? .?? ????? ????? ???? ??? ??? ???? ?????? ?????? ?? ??? ??????? ??????? ????? ?????? H0: τ1=τ2……=τk????? ?? ???? ??????? ???? ?????? i ???? ??????? ??????? Fi1≡Fi2≡…≡Fik≡Fi???? ??? ,n.....i=1, ( ?? ???? ??????? ???? ??? ????????? ?? ?????? ?????? ?????? ????? ?? ???? ??? ,n.....i=1, ? ,k.....j=1, ??? ??? ???????? ????? ?????? ?????? ???????? ???????? ???????? X ) .????? ????? ?????? Ha: τ1……τk not all equal????? ???? ??? ????? ???????? ?? ?????? ???????? .??? ????? ???? ?????? ?????? ???????????? k ?? ????????? ?? ???? ???? ??? ???? ???? ????? ?? ???? ??? ?????? ???? ??????? ? ???? ???? rij ???? ??? ( ?????? ) ??? ????????? Xij ??? ?????? i ??? ???? .Rj=i=1nrij , R.j=Rjn ???? ?????? ?????? ??????? ???? ??? ??????? ??????? S=12nk(k+1)i=1kR.j-(k+1)22?? ??????? ???????? ??????? S=12nk(k+1)j=1kRj2-3n(k+1)??? (k+1)/2 ???? ????? ????? ??????? ?? ???? ???????? ??????? .?????? ??? ????? ????? ??? ??? Reject H0 if S ≥Sα ???? ??? ?? ??? ????? ? ??? ?? ?????? Sα ????? ???? ???? ????? ?? ????? ????? ????? ??? ????? ???????? α . ??? ??? Sα ???? ?? ???? ( 5 ) .??? ?? ???? ??????? ??????? ?????? ????H0 ????? ????? ????? ? ??? ?????? ???????? S ??????? ??? ?????? n ????????? ?? ????????? ????? ????? ???? ???X2 ????? ???? k-1 .??? ??? ????? ????? ??? ???? ????????? ??????? Reject H0 if S ≥Xk-1,α2 ???? ??? ????? ????? .???? (1-55 ) :- ????? ?????? ??????? ??????? ?????? ?????? ????? ??? ????? ???????? . ??? ?? ????? ????? ????? ?????? ?? ?????? ???????? ??? ????? ???????? ??????? ? ???? ?????? ??????? ??????? ?????? ??????? . ????? ????? ??? ??? ?? ?????????????? (??????) ????? ??? ????? ????? ?????? ??? ?? ??????? ????????? ?????? ????? ????? ?????? .H0: all treatment effects are zero vs. H1: not all treatment effects are zero??? ????? ??? ???? ( ????? ?????? ) ????? ?????? ?????? ?? ????? ???????????? ????? ????? ??? ?? ????? ???? ??????? ???? ????? ? ????? ????? ???? ??????? ?????? ???? ??? (Rj ) ????? ????? ??? ?????? (?????? ??????) S=12nk(k+1)j=1kRj2-3nk+1=124.3.4(6.52+72+10.52)-3.4.4=2.375?2.38??? ???? ??????? ?? ???? ???? (MINITAB) ???? ?????? Friedman Test: Enzyme Activity versus Therapy blocked by Litter S = 2.38 DF = 2 P = 0.305S = 3.80 DF = 2 P = 0.150 (adjusted for ties) Sum ofTherapy N Est Median Ranks1 4 0.2450 6.52 4 0.3117 7.03 4 0.5783 10.5Grand median = 0.3783??????? ????????? S ???? ???????? p ????? (0.305) ?????? ?????????????????? ? ???????? (0.150) ? ???? ?? ????? ???? ?? ?????? ???????? (0.05) ??? (0.10)? ???? ??? ???? ???? ??? ???? ?????? ????? ????? ? ???????? ?? ??? ???? ????????? ?? ??????? ???????? ???? ???? ??? ?? ???????? ????????? ?????? ?? ????? .1-13 ?????????? ??????????????? ???? ?????? ?????? (sample space):- ?? ???? ?????? ?????? ???? ? S ? ??? ????? ?? ?????? ?? ??????? ??????? ??????? ? ????? ??? ??????? ??????? ??????? ????? ???? ???? ?????????? ?? ????? ??? ???? ????? ???????? ???? ????? ??? ????? ?????? ??? ?? ?????? ???? ???? (H) ?? ????? (T) ???? ?????? ????? ????? ??? ???? ?????? ??? ??? ??????? S=H,T ? ????? ??? ????? ??? ?????? ??? ????? ??????? ?? ???????? ??????? ???? ??? ??? ??? ?? ???? ??? ???? ?????? ???? S=M,F .??? ????? (event) ??? ????? ?? ????? (?????? ?????) ?? ???????? ??????? ????? ????? ?????? ? ??? ????? ???? ???? ??? ???? (simple) ??? ?? ????? ?????? ??? ????? ????? ??? ????? ?????? (compound) ???? ????? ??? ???? ?? ????? ?? ???? ?????? ?????? ????? ???? S ? ??? ?????????? ???? ?????? ???? ????? ??? ?? ??? A ???? ????? ?????? ????? A ????? ??????? P(A) ????? ???? ????? ????? ???? ????? A ??? ??? ???????? ?? ?? ?? ?? ???? ????????? ??????? (1) ??? ??? A ??? ???????? ???? ??? ???? ?? ????? ??? P(A)≥0 ? (2) ?? ?????? ???? ?????? ????? ???? PS=1 ? (3) ??? ??? A1,A2,A3…… ?????? ?? ??????? ??? ??????? ???? ???????? ??? ?????? ?????? ??????? ???? ?????? P(A1∪A2∪A3……)=i=1∞P(Ai)??? ?? ?????? ????? ?????? ????? ???? P?=0 ??? ? ???? ????? ?????? ??????? ??? ?????? ??? ??????? ??????? ??? ????? ???????? ???? ???? ???? ???? ?? ??????? ? ??? ??? ??? ??? ????? ????????? ?? ??? ???? ???? ???????? PA=1-P(A')???? ????? A ? B ??? ??????? (disjoint) ??? ???????? ????? ??????? ???? ??????? ???????? ??????? PA∪B=PA+PB??? ?? ???? ?????? ??????? (not disjoint) ???? ?????? PA∪B=PA+PB-PA∩B1-13-1 ??? ????????? ???????????? ???? ??? ??? ????????? ????????? ????? ????? ?????? (Binomial distribution)?? ?? ????? ????? ???? ?????? ??????? ??????? ???? ?????? ????? ?????? (1) ??????? ????? ??? ??????? ????? ?? n ?? ??????? ??????? ???? ?????????? ??? ?? n ????? ?? ??????? ?????? ????? .(2) ?? ????? ?? ?????? ???? ???? ?????? (????? ?????????) ????? ???? ??? ????(S) ?? ??? (F) . ???????? ????????? ?? ???? ?????? ????????? ???? ????????? ???? ??????????? ????????? ?????????? ??? .(3) ?? ????????? ?????? ? ??? ????? ?? ?????? ?????? ??? ?????? ?? ?? ????????? ?????? ??? ???? ????? ??????? ??? ??????? ????????? ?? ??? ???? ???? ?????? ?? ??????? ????????? ?? ??? ??? .(4) ?? ?????? ?????? ???? ?? ?????? ??? ????????????? (p) ?????? ????? ?????? ????? ??? n ?? ????????? ? ??? ????? ????? ?????? X ??????? ??? ??? ??????? ???? ??????? ??????? X = ??? ??????? ?????? S ?? ??? n ?? ????????? .??? ?? ???? ???? ???????? (pmf) ?????? ????? ????? ?????? X ????? ??? ??????? n ? p ? ????? ???? ???? ???????? ??????? b(x;n;p) bx;n;p=nxpx1-pn-x ,x=0,1,2,3,…..n0 otherwise??? ?????? ???????? ????????? ???????? ??????? ???? ??????EX=np , VarX=np1-p,σX=npq ??? ?????? ????? ?????? (Bernoulli distribution) ???? ???? ?? ????? ????? ?????? ????? n=1.????(1- 56 ):- ??? ??? ?????? ????? ??? ?? ???? ??? ????? ?? ???? ????? ?? ??? ???????? ???? (0.5) ???? ??? ??? ????????? ?????? ?????? (6 ) ??????? ??? ??????? ???? ????? ????? ?????? .p=Pa selected individual prefers S=0.5X = ??? ??????? ??????? ? ??? ????? ??????? X ???? ????? ????? ?????? .X~Bin(6,0.5) ??? ?????? ????? ???? ????? ?? ??? ?????? PX=3=b3;6;0.5=630.531-0.56-3=200.56=0.313 ??????? ????? ???? ????? ??? ????? ?? ?????? PX≥3=x=36bx;6;0.5=x=366x0.5x1-0.56-x=0.656??? ?? ?????? ?? ???? ?????? ?? ?????? ??? ????PX≤1=x=01bx;6;0.5=0.109????????? ??????? ????? ????? ????? ?????? ???? ??????? ????? ?????????? ???? ?? ?????? ????? ??? ????????? ??? ???? ??????? ????????? ( cdf ) ????? ??? ??? ?????? ???????? ???????? ??????? PX≤x=Bx;n;p=y=0xby;n;p ,x=0,1,2,…..n????? ?????? ?? ???? ??????? (?????????) n=5,10,15,20,25 ???? ?????? ???????? p.???? ( 1-57) :- ???? ???? ???????? ????????? ??? ????? ????? ????? ??????PX≤8=y=08by;15;0.2=B8;15;0.2??? ????? ????? ????? ?????? ??? ?? ???????? ???? B8;15;0.2=0.999??? ?????? PX=8=PX≤8-PX≤7=B8;15;0.2-B7;15;0.2?????????? ??? ??????? ?????????? ??? ?? ???? ???????? ???? 0.999-0.996=0.003 ????????? ????? ????? P4≤X≤7=PX=4,5,6, or7=PX≤7-PX≤3 =B7;15;0.2-B3;15;0.2 =0.996-0.648=0.348 ????? ?????? (poisson distribution)??? ??? X ????? ?????? ???? ????? ?????? ?????? ??????? λ(λ>0) ??? ???? ???? ???????? pmf ??????? X ???? fx,λ=e-λλx x! x=0,1,2,3,4,…..??????? ? ?? ???? ???? ???????? ?????? ????? ?????? bx;n;p ???????? ?? ??? ?????? ????? ?? ????????? n→∞ ????????? ????? ?? ??? p→0 ??? ??? ?????? ?? ? np ????? ??(λ>0) ? ????? ??? fx,λ bx;n;p→.? ??? ??? ??????? ????? ??? np<5 ? n>50 ??? ???? ????? ?????? ?? ??????? ???????? ????? ???? ??????? λ. Ex=VARx= λ?? ?????? ?????? (poisson process) ????? ??????? ???? ?????? ?????? ???? ?????? ???? ??????? ??????? ????? ???? ???? ???? ????? ? ???? ?? ????? ?????? ??????? ???? ??? t=0 ? ???? ???? ?????? ??? ??? ??????? ??????? ??? ????? ??????? ? ???? ????? ???????? ??????? ??? ??????? ???? ??? ??? ???? ????? . ???? ????? (α>0) ????? ???? ?????? ????? ????? ???? ?t? ?? ?????? ???? ??? ???? ??? ?????? ?????? ???? α?t + o( ?t ) .?? ?????? ???? ???? ?? ??? ???? ?????? ??????? ?t ???? o?t ? ????? ??????? ?????? ??? ???? ????? ???? ?t ????? 1-α. ?t-o?t.?? ??? ??????? ??????? ???? ????? ??????? ?t ???? ?????? ?? ??????? ??????? ?? ????? ??????? ??????? . ????? ???? ?? ???? ?????? ??????? fk,t=e-αtαtk k!????? ??? ??? ??????? ???? ???? ????? ???? t ???? ????? ?????? ?????? ?????? ????????? λ = αt ? ??? ????? ??????? ?? ??????? ???? αt ? ????? ????? ??????? ???? ???? ????? ????? ?? α .????(1-58 ):- ??? ?????? ???? ????? ???????? ???? ?? ???? ????? ????? ?? ???????? ?????? ?????? ?? ???? ?? ?????? ??? ??? ???? ??? ????? 0.005 ??? ??????? ?????? ?? ????? ?? ???? ????? ? ???? ?????? ????? ???? ?? 400 ???? ??? ????? ????? ????? ??? ??? ?? ???? ????? ? ??? ?? ?????? ??? ?????? ???? ????? ????? ??? ??????? . s = ???? ???? ????? ??? ????? ??? ??? ???? . F = ???? ???? ????? ?? ???????.?? x ????? ???? ??????? ????? ????? ??? ???? ??? ????? ??? ????? ?????? ?????? ????? ?????? ????????? n=400 ? p=0.005 ???? ?? np=2 ??? ???????? ??? ?????? ?????? ?????? ????? fx=1=b1,400,0.005 ? f1,2= e-2211!=0.2707?? ????? ?? ????? ????? ????? ?????? ????? B1,400,0.005=0.2706 ????? ?????? ??????? ?? ?????? ?????????? . ??? ?????? ??? ?????? ???? ???? ????? ????? ??? ??????? ????? ???????? fx≤3?x=03fx,2= x=03e-22xx!=0.8571?? ?? ???? ??????? ???????? ??????? fx≤3=0.85761-13-2 ??? ????????? ???????? (Continuous distribution)??? ??? x ????? ?????? ????? ? ??? ?????? ?????????? (probability distribution) (?? ???? ??????? ?????????? ) (((probability density function(pdf ??????? x ?? ?????? f(x) ????? ???? ?????? ??????? ??? ?????? a ? b ???? a ≤b .pa≤x ≤b= abf(x)dx??? ???? ?????? ?? x ???? ???? ?? ????? [a,b] ??? ??????? ??? ????? ???? ??????? ???? ????? ??? ????? ??????? ?????? ?????? f(x) ???? ???? ?????? ??????? (density curve). pa≤x ≤b ???? ??????? ??? ????? ??????? ??? ??? ?????? a, b .??? ?????? f(x) ???? ???? ????? ???????? (pdf ) ??? ????? ?????? ??????? fx≥0 for all x-∞∞f(x)dx= area under the entire graph offx=1 ??????? ??????? (uniform distribution) ??? ??? x ????? ?????? ????? ????? ????? ??? ???? ????? ???????? ???? fx=1b-a a ≤x≤b 0 otherwise ??? ???? ??????? x ~u(a,b) ????? ???? ?????? ??? ??? ??????? [a,b] ????? ????? ????? ?? ????? ?????? ?? ??? ????? ????? ??????? ??? a,b ??? ?????? ???? ????????? a<b ??? a ???? ????? ??????? (location parameter) ? ??????? ?????? ????? ??? Ex=a+ b/2varx=b-a2/12?? ??????? ????? ????? ???? ?? ????? ?????? ??????? ???? ???? ?? ????? ???? ????? (α=β=1) ???????? ??? ??? ??? ???????a=0,b=1 ???x ~u(0,1) ??????? ?????? [x ,x+?x ] ????? ???? ??? ???? ?? [0,1] ????? 0 > ?x ??? ?????? x ?? ????? ??? ??? ?????? ??????? ????? pr x ∈ x , x+ ?x= xx+?xdy= ?x??????? ??????? (normal distribution) ??????? ???????? ??????? x ???? ??? ????? ????? ????? ????????? μ ? σ ?? ( μ ? σ2 ) ??? ?? -∞≤μ≤ ∞ ? σ >0 ? ??? ?? ???? ???? ????? ???????? ??????? x ???? ??????? fx=1σ2πe-12( x-μσ)2 , for -∞ ≤x≤∞ ?? ??????? ???? ???? ??? ?? ??????? x ????? ????? ????? ????????? μ , σ2 ??? ????? x ~ N(μ,σ2) ? ?????? ???? ??????? ????? ????? ????????? ???????? ????? ???? (μ=0, σ2=1 ) ???? ??? ??????? ??????? ??????? ??????? (standard normal distribution) ???????? ???????? z ???? ???? ????? ??????? ??????? ??????? ????? ????? ?????? ?????????? ??????? fz=1σ2πe-z22 , for -∞ ≤z≤∞ ??? ???? ??????? ????????? ???? ??????? pZ≤z = -∞zf(y;0,1)dy ????? ??????? Φ(z) ? ????? ?????? ?? ????? ??????? ??????? ??????? . ?????? pz ≤1.25 ????? pz ≤1.25 =Φ(1.25) ??? ???? ??????? ??????? ??????? ????? ???? 1.2 ???? ???? 0.05 ????????? ???? ??? ?????? ?????????? 0.8944 ?????? ??????? ???? ????????????? p z >1.25 ????? pz >1.25 = 1- pz ≤1.25 =1-Φ(1.25) ???? ?? Φ1.25 =0.8944 ????????? ??? ?????? ?????????? ???? 0.1056??? ?????? p-0.38 ≤ z ≤1.25 ????? ???????? ???????p-0.38 ≤ z ≤1.25 =Φ1.25- Φ-0.38 =0.8944-0.3520=0.5424 ????? ?? ????????? ???????? zα ????? ???? ??????? ??? ?????? ?????? ????? ???? ??????? ??? ??????? ???? ??? ?????? ?? zα ?????? ??? α ??????? ?????? ??? ?? zα ???? ??????? ?????? z ( z critical values) ????? ??? ??? ??? ????? ?? ??????? ??????? ??????? ?????? ???????? .??????? 1001-0.05th=95 ?? ??????? ??????? ??????? ?? z0.05 ??? ?????? ????? z0.05=1.645 ? ???????? ??? ????? ??????? ??????? ??????? ???? ?????? ?? z0.05 ? ??????? ?? ?????? z0.05 - ???? 0.05 .???? ??? ??????? ???? ??????? ??????? ????????? μ ? σ ????? ?????? ?????? ???????? ??????? z=x-μσ ????? ???? ?? ???? ???????? ???????? ?? ??????? ??????? ??????? pa ≤x≤b=p a-μσ ≤z≤b-μσ = Φb-μσ-Φa-μσ ??? ???? ???????? ???? ??? x ????? ?????? ????? ????? ????? ?????? ? n ?? ????????? ???????? p ???? ??? ?????? ???????? ??? ????? ? ??? x ???? ???????? ??????? ?????? μ=np ??????? ?????? σ=npq ??? ?????? ?????? ???? ???????? ?????? ?????? pX≤x=Bx;n,p=Φx+0.5-np npq pX≤x=Bx;n,p??? ??????? ???? ??? ????? ????? ??????nq ≥10 ,np ≥10 .????(1-59 ) :- ???? ?? 25% ?? ?????? ?? ????? ?????? ???????? ????? ????? ??? ??????? ????????? ?????? ?? X ????? ?? ????? ?????? ???? ?????? ??? ???????? ??? ??????? ?? ???? ????? 50 ???? . ???? ?? p=0.25 ???????? μ=12.5 ???????? ?????? σ=3.062 ???? ?? np=50.25=12.5 ≥10 ? nq=37.5 ≥10 ??? ??????? ???? ?? ???? ????? ?????? ????? ??? ??? pX≤10=B10;50,0.25=Φ10+0.5-12.5 3.062= Φ- 0.65= 0.2578??? ?? ???????? ?????? ???? p 5 ≤X≤15=B15;50,0.25-B4;50,0.25 ?Φ15.5-12.5 3.062- Φ4.5-12.5 3.062= 0.8320 ????? ?? ????? ????? ?????????? ?? ????? ????? ?????? 0.8348 , 0.2622.?????? ????? ????? ?1:- ?? ?? ??? ?????? ??????? ??2:- ???? ??? ????? ??????????? ??????????3:- ?? ?? ??? ??????? ??????? ?????? ????? ????? ??4:- ???? ????? ???????? ????????? ??5:- ??? ?????? ??????? ??????? ? ????? ????? ??????? ????? ??????? ???????? ??6:- ???????? ??????? ???? ??????? ???? ??? ??? ?? ?????? ?????? ???? ???? ????? ??????? ????? ???? ????? ?????? .655288757498849659785947775997?7:- ???? ??????? ???????? ??????? ???????? ??????? ??????? ??????? ?52924477715293917?8:- ?????? ????? ?????? ????????? ???????? ??????? 78,74,82,66,94,74,82,75,96,78,84,77,71,83?9:- ???????? ??????? ?????? ?????? ???????? ??????? ?78,74,82,66,94,74,82,75,96,78,84,77,71,83,54,43,998,56,78?? ?????? ????? ?????? ?? ????? ??????? .?10:- ??? ??? ??? ????? (??????? ?????????) ??????? ?? ??????? ??????? ?? ???????????? ???? ??? ?? ??????? ???????? ??????18-20 16 >14 >12 >10 >8 >6 >class2918241352fi??????????? ???????? ???????? ??????? ?????? ??????? .?11:- ???? ????? ??????? ?????????????????? ???????? ??????? 52291824135122?12:- ???? ????? ???????? ?????? ??????? ???????? (11,22,34,46,50,56,67) .?13:- ???? ????? ???????? ?????? ?????? ?? ???? ???? ??????? ????? ???? ??? ??????? ???????? ?????? .2017141185classes222518352111fi?14:- ???? ????? ???????? ?? ??? ????????? ??????? ???? ?? ????????18 16 1412 10 8 6 x12251824131522Y?15:- ???????? ??????? ???? ???????????? (Y) ??????? ?????? (X) ????? ?? ???????? ??????? ???? ????? ?????? ????? ???????? ?????? ?????? ????? ?????? ?? ??????? ??? ??????? .???????????????????????? ?????????? ???X799570986266Y?16:- ?????? ???? (spss) ????? ????? ?????? ????? ???????? ?????? ?????? ????? ?????? ???????? ?? ?????? 15 ??17:- ??? ??? ?????? ????? ??? ?? ???? ??? ????? ?? ???? ????? ?? ??? ???????? ???? (0.7) ???? ??? ??? ????????? ?????? ?????? (12 ) ??????? ??? ??????? ???? ????? ????? ?????? .????? ?? ??? ?????? ????? ???? ????? ?? ??? ?????? .?????? ????? ???? ????? ??? ????? ?? ?????? .?????? ????? ???? ????? ??? ?????? ?? ?????? . ................
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