Frequently Used Statistics Formulas and Tables

Frequently Used Statistics Formulas and Tables

Chapter 2

Class Width =

highest value - lowest value

(increase to next integer)

number classes

Class Midpoint =

upper limit + lower limit

2

Chapter 3

Chapter 3

n = sample size

N = population size

f = frequency

¦² =sum

w = weight

Limits for Unusual Data

Below : ? - 2¦Ò

Above: ? + 2¦Ò

Empirical Rule

About 68%: ? -¦Ò to ? + ¦Ò

About 95%: ? -2¦Ò to ? + 2¦Ò

About 99.7%: ? -3¦Ò to ? + 3¦Ò

¡Æx

n

¡Æx

Population mean: ? =

N

¡Æ( w ? x)

Weighted mean: x =

¡Æw

Sample mean: x =

Sample coefficient of variation: CV =

s

? 100%

x

Population coefficient of variation: CV =

¡Æ( f ? x)

¡Æf

highest value + lowest value

Midrange =

2

Mean for frequency table: x =

Sample standard deviation for frequency table:

s=

n [ ¡Æ( f ? x 2 ) ] ? [ ¡Æ( f ? x) ]

n (n ? 1)

Range = Highest value - Lowest value

Sample standard deviation: s =

¡Æ( x ? x )

n ?1

Population standard deviation: ¦Ò =

Sample z-score: z =

2

¡Æ( x ? ? )

N

2

x?x

s

Population z-score: z =

x??

¦Ò

Interquartile Range: (IQR)

= Q3 ? Q1

Sample variance: s 2

Population variance: ¦Ò 2

¦Ò

? 100%

?

Modified Box Plot Outliers

lower limit: Q1 - 1.5 (IQR)

upper limit: Q3 + 1.5 (IQR)

2

Chapter 4

Chapter 5

Probability of the complement of event A

P (not A) = 1 - P ( A)

Discrete Probability Distributions:

Mean of a discrete probability distribution:

? =¡Æ[ x ? P( x)]

Multiplication rule for independent events

P ( A and

=

B)

P ( A) ? P ( B )

Standard deviation of a probability distribution:

General multiplication rules

P ( A and

=

B)

P ( A) ? P ( B, given A)

P ( A and

=

B)

P ( A) ? P ( A, given B )

¦Ò = ¡Æ[ x 2 ? P( x)] ? ? 2

Addition rule for mutually exclusive events

P ( A or B ) = P ( A) + P ( B )

Binomial Distributions

General addition rule

=

P ( A or B )

P ( A) + P ( B ) ? P ( A and B )

q = probability of failure

q=

1? p

p + q=1

r = number of successes (or x)

p = probability of success

Binomial probability distribution

P ( r ) = n Cr p r q n ? r

n!

Permutation rule: n Pr =

(n ? r )!

Combination rule:

n Cr

=

Mean: ? = np

n!

r !(n ? r )!

Standard deviation: ¦Ò = npq

Poisson Distributions

Permutation and Combination on TI 83/84

n

Math

PRB

nPr

enter

r = number of successes (or x)

? = mean number of successes

r

(over a given interval)

Poisson probability distribution

n

Math

PRB

nCr

enter

r

e? ? ? r

P(r ) =

r!

e ¡Ö 2.71828

Note: textbooks and formula

sheets interchange ¡°r¡± and ¡°x¡±

for number of successes

? = mean (over some interval)

¦Ò= ?

¦Ò2 = ?

2

Chapter 6

Chapter 7

Confidence Interval: Point estimate ¡À error

Normal Distributions

Raw score: =

x z¦Ò + ?

Standard score: z =

Point estimate = Upper limit + Lower limit

2

x??

¦Ò

Error = Upper limit - Lower limit

2

Mean of x distribution: ? x = ?

Sample Size for Estimating

Standard deviation of x distribtuion: ¦Ò x =

(standard error)

Standard score for x : z =

¦Ò

n

means:

?z ¦Ò ?

n = ? ¦Á /2 ?

? E ?

x ??

2

proportions:

¦Ò/ n

2

?z

?

? ? ? ¦Á / 2 ? with preliminary estimate for p

n = pq

? E ?

Chapter 7

2

?z

?

n = 0.25 ? ¦Á / 2 ? without preliminary estimate for p

? E ?

One Sample Confidence Interval

(np > 5 and nq > 5)

for proportions (p) :

variance or standard deviation:

*see table 7-2 (last page of formula sheet)

p? ? E < p < p? + E

where E = z¦Á / 2

p? =

p(1 ? p)

n

Confidence Intervals

Level of Confidence

r

n

for means (? ) when ¦Ò is known:

x?E ................
................

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