Percentiles & Quartiles:
Percentiles & Quartiles:
Due to some cases where our data distributions are heavily skewed or even bimodal, we are usually better off using the relative position of the data as opposed to exact values.
We have studied how the median is an average computed using relative position of the data. If we say that the median is 27, then we know that half (50%) of the data falls above 27 and half (50%) of the data falls below 27. The median is an example of a percentile (50th percentile).
Percentiles
For whole numbers P (where 1 < P < 99) the Pth percentile of a distribution is a value such that P% of the data fall at or below it.
*** Look at figure 3-10 (text p. 152) and Guided Exercise 11 (text p. 153).
There are 99 percentiles and in an ideal situation, the 99 percentiles divide the data set into 100 equal parts. However, if the number of data elements is not exactly divisible by 100, the percentiles will not divide the data into equal parts.
We will not be concerned about using the different procedures to evaluate percentiles. However, we will be more concerned with quartiles.
Quartiles – percentiles which divide the data into fourths.
Example: 1st quartile = 25th percentile
2nd quartile = median
3rd quartile = 75th percentile
View Figure 3-11 (text p. 153) and Figure 3-12 (text p. 154) to see how percentiles are broken up.
Procedure to Compute Quartiles:
1) Rank the data from smallest to largest.
2) Find the median (2nd quartile).
3) The first quartile (Q1) is then the median of the lower half of the data; that is, it is the median of the data falling below Q2 (and not including Q2).
4) The third quartile Q3 is the median of the upper half of the data; that is, it is the median of the data falling above Q2 (and not including Q2).
To help you find the median easily use the median rank for n pieces of data,
Median Rank = _n + 1_
2
If the rank is a whole number, the median is the value in that position. If the rank ends in .5, we take the mean of the data values in the adjacent positions to find the median.
Interquartile Range:
A useful measure of data spread utilizing relative position is the interquartile range (IQR). This is the difference between the 3rd and 1st quartiles.
IQR = Q3 – Q1
This range tells us the spread of the middle half of the data.
Examples:
For each of the following data sets, calculate the median rank, median, 1st quartile, 3rd quartile, and interquartile range:
1) 100 97 106 87 94 102 101 99 86 78 96 56 80 106 111 87 88 80 96 98 96 91
2) 78 89 56 67 45 67 89 78 55 44 78 55 34 90 66 54 78 97 67 89 76 78 89 88 90
3) 67 215 56 81 96 200 197 196 133 145 99 100 154 167 166 189 177 189 199 222 221 67 71 98 87 78
4) 333 456 399 345 390 411 400 405 415 388 327
378 345 377 389 378 322 267 400 409 467 422
5) 667 788 978 999 456 678 354 890 456 455 856 785 689 789 567 687 687 678 678 456 532 462
657 789 097 643 676 356 789 908 567 687 789
6) 673 678 567 445 558 782 892 982 378 378 899 378 289 890 890 567 367 395 278 378 378 279 729 280 208 289 730 672
7) 378 289 789 829 892 920 027 728 287 829 920 823 829 562 278 728 278 628 781 872 828 871 123 243 514 154 672 627 762 268 282
8) 362 873 879 040 983 284 908 567 348 624 362 847 632 467 843 927 498 327 984 728 947 983 729 478 473 982 794 872 389
9) 789 457 498 375 894 375 983 475 980 345 973 253 750 293 759 803 758 973 579 834 759 834 750 986 111 111 236
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