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
56
97
80
106
106
87
111
94
87
102
88
101
80
99
96
86
98
78
96
96
91
2)
78
55
89
89
34
88
56
90
90
67
66
45
54
67
78
89
97
78
67
55
89
44
76
78
78
3)
67
100
71
215
154
98
56
167
87
81
166
78
96
189
200
177
197
189
196
199
133
222
145
221
99
67
4)
333
378
456
345
399
377
345
389
390
378
411
322
400
267
405
400
415
409
388
467
327
422
5)
667
785
657
788
689
789
978
789
097
999
567
643
456
687
676
678
687
356
354
678
789
890
678
908
456
456
567
455
532
687
856
462
789
6)
673
378
729
678
289
280
567
890
208
445
890
289
558
567
730
782
367
672
892
395
982
278
378
378
378
378
899
279
7)
378
823
123
289
829
243
789
562
514
829
278
154
892
728
672
920
278
627
027
628
762
728
781
268
287
872
282
829
828
920
871
8)
362
847
729
873
632
478
879
467
473
040
843
982
983
927
794
284
498
872
908
327
389
567
984
348
728
624
947
362
983
9)
789
253
750
457
750
986
498
293
111
375
759
111
894
803
236
375
758
983
973
475
579
980
834
345
759
973
834