Quartiles in a Normal Distribution
There are other ways to divide data sets besides the percentile scheme. It is common to specify points or boundaries that divide data into quarters or into tenths.
A quartile or quartile point is a number that breaks a data set up into four intervals, each interval containing approximately 1/4 or 25% of the elements in the set. There are three quartiles, not four, because the quartiles represent boundaries where four intervals meet. Thus, quartiles are assigned values 1, 2, or 3. They are sometimes called the 1st quartile, the 2nd quartile, and the 3rd quartile.
Examine Fig. 4-1 again.
The location of the qth quartile point is found by locating the vertical line L such that the percentage n of the area beneath the curve is exactly equal to 25q, and then noting the point where L crosses the horizontal axis. In Fig. 4-1, imagine that line L can be moved freely back and forth. Let n be the percentage of the area under the curve that lies to the left of L. When n = 25%, line L crosses the horizontal axis at the 1st quartile point. When n = 50%, line L crosses the horizontal axis at the 2nd quartile point. When n = 75%, line L crosses the horizontal axis at the 3rd quartile point.
Quartiles in Tabular Data
Let's return to the 40-question test described above and in Table 4-1.
Where do we put the three quartile points in this data set? There are 41 different possible scores and 1000 actual data elements. We break these 1000 results up into four different groups with three different boundary points according to the following criteria:
- The highest possible boundary point representing the ''worst'' 250 or fewer papers, and the 1st quartile point at the top of that set.
- The highest possible boundary point representing the ''worst'' 500 or fewer papers, and the 2nd quartile point at the top of that set.
- The highest possible boundary point representing the ''worst'' 750 or fewer papers, and the 3rd quartile point at the top of that set.
The nomograph of Fig. 4-2A illustrates the positions of the quartile points for the test results shown by Table 4-1. The data in the table are unusual; they represent a coincidence because the quartiles are all clearly defined. There are obvious boundaries between the ''worst'' 250 papers and the ''2nd worst,'' between the ''2nd and 3rd worst,'' and between the ''3rd worst'' and the ''best.'' These boundaries occur at the transitions between scores of 16 and 17, 24 and 25, and 31 and 32 for the 1st, 2nd, and 3rd quartiles, respectively. If these same 1000 students are given another 40-question test, or if this 40-question test is administered to a different group of 1000 students, it's almost certain that the quartiles will not be so obvious.
Fig. 4-2A. At A, positions of quartiles in the test results described in the text.
Deciles in a Normal Distribution
A decile or decile point is a number that divides a data set into 10 intervals, each interval containing about 1/10 or 10% of the elements in the set. There are nine deciles, representing the points where the 10 sets meet. Deciles are assigned whole-number values between, and including, 1 and 9. They are sometimes called the 1st decile, the 2nd decile, the 3rd decile, and so on up to the 9th decile.
Refer again to Fig. 4-1.
The location of the dth decile point is found by locating the vertical line L such that the percentage n of the area beneath the curve is exactly equal to 10d, and then noting the point where L crosses the horizontal axis. Imagine again that L can be slid to the left or right at will. Let n be the percentage of the area under the curve that lies to the left of L. When n = 10%, L crosses the horizontal axis at the 1st decile point. When n = 20%, L crosses the horizontal axis at the 2nd decile point. When n = 30%, L crosses the horizontal axis at the 3rd decile point. This continues on up, until when n = 90%, line L crosses the horizontal axis at the 9th decile point.
Deciles in Tabular Data
One more time, let's scrutinize the 40-question test whose results are portrayed in Table 4-1.
Where do we put the decile points? We break the 1000 test papers into 10 different groups with nine different boundary points according to these criteria:
- The highest possible boundary point representing the ''worst'' 100 or fewer papers, and the 1st decile point at the top of that set.
- The highest possible boundary point representing the ''worst'' 200 or fewer papers, and the 2nd decile point at the top of that set.
- The highest possible boundary point representing the ''worst'' 300 or fewer papers, and the 3rd decile point at the top of that set.
- he highest possible boundary point representing the ''worst'' 900 or fewer papers, and the 9th decile point at the top of that set.
The nomograph of Fig. 4-2B illustrates the positions of the decile points for the test results shown by Table 4-1. As is the case with quartiles, the data in the table are coincidental, because the deciles are obvious. There are clear boundaries between the ''worst'' 100 papers and the ''2nd worst,'' between the ''2nd and 3rd worst,'' between the ''3rd and 4th worst,'' and so on up. If these same 1000 students are given another 40-question test, or if this 40- question test is administered to a different group of 1000 students, it's almost certain that the locations of the decile points will be less obvious. (By now you should be able to tell that this table has been contrived to make things come out neat.)
Fig. 4-2B. At B, positions of the deciles.
Quartiles and Deciles Practice Problems
Practice 1
Table 4-2 shows a portion of results for the same 40-question test, but with slightly different results from those shown in Table 4-1, so that the 1st quartile point is not ''cleanly'' defined. Where is the 1st quartile point here?
Solution 1
Interpret the definition literally. The 1st quartile is the highest possible boundary point at the top of the set of the ''worst'' 250 or fewer papers. In Table 4-2, that corresponds to the transition between scores of 16 and 17.
Table 4-2 Table for Practice 1.
Practice 2
Table 4-3 shows a portion of results for the same 40-question test, but with slightly different results from those portrayed in Table 4-1. Here, the 6th decile point is not ''cleanly'' defined. Where is that point in this case?
Table 4-3 Table for Practice 2.
Solution 2
Once again, interpret the definition literally. The 6th decile is the highest possible boundary point at the top of the set of the ''worst'' 600 or fewer papers. In Table 4-3, that corresponds to the transition between scores of 26 and 27.
Practice problems for these concepts can be found at:
Descriptive Measures Practice Test
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From Statistics Demystified: A Self-Teaching Guide. Copyright © 2004 by The McGraw-Hill Companies. All Rights Reserved.