**Introduction to Intervals by Element Quantitiy**

Percentiles, quartiles, and deciles can be confusing when statements are made such as, ''You are in the 99th percentile of this graduating class. That's the highest possible rank.'' Doubtless more than one student in this elite class has asked, upon being told this, ''Don't you mean to say that I'm in the 100th percentile?'' After all, the term ''percentile'' implies there should be 100 groups, not 99.

It's all right to think in terms of the intervals between percentile boundaries, quartile boundaries, or decile boundaries, rather than the intervals centered at the boundaries. In fact, from a purely mathematical standpoint, this approach makes more sense. The 99 percentile points in a ranked data set divide that set into 100 intervals, each of which has an equal number (or as nearly an equal number as possible) of elements. Similarly, the three quartile points divide a ranked set into four intervals, as nearly equal-sized as possible; the nine decile points divide a ranked data set into 10 intervals, as nearly equal-sized as possible.

**25% Intervals**

Go back to Table 4-1 one more time, and imagine that we want to express the scores in terms of the bottom 25%, the 2nd lowest 25%, the 2nd highest 25%, and the top 25%. Table 4-4 shows the test results with the 25% intervals portrayed. They can also be called the bottom quarter, the 2nd lowest quarter, the 2nd highest quarter, and the top quarter.

Again, this particular set of scores is special because the intervals are ''cleanly'' defined. If things weren't so neat, we would be obliged to figure out the quartile points, and then define the 25% intervals as the sets of scores between those boundaries.

**10% Intervals**

Once again, let's revisit the test whose results are portrayed in Table 4-1.

Suppose that, instead of thinking about percentiles, we want to express the scores in terms of the bottom 10%, the 2nd lowest 10%, the 3rd lowest 10%, and so forth up to the top 10%. Table 4-5 shows the test results with these intervals portrayed. The spans can also be called the 1st 10th, the 2nd 10th, the 3rd 10th, and so on up to the top 10th (or, if you want to be perverse, the 10th 10th).

This particular set of scores is special because the interval cutoff points are ''clean.'' If this set was not contrived to make the discussion as easy as possible, we'd have to find the decile points, and then define the 10% intervals as the sets of scores between those boundaries.

**Intervals by Element Quantity Practice Problems**

**Practice 1**

Table 4-6 shows a portion of results for a 40-question test given to 1000 students, but with slightly different results from those portrayed in Table 4-1. What range of scores represents the 2nd highest 10% in this instance?

**Table 4-6 **Table for Practice 1.

**Solution 1**

The 2nd highest 10% can also be thought of as the 9th lowest 10%. It is the range of scores bounded at the bottom by the 8th decile, and at the top by the 9th decile. The 8th decile is the *highest possible* boundary point at the top of the set of the ''worst'' 800 or *fewer* papers. In Table 4-6, that corresponds to the transition between scores of 31 and 32. The 9th decile is the *highest possible* boundary point at the top of the set of the ''worst'' 900 or *fewer* papers. In Table 4-6, that corresponds to the transition between scores of 34 and 35. The 9th lowest (or 2nd highest) 10% of scores is therefore the range of scores from 32 to 34, inclusive.

**Practice 2**

Table 4-7 shows a portion of results for a 40-question test given to 1000 students, but with slightly different results from those portrayed in Table 4-1. What range of scores represents the lowest 25% in this instance?

**Table 4-7 **Table for Practice 2.

**Solution 2**

The lowest 25% is the range of scores bounded at the bottom by the lowest possible score, and at the top by the 1st quartile. The lowest possible score is 0. The 1st quartile is the *highest possible* boundary point at the top of the set of the ''worst'' 250 or *fewer* papers. In Table 4-7, that corresponds to the transition between scores of 16 and 17. The lowest 25% of scores is therefore the range of scores from 0 to 16, inclusive.

Practice problems for these concepts can be found at:

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