**Introduction to Fixed Intervals**

In descriptive measures statistics, you can divided up data sets into subsets containing equal (or as nearly equal as possible) number of elements, and then observed the ranges of values in each subset. There's another approach: we can define fixed ranges of independent-variable values, and then observe the number of elements in each range.

**The Test Revisited**

Let's re-examine the test whose results are portrayed in Table 4-1. This time, let's think about the ranges of scores. There are many ways we can do this, three of which are shown in Tables 4-8, 4-9, and 4-10.

In Table 4-8, the results of the test are laid out according to the number of papers having scores in the following four ranges: 0–10, 11–20, 21–30, and 31–40. We can see that the largest number of students have scores in the range 21–30, followed by the ranges 31–40, 11–20, and 0–10.

In Table 4-9, the results are shown according to the number of papers having scores in 10 ranges. In this case, the most ''popular'' range is 29–32. The next most ''popular'' range is 21–24. The least ''popular'' range is 0–4.

Both Tables 4-8 and 4-9 divide the test scores into equal-sized ranges (except the lowest range, which includes one extra score, the score of 0). Table 4-10 is different. Instead of breaking the scores down into ranges of equal size, the scores are tabulated according to letter grades A, B, C, D, and F. The assignment of letter grades is often subjective, and depends on the performance of the class in general, the difficulty of the test, and the disposition of the teacher. (The imaginary teacher grading this test must be a hardnosed person.)

**Pie Graph**

The data in Tables 4-8, 4-9, and 4-10 can be portrayed readily in graphical form using broken-up circles. This is a *pie graph*, also sometimes called a *pie chart*. The circle is divided into wedge-shaped sections in the same way a pie is sliced. As the size of the data subset increases, the angular width of the pie section increases in direct proportion.

In Fig. 4-3, graph A portrays the data results from Table 4-8, graph B portrays the results from Table 4-9, and graph C portrays the results from Table 4-10. The angle at the tip or apex of each pie wedge, in degrees, is directly proportional to the percentage of data elements in the subset. Thus if a wedge portrays 10% of the students, its apex angle is 10% of 3608, or 368; if a wedge portrays 25% of the students, its apex angle is 25% of 3608, or 908. In general, if a wedge portrays *x*%of the elements in the population, the apex angle *θ* of its wedge in a pie graph, in degrees, is 3.6*x*.

The sizes of the wedges of each pie can also be expressed in terms of the area percentage. The wedges all have the same radius – equal to the radius of the circle – so their areas are proportional to the percentages of the data elements in the subsets they portray. Thus, for example, in Fig. 4-3A, the range of scores 31–40 represents a slice containing ''30% or 3/10 of the pie,'' while in Fig. 4-3C, we can see that the students who have grades of C represent ''25% or 1/4 of the pie.''

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