Fixed Intervals Help (page 3)
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.)
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 3.6x.
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.''
Histograms were introduced back in Chapter 1. The example shown in that chapter is a bit of an oversimplification, because it's a fixed-width histogram. There exists a more flexible type of histogram, called the variable-width histogram. This sort of graph is ideal for portraying the results of our hypothetical 40-question test given to 1000 students in various ways.
Figure 4-4 shows variable-width histograms that express the same data as that in the tables and pie graphs. In Fig. 4-4, 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 width of each vertical bar is directly proportional to the range of scores. The height of each bar is directly proportional to the percentage of students who received scores in the indicated range.
Percentages are included in the histogram of Fig. 4-4A, because there's room enough to show the numbers without making the graph look confusing or cluttered. In Figs. 4-4B and C, the percentages are not written at the top of each bar. This is a matter of preference. Showing the numbers in graph B would make it look too cluttered to some people. In graph C, showing the percentage for the grade of A would be difficult and could cause confusion, so they're all left out. It's a good idea to include tabular data with histograms when the percentages aren't listed at the tops of the bars.
Fixed Intervals Practice Problems
Imagine a large corporation that operates on a five-day work week (Monday through Friday). Suppose the number of workers who call in sick each day of the week is averaged over a long period, and the number of sick-person-days per week is averaged over the same period. (A sick-person-day is the equivalent of one person staying home sick for one day. If the same person calls in sick for three days in a given week, that's three sick-person-days in that week, but it's only one sick person.) For each of the five days of the work week, the average number of people who call in sick on that day is divided by the average number of sick-person-days per week, and is tabulated as a percentage for that work-week day. The results are portrayed as a pie graph in Fig. 4-5. Name two things that this graph tells us about Fridays. Name one thing that this graph might at first seem to, but actually does not, tells us about Fridays.
The pie graph indicates that more people (on the average) call in sick on Fridays than on any other day of the work week. It also tells us that, of the total number of sick-person-days on a weekly basis, an average of 33% of them occur on Fridays. The pie graph might at first seem to, but in fact does not, indicate that an average of 33% of the workers in the corporation call in sick on Fridays.
Suppose that, in the above described corporation and over the survey period portrayed by the pie graph of Fig. 4-5, there are 1000 sick-person-days per week on average. What is the average number of sick-person-days on Mondays? What is the average number of people who call in sick on Mondays?
For a single day, a sick-person-day is the equivalent of one person calling in sick. But this is not necessarily true for any period longer than one day. In this single-day example, we can multiply 1000 by 17.8%, getting 178, and this gives us both answers. There are, on the average, 178 sick-person-days on Mondays. An average of 178 individuals call in sick on Mondays.
Given the same scenario as that described in the previous two problems, what is the average number of sick-person-days on Mondays and Tuesdays combined? What is the average number of individuals who call in sick on Mondays and Tuesdays combined?
An average of 178 sick-person-days occur on Mondays, as we have determined in the solution to the previous problem. To find the average number of sick-person-days on Tuesdays, multiply 1000 by 14.4%, getting 144. The average number of sick-person-days on Mondays and Tuesdays combined is therefore 178 + 144, or 322. It is impossible to determine the average number of individuals who call in sick on Mondays and Tuesdays combined, because we don't know how many of the Monday–Tuesday sick-person-day pairs represent a single individual staying out sick on both days (two sickperson-days but only one sick person).
Practice problems for these concepts can be found at:
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