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# Histogram for AP Statistics

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Practice problems for these concepts can be found at:

A bar graph is used to illustrate qualitative data, and a histogram is used to illustrate quantitative data. The horizontal axis in a bar graph contains the categories, and the vertical axis contains the frequencies, or relative frequencies, of each category. The horizontal axis in a histogram contains numerical values, and the vertical axis contains the frequencies, or relative frequencies, of the values (often intervals of values).

example: Twenty people were asked to state their preferences for candidates in an upcoming election. The candidates were Arnold, Betty, Chuck, Dee, and Edward. Five preferred Arnold, three preferred Betty, six preferred Chuck, two preferred Dee, and four preferred Edward. A bar graph of their preferences is shown below:

A histogram is composed of bars of equal width, usually with common edges. When you choose the intervals, be sure that each of the datapoints fits into a category (it must be clear in which class any given value is). A histogram is much like a stemplot that has been rotated 90 degrees.

Consider again the quiz scores we looked at when we discussed dotplots:

Because the data are integral and range from 15 to 50, reasonable intervals might be of size 10 or of size 5. The graphs below show what happens with the two choices:

Typically, the interval with midpoint 15 would have class boundaries 12.5 ≤ x < 17.5; the interval with midpoint 20 would have class boundaries 17.5 ≤ x < 22.5, etc.

There are no hard and fast rules for how wide to make the bars (called "class intervals"). You should use computer or calculator software to help you find a picture to which your eye reacts. In this case, intervals of size 5 give us a better sense of the data. Note that some computer programs will label the boundaries of each interval rather than the midpoint.

The following is the histogram of the same data (with bar width 5) produced by the TI-83/84 calculator:

example: For the histogram below, identify the boundaries for the class intervals.

solution: The midpoints of the intervals begin at 15 and go by increments of 5. So the boundaries of each interval are 2.5 above and below the midpoints. Hence, the boundaries for the class intervals are 12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5, and 52.5.

example: For the histogram given in the previous example, what proportion of the scores are less than 27.5?

solution: From the graph we see that there is 1 value in the first bar, 2 in the second, 6 in the third, etc., for a total of 31 altogether. Of these, 1 + 2 + 6 = 9 are less than 27.5. 9/31 = 0.29.

example: The following are the heights of 100 college-age women, followed by a histogram and stemplot of the data. Describe the graph of the data using either the histogram, the stemplot, or both.

solution: Both the stemplot and the histogram show symmetric, bell-shaped distributions. The graph is symmetric and centered about 66 inches. In the histogram, the boundaries of the bars are 59 ≤ x < 61, 61 ≤ x < 63, 63 ≤ x < 65,…, 71 ≤ x < 73. Note that, for each interval, the lower bound is contained in the interval, while the upper bound is part of the next larger interval. Also note that the stemplot and the histogram convey the same visual image for the shape of the data.

Practice problems for these concepts can be found at:

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