1. Histograms Lecture 14 Sec. 4.4.4 Tue, Sep 26, 2006

2. Histograms Histogram – A display of quantitative data in which the data are divided into classes, with rectangles representing the frequencies of the classes. Histogram – A display of quantitative data in which the data are divided into classes, with rectangles representing the frequencies of the classes.

3. Histograms vs. Bar Graphs Histograms appear to be very similar to bar graphs. Histograms appear to be very similar to bar graphs. However, bar graphs are for qualitative data, while histograms are for quantitative data. However, bar graphs are for qualitative data, while histograms are for quantitative data. We indicate this difference by leaving a gap between the bars of a bar graph and no gap between the rectangles of a histogram. We indicate this difference by leaving a gap between the bars of a bar graph and no gap between the rectangles of a histogram.

4. Example Draw a histogram of the following data. Draw a histogram of the following data. 2.9462.3353.4181.890 2.7313.8551.3442.126 2.8812.5422.5043.367 1.9502.3922.4433.053

5. Drawing Histograms Find the maximum value, the minimum value, and the range. Find the maximum value, the minimum value, and the range. Minimum = 1.344 Minimum = 1.344 Maximum = 3.855 Maximum = 3.855 Range = Max – Min = 3.855 – 1.344 = 2.511. Range = Max – Min = 3.855 – 1.344 = 2.511.

6. Drawing Histograms Divide the data into classes of equal width. Divide the data into classes of equal width. The classes must not overlap. The classes must not overlap. Choose a convenient starting point. Choose a convenient starting point. Choose a convenient class width. Choose a convenient class width. It works best if the starting point is a multiple of the class width, but that is not necessary. It works best if the starting point is a multiple of the class width, but that is not necessary. Write the endpoints of each class. Write the endpoints of each class.

7. Drawing Histograms Let’s use 6 classes Let’s use 6 classes Then the width must be at least 2.511/6 = 0.4185. Then the width must be at least 2.511/6 = 0.4185. Let’s use 0.5 (other choices are possible). Let’s use 0.5 (other choices are possible). Starting point = 1.0 (other choices are possible). Starting point = 1.0 (other choices are possible).

8. Drawing Histograms Classes: Classes: 1.0 up to 1.5 (but not including 1.5) 1.0 up to 1.5 (but not including 1.5) 1.5 up to 2.0 1.5 up to 2.0 2.0 up to 2.5 2.0 up to 2.5 2.5 up to 3.0 2.5 up to 3.0 3.0 up to 3.5 3.0 up to 3.5 3.5 up to 4.0 3.5 up to 4.0

9. Drawing Histograms We may write the classes in either of two ways. We may write the classes in either of two ways. Interval notation: [low, high) Interval notation: [low, high) [1.0, 1.5), [1.0, 1.5), [1.5, 2.0), [1.5, 2.0), [2.0, 2.5), etc. [2.0, 2.5), etc. [ and ] mean “include endpoints.” [ and ] mean “include endpoints.” ( and ) mean “exclude endpoints.” ( and ) mean “exclude endpoints.”

10. Drawing Histograms Range notation: low – high Range notation: low – high 1.000 – 1.499, 1.000 – 1.499, 1.500 – 1.999, 1.500 – 1.999, 2.000 – 2.499, etc. 2.000 – 2.499, etc. With this notation, the endpoints are assumed to be included. With this notation, the endpoints are assumed to be included. Therefore, be sure the endpoints do not overlap. Therefore, be sure the endpoints do not overlap. Yet be sure that no possible values are left out. Yet be sure that no possible values are left out.

11. Drawing Histograms Count the number of observations in each class. This is the frequency of the class. Count the number of observations in each class. This is the frequency of the class. ClassFrequency 1.000 – 1.499 1 1.499 – 1.999 2 2.000 – 2.499 4 2.500 – 2.999 5 3.000 – 3.499 3 3.500 – 3.999 1

12. Drawing Histograms Draw horizontal and vertical axes. Draw horizontal and vertical axes. On the horizontal axis, show the class limits. On the horizontal axis, show the class limits. On the vertical axis, show uniform reference points representing frequencies or precentages that are appropriate for the data, starting at 0. On the vertical axis, show uniform reference points representing frequencies or precentages that are appropriate for the data, starting at 0. For example, 0, 2, 4, … For example, 0, 2, 4, …

13. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

14. Drawing Histograms Over each class, draw a rectangle whose height is the frequency, or relative frequency, of that class. Over each class, draw a rectangle whose height is the frequency, or relative frequency, of that class.

15. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

16. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

17. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

18. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

19. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

20. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

21. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

22. Drawing Histograms 1.01.52.02.53.03.5 GPA 2 4 6 8 0 Frequency 4.0

23. Drawing Histograms Never use too few or too many classes. Never use too few or too many classes. Usually 5 to 12 classes is about right. Usually 5 to 12 classes is about right. Use simple round numbers for the class boundaries. Use simple round numbers for the class boundaries. Mark off the vertical axis uniformly, showing regular reference points, not the actual frequencies. Mark off the vertical axis uniformly, showing regular reference points, not the actual frequencies. The vertical scale must start at 0. The vertical scale must start at 0.

24. TI-83 – Histograms Enter the data into list L 1. Enter the data into list L 1. {2.946, 2.731, 2.881, …, 3.053}  L 1 {2.946, 2.731, 2.881, …, 3.053}  L 1 Press STAT PLOT Press STAT PLOT Select Plot1. Select Plot1. Press Enter. Press Enter. Turn Plot1 On. Turn Plot1 On. Select Histogram Type. Select Histogram Type. Specify List L 1. Specify List L 1.

25. TI-83 – Histograms Press WINDOW Press WINDOW Set Xmin to the starting point. Set Xmin to the starting point. Set Xmax to the last endpoint. Set Xmax to the last endpoint. Set Xscl to the class width. Set Xscl to the class width. Set Ymin to 0 (or -1 for a margin). Set Ymin to 0 (or -1 for a margin). Set Ymax to the maximum frequency. Set Ymax to the maximum frequency. Press GRAPH Press GRAPH The histogram appears. The histogram appears.

26. TI-83 – Frequency Distributions After getting the histogram, press TRACE. After getting the histogram, press TRACE. The display shows the first class and its frequency. The display shows the first class and its frequency. Use the left arrow to see the other class frequencies. Use the left arrow to see the other class frequencies.