1. Histogram Differences from a bar chart: bars have equal width and always touch width of bars represents quantity heights of bars represent frequency f Measured quantity

2. To construct a histogram from raw data: Decide on the number of classes (5 to 15 is customary). Find a convenient class width. Organize the data into a frequency table. Find the class midpoints and the class boundaries. Sketch the histogram.

3. Finding class width 1.Compute: 2.Increase the value computed to the next highest whole number

4. Class Width Raw Data: 10.2 18.7 22.3 20.0 6.3 17.8 17.1 5.0 2.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.7 11.4 Use 5 classes. 24.7 – 0.0 5 = 4.94 Round class width up to 5.

5. Frequency Table Determine class width. Create the classes. May use smallest data value as lower limit of first class and add width to get lower limit of next class. Tally data into classes. Compute midpoints for each class. Determine class boundaries.

6. Tallying the Data # of miles tally frequency 0.0 - 4.9|||| |6 5.0 - 9.9||||5 10.0 - 14.9||||4 15.0 - 19.9||||5 20.0 - 24.9||||5

7. Grouped Frequency Table # of miles f 0.0 - 4.96 5.0 - 9.95 10.0 - 14.94 15.0 - 19.95 20.0 - 24.95 Class limits: lower - upper

8. Computing Class Width difference between the lower class limit of one class and the lower class limit of the next class

9. # of miles fclass widths 0.0 - 4.96 5 5.0 - 9.965 10.0 - 14.945 15.0 - 19.955 20.0 - 24.955 Finding Class Widths

10. Computing Class Midpoints lower class limit + upper class limit 2

11. # of miles fclass midpoints 0.0 - 4.96 2.45 5.0 - 9.95 10.0 - 14.94 15.0 - 19.95 20.0 - 24.95 Finding Class Midpoints

12. # of miles fclass midpoints 0.0 - 4.96 2.45 5.0 - 9.957.45 10.0 - 14.94 15.0 - 19.95 20.0 - 24.95 Finding Class Midpoints

13. # of miles fclass midpoints 0.0 - 4.96 2.45 5.0 - 9.957.45 10.0 - 14.9412.45 15.0 - 19.9517.45 20.0 - 24.9522.45 Finding Class Midpoints

14. Class Boundaries (Upper limit of one class + lower limit of next class) divided by two

15. Finding Class Boundaries # of miles fclass boundaries 0.0 - 4.96 5.0 - 9.954.95 - 9.95 10.0 - 14.94 15.0 - 19.95 20.0 - 24.95

16. Finding Class Boundaries # of miles fclass boundaries 0.0 - 4.96 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.95 20.0 - 24.95

17. # of miles fclass boundaries 0.0 - 4.96 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.9514.95 - 19.95 20.0 - 24.95 Finding Class Boundaries

18. # of miles fclass boundaries 0.0 - 4.96 ?? 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.9514.95 - 19.95 20.0 - 24.9519.95 - 24.95 Finding Class Boundaries

19. # of miles fclass boundaries 0.0 - 4.96 ?? - 4.95 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.9514.95 - 19.95 20.0 - 24.9519.95 - 24.95 Finding Class Boundaries

20. # of miles fclass boundaries 0.0 - 4.96  0.05 - 4.95 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.9514.95 - 19.95 20.0 - 24.9519.95 - 24.95 Finding Class Boundaries

21. # of miles f 0.0 - 4.96 5.0 - 9.95 10.0 - 14.94 15.0 - 19.95 20.0 - 24.95 Constructing the Histogram f | | | | | | 65432106543210 -------------- -0.05 4.95 9.95 14.95 19.95 24.95 mi.

22. Relative Frequency Relative frequency = f = class frequency n total of all frequencies

23. Relative Frequency f = 6 = 0.24 n 25 f = 5 = 0.20 n 25

24. # of miles f relative frequency 0.0 - 4.9 6 0.24 5.0 - 9.9 50.20 10.0 - 14.9 40.16 15.0 - 19.9 50.20 20.0 - 24.9 50.20 Relative Frequency Histogram | | | | | |.24.20.16.12.08.04 0 -------------- -0.05 4.95 9.95 14.95 19.95 24.95 mi. Relative frequency f/n

25. Common Shapes of Histograms Symmetrical f When folded vertically, both sides are (more or less) the same.

26. Common Shapes of Histograms Also Symmetrical f

27. Common Shapes of Histograms Uniform f

28. Common Shapes of Histograms Non-Symmetrical Histograms skewed. These histograms are skewed.

29. Common Shapes of Histograms Skewed Histograms Skewed leftSkewed right

30. Common Shapes of Histograms Bimodal f The two largest rectangles are approximately equal in height and are separated by at least one class.

31. Frequency Polygon A frequency polygon or line graph emphasizes the continuous rise or fall of the frequencies.

32. Constructing the Frequency Polygon Dots are placed over the midpoints of each class. Dots are joined by line segments. Zero frequency classes are included at each end.

33. Weights (in pounds) f 2 - 46 5 - 75 8 - 104 11 - 135 Constructing the Frequency Polygon f | | | | | | 65432106543210 -------------- 0 3 6 9 12 15 pounds

34. Cumulative Frequency The sum of the frequencies for that class and all previous or later classes

35. Weights (in pounds) f Greater than 1.5 20 Greater than 4.5 14 Greater than 7.5 9 Greater than 10.5 5 Greater than 13.5 0 Cumulative Frequency Table Weights (in pounds) f 2 - 46 5 - 75 8 - 104 11 - 135 20

36. Ogive Graph of a cumulative frequency table

37. Weights (in pounds) f Greater than 1.5 20 Greater than 4.5 14 Greater than 7.5 9 Greater than 10.5 5 Greater than 13.5 0 Constructing the Ogive Cumulative frequency | | | | | | 20 15 10 5 0 ---------- 1.5 4.5 7.5 10.5 13.5 pounds

38. Exploratory Data Analysis A field of statistical study useful in detecting patterns and extreme data values Tools used include histograms and stem- and-leaf displays