Histograms, Frequency Distributions and Related Topics

1. Histograms, Frequency Distributions and Related Topics These are constructions that will allow us to represent large sets of data in ways that may be more meaningful to the reader.

2. Histograms provide graphical representation of data with bars whose heights indicate the number of data in a certain range. A frequency table shows the distribution of data in classes (intervals). The classes are constructed so that each data values falls into exactly one class, and the class frequency is the number of data in the class.

3. How long does the 1161 mile Iditarod take? (p. 47, problem 7). Can you easily see what the maximum and minimum times are? Is it easy to tell how the times are distributed?

4. To find the class width, First compute: Largest value - smallest Value Desired number of classes Increase the value computed to the next highest whole, number even if the first value was a whole number. This will ensure the classes cover the data. The lower class limit of a class is the lowest data that can fit into the class, the upper class limit is the highest data value that can fit into the class. The class width is the difference between lower class limits of adjacent classes.

5. In a frequency table, divide the data range into classes equal width, compute: Largest value - smallest Value Desired number of classes Increase the value computed to the next highest whole, number even if the first value was a whole number. This will ensure the classes cover the data. The lower class limit of a class is the lowest data that can fit into the class, the upper class limit is the highest data value that can fit into the class. The class width is the difference between lower class limits of adjacent classes.

6. Class Boundaries • Class boundaries cannot belong to any class. • Class boundaries between adjacent classes are the midpoint between the upper limit of the first class, and the lower limit of the higher class. • Differences between upper and lower boundaries of a given class is the class width. • The midpoint of a class (class mark) is the average of its upper and lower boundaries, which is also the average of its upper and lower limits.

7. It is easier to make the histogram if the data is sorted:

8. The class width is computed as (360-236)/5 which is 24.8. Hence the class width is 25.

9. Histograms A histogram is a bar graph that can be constructed using a frequency table: • Put the class boundaries on the horizontal axis • The bars have the same width and always touch and the edges of the bars are on class boundaries. • The height of the bar is the class frequency.

10. Histogram for Iditarod Data

11. Relative Frequencies The relative frequency of a class is f/n where f is the frequency of the class, and n is the total of all frequencies. Relative frequency tables are like frequency tables except the relative frequency is given. Relative frequency histograms are like frequency histograms except the height of the bars represent relative frequencies.

12. Systolic blood pressures of 50 subjectsMake a histogram with 8 classes

13. Systolic blood pressures of 50 subjectsClass Width = (208-100)/8 = 13.5, thus use 14

14. Frequency Histogram for Blood Pressure Data

15. Relative Frequency Histogram for Blood Pressure Data

16. Cumulative Frequencies & Ogives • The cumulative frequency of a class is the frequency of the class plus the frequencies for all previous classes. • An ogive is a line graph that displays cumulative frequencies.

17. Constructing Ogives • Make a frequency table showing class boundaries and cumulative frequencies. • For each class, put a dot over the upper class boundary at the height of the cumulative class frequency. • Place dot on horizontal axis at the lower class boundary of the first class. • Connect the dots.

18. Ogive for Blood Pressure Data

19. (a) What number, and percentage, of winning times are under 2:07.15? (b) Estimate number, and percentage, of winning times between 2:05.15 and 2:11.15.

20. Distribution Shapes • Symmetrical • Uniform (it has a rectangular histogram) • Skewed left – the longer tail is on the left side. • Skewed right – the longer tail is on the right side. • Bimodal (the two classes with the largest frequencies are separated by at least one class)