1. Chapter 2 Data Presentation Using Descriptive Graphs

2. 2 2.1 Frequency Distributions The tabulation of data by dividing it into classes and computing the number of data points (or their fraction out of the total) falling within each class. Example 2.1.1: Grades on Business Statistics Exam Classes (Exam Score)Frequency (Number of Students) Below 5028 50 and under 6030 60 and under 7036 70 and under 8020 80 and under 9090 90 and over16 220

3. 3 Constructing a Frequency Distribution Gather the sample data Gather the sample data Arrange the data in an ordered array Arrange the data in an ordered array Ascending Order: Lowest to highest Ascending Order: Lowest to highest Descending Order: Highest to lowest Descending Order: Highest to lowest Select the number of classes, K, to be used Select the number of classes, K, to be used There is no “correct” number of classes. There is no “correct” number of classes. Determine the class width, CW. Determine the class width, CW. Determine the class limits for each class Determine the class limits for each class Count the number of data values in each class (the class frequencies) Count the number of data values in each class (the class frequencies) Summarize the class frequencies in a frequency distribution table Summarize the class frequencies in a frequency distribution table Where:H = Highest Value L = Lowest Value Rounded up or down to a value that is easy to interpret.

4. 4 Constructing a Frequency Distribution (cont.) Example 2.1.2: Frequency Distribution for Continuous Data Fifty starting salaries for business majors at Bellaire College Raw Data 41.539.440.935.937.4 39.540.339.341.636.6 41.135.743.737.041.3 40.638.042.435.741.4 39.236.839.343.838.5 43.036.335.636.238.1 34.838.135.736.539.5 37.934.336.833.835.0 37.838.737.232.838.2 37.039.738.835.236.2

5. 5 Constructing a Frequency Distribution (cont.) Example 2.1.2: Arrange the data in an ordered array Arrange the data in an ordered array Ordered Array 32.833.834.334.835.0 35.235.635.7 35.936.2 36.336.5 36.636.8 37.0 37.237.437.837.938.0 38.1 38.238.538.7 38.839.239.3 39.4 39.5 39.740.340.6 40.941.141.341.441.5 41.642.443.043.743.8

6. 6 Constructing a Frequency Distribution (cont.) Example 2.1.2: Number of classes, K = 6 Class Width: Class NumberClassFrequency 132 and under 342 234 and under 369 336 and under 3813 438 and under 4014 540 and under 428 642 and under 444 50

7. 7 Constructing a Frequency Distribution (cont.) Example 2.1.3: Frequency Distribution for Discrete Data and Categorical Data Class NumberClassFrequencyClass NumberClassFrequency 1 4 - 6 91Accounting26 2 7 - 9 102IT10 3 10 - 12 83Marketing14 4 13 - 15 8 5 16 - 18 6 50 6 19 - 21 9 50

8. 8 Frequency Distributions Relative Frequency Distribution The ratio of each class frequency to the total number of data points in a frequency distribution. Cumulative Frequency Distribution The cumulative frequency corresponding to the upper limit of any class is the total frequency of all values less than that upper limit. Relative Cumulative Frequency Distribution The ratio of the cumulative frequency of each class to the total number of data points in a frequency distribution.

9. 9 Frequency Distributions (cont.) Example 2.1.4: Frequency Distributions ClassFrequency Relative Frequency Cumulative Frequency Relative Cumulative Frequency 32 and under 3420.042 34 and under 3690.18110.22 36 and under 38130.26240.48 38 and under 40140.28380.76 40 and under 4280.16460.92 42 and under 4440.08501.00 501.00

10. 10 Comments on Frequency Distribution Outliers Very small or very large numbers quite unlike the remaining data values. Open-ended Classes Example 2.1.1 (Revisited): Grades on Business Statistics Exam Classes (Exam Score)Frequency (Number of Students) Below 5028 50 and under 6030 60 and under 7036 70 and under 8020 80 and under 9090 90 and over16 220

11. 11 Comments on Frequency Distribution (cont.) Class Limits The highest and lowest values describing a class Lower Limit Upper Limit Class Midpoints (also called Class Marks) Values in the center of the classes. Example 2.1.5: Finding Class Midpoints ClassClass MidpointsFrequency 32 and under 34(32+34)/2 = 332 34 and under 36(34+36)/2 = 359 36 and under 38(36+38)/2 = 3713 38 and under 40(38+40)/2 = 3914 40 and under 42(40+42)/2 = 418 42 and under 44(42+44)/2 = 434

12. 12 Class Midpoints - Example Three midpoints of adjoining classes in a frequency distribution are 16.5, 19.5, and 22.5. How wide are the classes? Note: In a frequency distribution, all classes usually have the same class width unless we have open-ended classes to accommodate outliers. ClassClass MidpointsFrequency ::: A and under B16.5: B and under C19.5: C and under D22.5: ::: 16.519.522.5 A B C D The three adjoining classes and their midpoints can be shown below in a frequency distribution form. If we know A and B, or B and C, or C and D, we can get the class width. We can also put the midpoints in the following line graph. A-B is a class, B-C is a class, and C-D is a class. Since all classes have the same class width, B is equidistant from 16.5 and 19.5. Same goes for C. I am taking B and C, because they are closed by the midpoints. If B is equidistant from 16.5 and 19.5, what is the value of B? It’s 18. Same way C is equidistant from 19.5 and 22.5. Then the value of C is 21. So class width = 21-18. Remember, class width is simply the difference between the upper limit and the lower limit of a class.

13. 13 Using Excel KPK Data Analysis > Quantitative Data Charts/Tables > Histogram/Freq. Charts. Frequency Distribution Table CLASSCLASS LIMITSFREQUENCY RELATIVE FREQ CUMULATIVE FREQ CUM REL FREQ 132 and under 3420.042.000.04 234 and under 3690.1811.000.22 336 and under 38130.2624.000.48 438 and under 40140.2838.000.76 540 and under 4280.1646.000.92 642 and under 4440.0850.001.00 TOTAL50

14. 14 2.2 Histograms and Stem-and-Leaf Diagrams Histogram A Histogram is a graphical representation of a frequency distribution for continuous data. A Histogram is a graphical representation of a frequency distribution for continuous data. Drawn by putting class limits on X-axis and frequencies on Y-axis. Drawn by putting class limits on X-axis and frequencies on Y-axis. Describes the shape of the data. Describes the shape of the data. Relative Frequency Histogram: Constructed using relative frequencies rather than the frequencies. Relative Frequency Histogram: Constructed using relative frequencies rather than the frequencies.

15. 15 Stem-and-Leaf Diagrams Summarizing reasonably sized data (under 150 values as a general rule) without loss of information. Each observation is represented by a stem to the left of a vertical line and a leaf to the right of the vertical line. The leaf for each observation is generally the last digit (or possibly the last two digits) of the data value, with the stem consisting of the remaining first digits. Example 2.2.1: Constructing Stem-and-Leaf Diagrams Example 2.2.1: Constructing Stem-and-Leaf Diagrams Reports of the after-tax profits of 12 companies are (recorded as cents per dollar of revenue) as follows: Reports of the after-tax profits of 12 companies are (recorded as cents per dollar of revenue) as follows: 3.4, 4.5, 2.3, 2.7, 3.8, 5.9, 3.4, 4.7, 2.4, 4.1, 3.6, 5.1 StemLeaf (unit =.1) 2347 34468 4157 519 What percentage of the companies pays tax more than 4.5 cents per dollar of revenue? What is the range of these data in cents?

16. 16 2.3 Frequency Polygons Constructed by connecting the centers of the tops of the histogram bars (located at the class midpoints) with a series of straight lines. Relative Frequency Polygons use relative frequencies rather than frequencies.

17. 17 Frequency Polygons (cont.) Better than histograms for comparing the shape of two (or more) different frequency distributions. College degree No college degree |10|20|30|40|50|60|70|80|90 Annual salaries (thousands of dollars) Number of employees |100

18. 18 Frequency Polygons (cont.) For open-ended class, place a footnote at each open-ended class location indicating the frequency of that particular class. *4 cities had populations of less than 10,000 **5 cities had populations of 50,000 or greater 100 – 90 – 80 – 70 – 60 – 50 – 40 – 30 – 20 – 10 – |10|15|20|25|30|35|40|45|50 Population (thousands) ** * Frequency

19. 19 2.4 Cumulative Frequencies (Ogives) Constructed by putting upper class limits on X-axis and cumulative frequencies (or cumulative relative frequencies) on the Y-axis. Useful in determining what percentage of the data lies below a certain value.

20. 20 2.5 Bar Charts Bar Charts are used for graphical representation of nominal and ordinal data Bar Charts are used for graphical representation of nominal and ordinal data As with a histogram the height of the bar is proportional to the number of values in the category As with a histogram the height of the bar is proportional to the number of values in the category2610 14 Accounting Information systems Marketing Number of majors 30 – 25 – 20 – 15 – 10 – 5 –

21. 21 2.6 Pie Charts The Pie Chart is an alternative to the bar chart for nominal and ordinal data The Pie Chart is an alternative to the bar chart for nominal and ordinal data The proportion of the Pie represents the category’s percentage in the population or sample The proportion of the Pie represents the category’s percentage in the population or sample Dept.# of Students Accounting26 IT10 Marketing14 Management25 75

22. 22 Bar and Pie Charts

23. 23 2.7 Deceptive Graphs If care is not taken in constructing graphs, the graph may not properly present the data If care is not taken in constructing graphs, the graph may not properly present the data Also, graphs can be purposely manipulated to provide false impressions of the data Also, graphs can be purposely manipulated to provide false impressions of the data Women A Men B 30 – 25 – 20 – 15 – 10 – 5 – – Number of employees (thousands)

24. 24 Deceptive Graphs (cont.) 199920002001 Year Salary (thousands of dollars) 32 – 31 – 30 – 0 199920002001 Year Salary (thousands of dollars) 32 – 31 – 30 –