Summarizing Quantitative Data

1. Summarizing Quantitative Data • Frequency Distribution • Relative Frequency and Percent Frequency Distributions • Histogram • Cumulative Distributions • Ogive

2. Constructing a Frequency Distribution for Quantitative Data • 3 initial steps • Determine the number of nonoverlapping classes. • Determine the width of each class. • Determine the class limits

3. Example: Hudson Auto Repair The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.

4. Example: Hudson Auto Repair • Sample of Parts Cost for 50 Tune-ups

5. Frequency Distribution • Guidelines for Selecting Number of Classes • Use between 5 and 20 classes. • Data sets with a larger number of elements • usually require a larger number of classes. • Smaller data sets usually require fewer classes

6. Frequency Distribution • Guidelines for Selecting Width of Classes • Use classes of equal width. • Approximate Class Width =

7. Frequency Distribution • We decide that SIX (6) classes are appropriate for our purposes • Thus the approximate class width is given by: Approximate Class Width = (109 - 52)/6 = 9.5   10

8. Thus we have:

9. Using Excel’s Frequency Function to Construct a Frequency Distribution Formula Worksheet (showing data entered) Note: Rows 9-51 are not shown.

10. Using Excel’s Frequency Function to Construct a Frequency Distribution Formula Worksheet Note: Rows 9-51 are not shown.

11. Relative Frequency and Percent Frequency .04(100) 2/50

12. Using Excel’s Frequency Function to Construct a Frequency Distribution Value Worksheet Note: Rows 9-51 are not shown.

13. Audit example • 5 classes should be sufficient, given the size of our data set. • Approximate class width is given by: We round up to the next highest integer, so our class interval is 5.

14. Using Excel’s Frequency Function to Construct a Frequency Distribution • Step 1: Select cells D2:D6, the cells in which we want the frequencies to appear • Step 2: Type the following formula in the formula bar:=FREQUENCY(A2:A21,{14,19,24,29,34}) • Step 3: Press Ctrl-Shift-Enter

15. The Audit Example

16. The Histogram This is a bar graph of a frequency distribution for quantitative data. • The variable of interest is measured on the horizontal axis. • Frequency, relative frequency, or percent frequency for each class is measured on the vertical axis. • There are no spaces between the bars on a histogram—the rectangles are adjacent.

17. 18 16 14 12 10 8 6 4 2 Histogram Tune-up Parts Cost Frequency Parts Cost ($) 50 60 70 80 90 100 110 120

18. Using Excel’s Chart Wizard to Construct a Histogram: Audit Example • Step 1: Select cells C1:D6. • Click the Chart Wizard button. • When Chart Type dialog box appears, select Clustered Column from the Chart sub-type display. Click Next>. • At Step 2 —Chart Data Source —Click Next>. • At Step 3 – Chart Options – select Titles Tab and then type Histogram for Audit Time Data in the Chart title box. Type Audit Time in Days in the Category (X) axis box and type Frequency in the Value (Y) axis box. Select the Legend tab and then remove the check in the Show Legend box. Click Next>. • Step 6: Click Finish

19. Removing the Gaps Between Rectangles (bars) • Step 1: Click right on any rectangle (bar) in the chart to produce a list of options. • Step 2: Select the Format Data Series option • Step 3: Select the Options tab and then enter 0 in the Gap width box.Click OK

20. It looks like this:

21. Cumulative Distributions These show the number of data items with values less than or equal to the class limit of each class. • Cumulative relative frequency distributions show the proportion of data items with values less than or equal to the class limit of each class. • Cumulative frequency distributions show the proportion of data items with values less than or equal to the class limit of each class.

22. Cumulative Distributions • Hudson Auto Repair Cumulative Relative Frequency Cumulative Percent Frequency Cumulative Frequency < 59 < 69 < 79 < 89 < 99 < 109 Cost ($) .04 .30 .62 .76 .90 1.00 2 15 31 38 45 50 4 30 62 76 90 100 2 + 13 15/50 .30(100)

23. 100 80 60 40 20 Ogive with Cumulative Percent Frequencies Tune-up Parts Cost (89.5, 76) Cumulative Percent Frequency Parts Cost ($) 50 60 70 80 90 100 110

24. Ogive for Audit Example