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 1.Determine the number of nonoverlapping classes. 2.Determine the width of each class. 3.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 n Sample of Parts Cost for 50 Tune-ups

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

6. Frequency Distribution Guidelines for Selecting Width of Classes Use classes of equal width. Use classes of equal width. Approximate Class 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: Parts Cost($)Frequency 50-592 60-6913 70-7916 80-897 90-997 100-1095 Total50

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

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

11. Relative Frequency and Percent Frequency Parts Cost ($) Relative Frequency Percent Frequency 50-59.044 60-69.2626 70-79.3232 80-89.1414 90-99.1414 100-109.1010 Total1.00100 2/50.04(100 )

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

13. Audit Time Example We want to construct a frequency distribution to describe the days required to complete year end audits for our clients. Here we have data for 20 of our clients Year End Audit Times (in Days) 12141918 15 1817 20272223 22213328 14181630.5

14. 5 classes should be sufficient, given the size of our data set. Approximate class width is given by: Audit example We round up to the next highest integer, so our class interval is 5.

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

16. The Audit Example Audit TimeFrequencyRelative FrequencyPercent Frequency 10-1440.220 15-1980.440 20-2450.2525 25-2920.110 30-3410.055 Total201100

17. 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.

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

19. Using Excel’s Chart Wizard to Construct a Histogram: Audit Example 1.Step 1: Select cells C1:D6. 2.Click the Chart Wizard button. 3.When Chart Type dialog box appears, select Clustered Column from the Chart sub-type display. Click Next>. 4.At Step 2 —Chart Data Source —Click Next>. 5.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>. 6.Step 6: Click Finish

20. 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

21. It looks like this:

22. 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.

23. Cumulative Distributions Hudson Auto Repair <59 <69 <79 <89 <99 <109 Cost ($) Cumulative CumulativeFrequency RelativeFrequencyCumulativePercent Frequency Frequency 2 15 31 38 45 50.04.30.62.76.90 1.00 4 30 62 76 90 100 2 + 13 15/50.30(100)

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

25. Ogive for Audit Example