1. 1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

2. 2 2 Slide IS 310 – Business Statistics Understanding Data n In order to make sense from a group of data, we need to summarize the data into a form that can be easily understood. Once summarized, the data provides meaningful information to a reader. n There are various methods to summarize data. We will cover some of these methods in chapter 2.

3. 3 3 Slide IS 310 – Business Statistics Chapter 2, Part A Descriptive Statistics: Tabular and Graphical Presentations Summarizing Quantitative Data Summarizing Quantitative Data n Frequency Distribution n Relative Frequency Distribution n Percent Frequency Distribution n Bar Graph n Pie Chart n Histogram n Cumulative Distributions n Ogive

4. 4 4 Slide IS 310 – Business Statistics Frequency Distribution of Quantitative Data n Let’s look at Table 2.4 (10-Page 34; 11-Page 39). This data represents the time (in days) that it took Sanderson and Clifford to complete audits of 20 customers. This is what we call Raw Data. It does not provide any meaningful information; it’s just a bunch of data. n Once summarized as a Frequency Distribution table, it will convey important and useful information. Let’s look at Table 2.5 (10-Page 35; 11-Page 40). For example, we know how many audits took between 10 and 14 days, how many between 15 and 19 days, and so on.

5. 5 5 Slide IS 310 – Business Statistics A frequency distribution is a tabular summary of A frequency distribution is a tabular summary of data showing the frequency (or number) of items data showing the frequency (or number) of items in each of several non-overlapping classes. in each of several non-overlapping classes. A frequency distribution is a tabular summary of A frequency distribution is a tabular summary of data showing the frequency (or number) of items data showing the frequency (or number) of items in each of several non-overlapping classes. in each of several non-overlapping classes. The objective is to provide insights about the data The objective is to provide insights about the data that cannot be quickly obtained by looking only at that cannot be quickly obtained by looking only at the original data. the original data. The objective is to provide insights about the data The objective is to provide insights about the data that cannot be quickly obtained by looking only at that cannot be quickly obtained by looking only at the original data. the original data. Frequency Distribution

6. 6 6 Slide IS 310 – Business Statistics Frequency Distribution (continued) n Let’s review Table 2.5 (10-Page 35; 11-Page 40). n Audit Time Frequency n 10-14 4 n 15-19 8 n 20-24 5 Understand classes, width n 25-29 2 of classes, class limits, and n 30-34 1 class midpoint n Total 20

7. 7 7 Slide IS 310 – Business Statistics The relative frequency of a class is the fraction or The relative frequency of a class is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to the class. belonging to the class. The relative frequency of a class is the fraction or The relative frequency of a class is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to the class. belonging to the class. A relative frequency distribution is a tabular A relative frequency distribution is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class. frequency for each class. A relative frequency distribution is a tabular A relative frequency distribution is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class. frequency for each class. Relative Frequency Distribution

8. 8 8 Slide IS 310 – Business Statistics Percent Frequency Distribution The percent frequency of a class is the relative The percent frequency of a class is the relative frequency multiplied by 100. frequency multiplied by 100. The percent frequency of a class is the relative The percent frequency of a class is the relative frequency multiplied by 100. frequency multiplied by 100. A percent frequency distribution is a tabular A percent frequency distribution is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class. frequency for each class. A percent frequency distribution is a tabular A percent frequency distribution is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class. frequency for each class.

9. 9 9 Slide IS 310 – Business Statistics Relative Frequency and Percent Frequency n Let’s look at Table 2.6 (10-Page 36; 11-Page 41) This table is an extension of Table 2.5. Table 2.6 presents both Relative Frequency and Percent Frequency. n Audit Time Relative Frequency Percent Frequency n 10-14 0.20 20 n 15-19 0.40 40 n 20-24 0.25 25 n 25-29 0.10 10 n 30-34 0.05 5

10. 10 Slide IS 310 – Business Statistics Bar Graph A bar graph is a graphical device for depicting A bar graph is a graphical device for depicting qualitative data. qualitative data. On one axis (usually the horizontal axis), we specify On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes. the labels that are used for each of the classes. A frequency, relative frequency, or percent frequency A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the scale can be used for the other axis (usually the vertical axis). vertical axis). Using a bar of fixed width drawn above each class Using a bar of fixed width drawn above each class label, we extend the height appropriately. label, we extend the height appropriately. The bars are separated to emphasize the fact that each The bars are separated to emphasize the fact that each class is a separate category. class is a separate category.

11. 11 Slide IS 310 – Business Statistics Pie Chart The pie chart is a commonly used graphical device The pie chart is a commonly used graphical device for presenting relative frequency distributions for for presenting relative frequency distributions for qualitative data. qualitative data. n First draw a circle; then use the relative frequencies to subdivide the circle frequencies to subdivide the circle into sectors that correspond to the into sectors that correspond to the relative frequency for each class. relative frequency for each class. n Since there are 360 degrees in a circle, a class with a relative frequency of.25 would a class with a relative frequency of.25 would consume.25(360) = 90 degrees of the circle. consume.25(360) = 90 degrees of the circle.

12. 12 Slide IS 310 – Business Statistics 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.

13. 13 Slide IS 310 – Business Statistics Example: Hudson Auto Repair n Sample of Parts Cost($) for 50 Tune-ups

14. 14 Slide IS 310 – Business Statistics Frequency Distribution n 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. Use enough classes to show the variation in Use enough classes to show the variation in the data. the data. Do not use so many classes that some contain Do not use so many classes that some contain only a few data items. only a few data items.

15. 15 Slide IS 310 – Business Statistics Frequency Distribution n Guidelines for Selecting Width of Classes Use classes of equal width. Use classes of equal width. Approximate Class Width = Approximate Class Width =

16. 16 Slide IS 310 – Business Statistics Frequency Distribution For Hudson Auto Repair, if we choose six classes: 50-59 60-69 70-79 80-89 90-99 100-109 2 13 16 7 7 5 Total 50 Parts Cost ($) Frequency Approximate Class Width = (109 - 52)/6 = 9.5  10

17. 17 Slide IS 310 – Business Statistics Relative Frequency and Percent Frequency Distributions 50-59 50-59 60-69 60-69 70-79 70-79 80-89 80-89 90-99 90-99 100-109 100-109 Parts Cost ($).04.26.32.14.14.10 Total 1.00 Relative RelativeFrequency 4 26 32 14 14 10 100 Percent Frequency Frequency 2/502/50.04(100).04(100)

18. 18 Slide IS 310 – Business Statistics Only 4% of the parts costs are in the $50-59 class. Only 4% of the parts costs are in the $50-59 class. The greatest percentage (32% or almost one-third) The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class. of the parts costs are in the $70-79 class. 30% of the parts costs are under $70. 30% of the parts costs are under $70. 10% of the parts costs are $100 or more. 10% of the parts costs are $100 or more. n Insights Gained from the Percent Frequency Distribution Relative Frequency and Percent Frequency Distributions

19. 19 Slide IS 310 – Business Statistics Histogram Another common graphical presentation of Another common graphical presentation of quantitative data is a histogram. quantitative data is a histogram. The variable of interest is placed on the horizontal The variable of interest is placed on the horizontal axis. axis. A rectangle is drawn above each class interval with A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, its height corresponding to the interval’s frequency, relative frequency, or percent frequency. relative frequency, or percent frequency. Unlike a bar graph, a histogram has no natural Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. separation between rectangles of adjacent classes.

20. 20 Slide IS 310 – Business Statistics Histogram 2 2 4 4 6 6 8 8 10 12 14 16 18 Parts Cost ($) Parts Cost ($) Frequency 50  59 60  69 70  79 80  89 90  99 100-110 Tune-up Parts Cost

21. 21 Slide IS 310 – Business Statistics n Symmetric Left tail is the mirror image of the right tail Left tail is the mirror image of the right tail Examples: heights and weights of people Examples: heights and weights of people Histogram Relative Frequency.05.10.15.20.25.30.35 0 0

22. 22 Slide IS 310 – Business Statistics Histogram n Moderately Skewed Left A longer tail to the left A longer tail to the left Example: exam scores Example: exam scores Relative Frequency.05.10.15.20.25.30.35 0 0

23. 23 Slide IS 310 – Business Statistics n Moderately Right Skewed A longer tail to the right A longer tail to the right Example: housing values Example: housing values Histogram Relative Frequency.05.10.15.20.25.30.35 0 0

24. 24 Slide IS 310 – Business Statistics Histogram n Highly Skewed Right A very long tail to the right A very long tail to the right Example: executive salaries Example: executive salaries Relative Frequency.05.10.15.20.25.30.35 0 0

25. 25 Slide IS 310 – Business Statistics Cumulative frequency distribution  shows the Cumulative frequency distribution  shows the number of items with values less than or equal to the number of items with values less than or equal to the upper limit of each class.. upper limit of each class.. Cumulative frequency distribution  shows the Cumulative frequency distribution  shows the number of items with values less than or equal to the number of items with values less than or equal to the upper limit of each class.. upper limit of each class.. Cumulative relative frequency distribution – shows Cumulative relative frequency distribution – shows the proportion of items with values less than or the proportion of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative relative frequency distribution – shows Cumulative relative frequency distribution – shows the proportion of items with values less than or the proportion of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative Distributions Cumulative percent frequency distribution – shows Cumulative percent frequency distribution – shows the percentage of items with values less than or the percentage of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative percent frequency distribution – shows Cumulative percent frequency distribution – shows the percentage of items with values less than or the percentage of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class.

26. 26 Slide IS 310 – Business Statistics Cumulative Distributions n Hudson Auto Repair <59 <69 <79 <89 <99 <109 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency 2 15 31 38 45 50.04.30.62.76.90 1.00 4 30 62 76 90 100 2 + 13 15/5015/50.30(100).30(100)

27. 27 Slide IS 310 – Business Statistics Ogive n An ogive is a graph of a cumulative distribution. n The data values are shown on the horizontal axis. n Shown on the vertical axis are the: cumulative frequencies, or cumulative frequencies, or cumulative relative frequencies, or cumulative relative frequencies, or cumulative percent frequencies cumulative percent frequencies n The frequency (one of the above) of each class is plotted as a point. n The plotted points are connected by straight lines.

28. 28 Slide IS 310 – Business Statistics Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on. Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on. Ogive These gaps are eliminated by plotting points halfway between the class limits. These gaps are eliminated by plotting points halfway between the class limits. Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on. Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on. n Hudson Auto Repair

29. 29 Slide IS 310 – Business Statistics 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

30. 30 Slide IS 310 – Business Statistics End of Chapter 2, Part A