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• ### Assessment

1. 1. Task for the Day • Work with a partner and answer the activity. • Topic: ITEM ANALYIS
2. 2. STATISTICS Descriptive Statistics Inferential Statistics Gives numerical and • Provides procedures graphic procedures to to draw inferences summarize a collection of about a population data in a clear and from a sample understandable way
3. 3. Statistics: Tabular and Graphical Presentations    Summarizing Qualitative Data Summarizing Quantitative Data Recall − − Qualitative Quantitative
4. 4. Summarizing Qualitative Data       Frequency Distribution (shows how many) Relative Frequency Distribution (shows what fraction) Percent Frequency Distribution (shows what percentage) Bar Graph Pie Chart Both these are graphical means for displaying any of above.
5. 5. Data – any set of information that describes a given identity • It an be • GROUPED DATA is a data that has been organized into classes. This data is no longer “raw” • UNGROUPED DATA is simply an arrangement of data from lowest to highest. A data class is a group of data which is related by some user defined property Each of those classes is of a certain width and this is referred to as class width or class size.
6. 6. Age (years) 0-9 12 10-19 30 20-29 Class Frequency 18 30-39 12 Age (years) Frequency 1 12 2 30 3 18 4 6
7. 7. Calculating Class interval or Class Size • Class interval = Higest Value – Lowest Value Number of classes you want to have • or • Class interval = HV - LV = Range • k k • Where k is equal to 1 + 3.3 log n
8. 8. Frequency Distribution 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 nonoverlapping classes. in each of several nonoverlapping 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.
9. 9. Example: Miranda Inn • • • • • Guests staying at Miranda Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 guests are: Below Average Above Average Above Average Average Above Average Average Above Average Average Above Average Below Average Poor Excellent Above Average Average Above Average Above Average Below Average Poor Above Average Average Average
10. 10. Frequency Distribution Rating Frequency 2 Poor 3 Below Average 6 Average 9 Above Average 1 Excellent Total 21
11. 11. Relative Frequency Distribution 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.
12. 12. 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. 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.
13. 13. Relative Frequency and Percent Frequency Distributions Relative Frequency Rating .10 Poor .15 Below Average .25 Average .45 Above Average .05 Excellent Total 1.00 Percent Frequency 10 15 25 .10(100) = 10 45 5 100 1/20 = .05
14. 14. Bar Graph  A bar graph is a graphical device for depicting qualitative data.  On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes.  A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the vertical axis).  Using a bar of fixed width drawn above each class label, we extend the height appropriately.  The bars are separated to emphasize the fact that each class is a separate category.
15. 15. Bar Graph Good? Bad? Miranda Inn Quality Ratings 10 9 Frequency 8 7 6 5 4 3 2 1 Poor Below Average Above Excellent Average Average Rating
16. 16. Pie Chart  The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle.
17. 17. Pie Chart Miranda Inn Quality Ratings Excellent 5% Poor 10% Above Average 45% Below Average 15% Average 25%
18. 18. Example: Miranda Inn Insights Gained from the Preceding Pie Chart • One-half of the customers surveyed gave Miranda a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager.
19. 19. Summarizing Quantitative Data       Frequency Distribution Relative Frequency and Percent Frequency Distributions Dot Plot Histogram Cumulative Distributions Ogive
20. 20. Example: Juson Auto Repair The manager of Juson 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.
21. 21. Example: Juson Auto Repair Sample of Parts Cost for 50 Tune-ups 91 71 104 85 62 78 69 74 97 82 93 72 62 88 98 57 89 68 68 101 75 66 97 83 79 52 75 105 68 105 99 79 77 71 79 Including a line in the table for every possible cost is not a good idea. Need to categorize. 80 75 65 69 69 97 72 80 67 62 62 76 109 74 73
22. 22. 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
23. 23. Frequency Distribution  Guidelines for Selecting Width of Classes •Use classes of equal width. •Approximate Class Width = Largest Data Value − Smallest Data Value Number of Classes
24. 24. Frequency Distribution • For Juson Auto Repair, if we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 ≅ 10 Parts Cost (\$) Frequency 50-59 2 60-69 13 70-79 16 80-89 7 90-99 7 100-109 5 Total 50
25. 25. Preview cumulative frequencies here. Relative Frequency and Percent Frequency Distributions Parts Relative Percent Cost (\$) Frequency Frequency 50-59 .04 4 60-69 .26 2/50 26 .04(100) 70-79 .32 32 80-89 .14 14 90-99 .14 14 100-109 .10 10 Total 1.00 100
26. 26. Relative Frequency and Percent Frequency Distributions Insights Gained from the Percent Frequency Distribution • Only 4% of the parts costs are in the \$50-59 class. • 30% of the parts costs are under \$70. • The greatest percentage (32% or almost one-third) of the parts costs are in the \$70-79 class. • 10% of the parts costs are \$100 or more.
27. 27. Dot Plot    One of the simplest graphical summaries of data is a dot plot. A horizontal axis shows the range of data values. Then each data value is represented by a dot placed above the axis.
28. 28. Dot Plot Tune-up Parts Cost . 50 . . .. . . . . .. .. .. .. . . . . ..... .......... .. . .. . . ... . .. . 60 70 80 90 Cost (\$) Not used much anymore. Common when graphical drawing tools were primitive. 100 110
29. 29. Histogram  Another common graphical presentation of quantitative data is a histogram.  The variable of interest is placed on the horizontal axis.  A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.  Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. In informal discussions bar graphs and histograms are often equated. In this class you should be careful to keep them straight.
30. 30. Histogram Tune-up Parts Cost 18 16 Frequency 14 12 10 8 6 4 2 Parts 50−59 60−69 70−79 80−89 90−99 100-110 Cost (\$)
31. 31. Histogram (Common categories) Symmetric − − Left tail is the mirror image of the right tail Examples: heights and weights of people .35 Relative Frequency  .30 .25 .20 .15 .10 .05 0
32. 32. Histogram Moderately Skewed Left − − A longer tail to the left Example: exam scores .35 Relative Frequency  .30 .25 .20 .15 .10 .05 0
33. 33. Histogram Moderately Right Skewed − − A Longer tail to the right Example: housing values .35 Relative Frequency  .30 .25 .20 .15 .10 .05 0
34. 34. Histogram Highly Skewed Right − − A very long tail to the right Example: executive salaries .35 Relative Frequency  .30 .25 .20 .15 .10 .05 0
35. 35. Cumulative Distributions Cumulative frequency distribution − shows the Cumulative frequency distribution − shows the number of items with values less than or equal to number of items with values less than or equal to the upper limit of each class.. 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 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.
36. 36. Cumulative Distributions  Hudson Auto Repair Cost (\$) < 59 < 69 < 79 < 89 < 99 < 109 Cumulative Cumulative Cumulative Relative Percent Frequency Frequency Frequency 2 .04 4 15 .30 30 31 2 + 13 .62 15/50 62 .30(100) 38 .76 76 45 .90 90 50 1.00 100 Cumulative frequency distribution − shows the Cumulative frequency distribution − shows the number of items with values less than or equal to number of items with values less than or equal to the upper limit of each class.. the upper limit of each class..
37. 37. Ogive An ogive is a graph of a cumulative distribution. The data values are shown on the horizontal axis. Shown on the vertical axis are the: • cumulative frequencies, or • cumulative relative frequencies, or • cumulative percent frequencies The frequency (one of the above) of each class is plotted as a point. The plotted points are connected by straight lines.
38. 38. Ogive Hudson Auto Repair • 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. • 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.
39. 39. Ogive with Cumulative Percent Frequencies Cumulative Percent Frequency Tune-up Parts Cost Tune-up Parts Cost 100 80 60 (89.5, 76) 40 20 50 60 70 80 90 100 110 Parts Cost (\$)
40. 40. Class Limits f ˂cf ˃cf ˂cpf ˃cpf 46-48 1 35 1 100 2.86 43-45 1 34 2 97.14 5.70 40-42 2 33 4 94.29 11.43 37-39 3 31 7 88.57 17.14 34-36 3 28 10 80.00 28.57 31-33 4 25 14 71.43 40.00 28-30 7 21 21 60.00 60.00 25-27 5 14 26 40.00 74.29 22-24 3 9 29 25.71 82.86 19-21 2 6 31 17.14 88.57 16-18 2 4 33 11.43 94.29 13-15 1 2 34 5.70 97.14 10-12 1 1 35 2.86 100.0 N = 35