1 1 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Chapter 2 Descriptive Statistics

Descriptive Statistics Learning Objectives: 1.Learn how to construct (procedures) and interpret summarized qualitative and quantitative data using : frequency and relative frequency distributions, bar graphs and pie charts, a dot plot, a histogram, and an ogive 2.Learn how the shape of a data distribution (negatively skewed, symmetric, and positively skewed).

3 3 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Descriptive Statistics: Tabular and Graphical Presentations Part A Exploratory Data Analysis Cross-tabulations; Scatter Diagrams Part-B

4 4 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Tabular and Graphical Presentations: Visual Description of Data Part A

5 5 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Tabular and Graphical Procedures Qualitative Data Quantitative Data Tabular TabularMethods Methods Methods MethodsGraphical Methods MethodsGraphical Graphical Graphical FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Percent Freq.Percent Freq. Distribution Distribution CrosstabulationCrosstabulation Bar GraphBar Graph Pie ChartPie Chart FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Cum. Freq. Dist.Cum. Freq. Dist. Cum. Rel. Freq.Cum. Rel. Freq. Distribution Distribution Stem-and-LeafStem-and-Leaf Display Display CrosstabulationCrosstabulation Dot PlotDot Plot HistogramHistogram OgiveOgive ScatterScatter Diagram Diagram Two Types of Data Overview of Tabular and Graphical Methods Summarizing Data

6 6 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) n Summarizing Qualitative Data

7 7 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.1) Summarizing Qualitative Data 1. Frequency Distribution 2. Relative Frequency Distribution 3. Percent Frequency Distribution 4. Bar Graph 5. Pie Chart Graphical Methods Tabular Methods

8 8 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) The objective is to provide insights about the data that cannot be quickly obtained by looking at the original data. The objective is to provide insights about the data that cannot be quickly obtained by looking at the original data. 2.1) Summarizing Qualitative Data

9 9 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 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 presented in several non-overlapping classes. presented in 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 presented in several non-overlapping classes. presented in several non-overlapping classes Frequency Distribution

10 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Example: Days Inn Guests staying at Days Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 guests are: Below Average Below Average Above Average Above Average Average Average Above Average Above Average Average Average Above Average Above Average Average Average Above Average Above Average Below Average Below Average Poor Poor Excellent Excellent Above Average Above Average Average Average Above Average Above Average Below Average Below Average Poor Poor Above Average Above Average Average Average

11 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Frequency Distribution Poor Below Average Average Above Average Excellent Total 20 RatingFrequency

12 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) The relative frequency is the fraction or The relative frequency is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to a given class (category). belonging to a given class (category). The relative frequency is the fraction or The relative frequency is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to a given class (category). belonging to a given class (category). 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

13 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 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.

14 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Relative Frequency and Percent Frequency Distributions Poor Below Average Average Above Average Excellent Total Relative RelativeFrequency Percent PercentFrequency Rating.10(100) = 10 1/20 =.05

15 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 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 represent each of class (Category). the labels that represent each of class (Category). 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.

16 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Recall The Frequency Distribution for Days Inn Poor Below Average Average Above Average Excellent Total 20 RatingFrequency

17 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating Bar Graph Days Inn Quality Ratings

18 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 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 into sectors that correspond to the subdivide the circle into sectors that correspond to the relative frequency for each class. relative frequency for each class. n Example: As there are 360 degrees in a circle, a class with a relative frequency of.25 would consume a class with a relative frequency of.25 would consume.25(360) = 90 degrees of the circle..25(360) = 90 degrees of the circle.

19 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Recall The Relative Frequency for Days Inn Poor Below Average Average Above Average Excellent Total 1.00 Relative RelativeFrequency Rating

20 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Below Average 15% Below Average 15% Average 25% Average 25% Above Average 45% Above Average 45% Poor 10% Poor 10% Excellent 5% Excellent 5% InnQuality Ratings Days Inn Quality Ratings Pie Chart

21 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) n Some Insights about Days Inn—from the Preceding Pie Chart Example: Days Inn One-half of the customers surveyed gave Days Inn One-half of the customers surveyed gave Days Inn a quality rating of “above average” or “excellent” a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might (looking at the left side of the pie). This might please the manager. please the manager. For each customer who gave an “excellent” rating, For each customer who gave an “excellent” rating, there were two customers who gave a “poor” there were two customers who gave a “poor” rating (looking at the top of the pie). This should rating (looking at the top of the pie). This should displease the manager. displease the manager.

22 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) n Summarizing Quantitative Data Summarizing Data

23 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2) Summarizing Quantitative Data 1. Frequency Distribution 2. Relative Frequency and Percent Frequency Distributions 3. Dot Plot 4. Histogram 5. Cumulative Distributions 6. Ogive TabularMethods Graphical Methods

24 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) n Three Important things to remember when working on Frequency Distribution for Quantitative Data: 1. Determine the number of non-overlapping classes (categories)—Quantitative data do not naturally come in categories 2. Determine the Width of each class (Equal class widths are preferred) 3. Specify the Class Limits: (a given observation should belong to one and only one class) 2.2.1) Frequency Distribution

25 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.1) Frequency Distribution n In Determining the Number of CLASSES Use a thumb rule of 5 to 20 classes. Use a thumb rule of 5 to 20 classes. Larger data sets usually require a larger Larger data sets usually require a larger number of classes. Smaller data sets usually require fewer classes Smaller data sets usually require fewer classes

26 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.1) Frequency Distribution n To Determine the WIDTH of Classes:

27 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Example: Hudson Auto Repair n Sample of Parts Cost for 50 Tune-ups

28 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.1) Frequency Distribution For Hudson Auto Repair, if we choose six classes: Total 50 Parts Cost ($) Frequency Approximate Class Width = ( )/6 = 9.5  10

29 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.2) Relative Frequency and Percent Frequency Distributions 2.2.2) Relative Frequency and Percent Frequency Distributions Parts Cost ($) Total 1.00 Relative RelativeFrequency Percent Frequency Frequency 2/50.04(100)

30 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 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 Some Insights ….on the Parts Cost Data 2.2.2) Relative Frequency and Percent Frequency Distributions

31 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.3) Dot Plot n Is one of the simplest graphical summaries of data. It has two components: n A horizontal axis that shows the range of data values. n And a dot placed above the axis, that represents each data value.

32 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Cost ($) Tune-up Parts Cost 2.2.3) Dot Plot

33 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.4) Histogram Histogram is a common graphical presentation of Histogram is a common graphical presentation of quantitative data. quantitative data. In describing data using a Histogram, place the variable In describing data using a Histogram, place the variable of interest on the horizontal axis. of interest on the horizontal axis. Then we draw a rectangle above each class interval with Then we draw a rectangle 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.

34 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.4) Histogram Parts Cost ($) Parts Cost ($) Frequency 50      Tune-up Parts Cost

35 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Shapes of Histogram Depending upon the data set, we may see different shapes when we summarize a data using histogram.

36 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 1. 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 Various Shapes of Histogram Relative Frequency

37 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Various Shapes of Histogram 2. Moderately Skewed Left A longer tail to the left A longer tail to the left Example: exam scores Example: exam scores Relative Frequency

38 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 3. Moderately Right Skewed A Longer tail to the right A Longer tail to the right Example: housing values Example: housing values Various Shapes of Histogram Relative Frequency

39 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Various Shapes of Histogram 4. 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

40 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 1. Cumulative frequency distribution- shows the number of items with values less than or equal to the upper limit of each class.. 1. Cumulative frequency distribution- shows the number of items with values less than or equal to the upper limit of each class.. 2.Cumulative relative frequency distribution – shows the proportion of items with values less than or equal to the upper limit of each class. 2.Cumulative relative frequency distribution – shows the proportion of items with values less than or equal to the upper limit of each class ) Cumulative Distributions 3. Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class. or equal to the upper limit of each class. 3. Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class. or equal to the upper limit of each class.

41 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Example Recall the Hudson Auto Repair Data: Total 50 Parts Cost ($) Frequency

42 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.5) Cumulative Distributions n Hudson Auto Repair < 59 < 59 < 69 < 69 < 79 < 79 < 89 < 89 < 99 < 99 < 109 Cost ($) Cumulative CumulativeFrequency RelativeFrequency CumulativePercent Frequency Frequency /50.30(100)

43 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) 2.2.6) Ogive n An ogive is a graph of a cumulative distribution. Can be constructed using one of the following 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 plotted points are connected by straight lines.

44 Slide © University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse) Parts Parts Cost ($) Parts Parts Cost ($) Cumulative Percent Frequency (89.5, 76) Example: Ogive with Cumulative Percent Frequencies Cumulative Percent Frequencies Tune-up Parts Cost