Chapter 2 Descriptive Statistics

Descriptive Statistics Learning Objectives: Methods of Summarizing qualitative and quantitative data: Frequency and relative frequency distributions, bar graphs and pie charts, a dot plot, a histogram, and an ogive Shapes of data distribution: Negatively skewed, symmetric, and positively skewed.

Descriptive Statistics: Tabular and Graphical Presentations Part A Exploratory Data Analysis Cross-tabulations; Scatter Diagrams Part-B

Tabular and Graphical Presentations: Part A Tabular and Graphical Presentations: Visual Description of Data

Tabular and Graphical Procedures Overview of Tabular and Graphical Methods Two Types of Data Qualitative Data Quantitative Data Tabular Methods Graphical Methods Tabular Methods Graphical Methods Summarizing Data Bar Graph Pie Chart Frequency Distribution Rel. Freq. Dist. Percent Freq. Crosstabulation Dot Plot Histogram Ogive Scatter Diagram Frequency Distribution Rel. Freq. Dist. Cum. Freq. Dist. Cum. Rel. Freq. Stem-and-Leaf Display Crosstabulation

Summarizing Qualitative Data

2.1) Summarizing Qualitative Data The main objective of summarizing data is to provide insights about the data that cannot be quickly obtained by looking at the original data.

2.1) Summarizing Qualitative Data Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Bar Graph Pie Chart Tabular Methods Graphical Methods

2.1. Frequency Distribution A frequency distribution is a tabular summary of data showing the frequency (or number) of items presented in several non-overlapping classes.

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 poor. The ratings provided by a sample of 20 guests are: Below Average Above Average Average Average Above Average Below Average Poor Excellent Above Average Below Average Poor Average

Frequency Distribution Poor Below Average Average Above Average Excellent Rating 2 3 5 9 1 Total 20 Frequency

Relative Frequency Distribution A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class---the fraction or proportion of the total number of data items belonging to a given class (category).

Relative Frequency Distributions Rating Poor Below Average Average Above Average Excellent 2 3 5 9 1 Total 20 .10 .15 .25 .45 .05 1.00 1/20 = .05

Percent Frequency Distribution A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. Percent frequency of a class is the relative frequency multiplied by 100.

Relative Frequency and Percent Frequency Distributions Rating Poor Below Average Average Above Average Excellent 2 3 5 9 1 Total 20 .10 .15 .25 .45 .05 1.00 10 15 25 45 5 100 .10(100) = 10 1/20 = .05

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 represent each of class (Category). 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.

Recall the Frequency Distribution for Days Inn? Poor Below Average Average Above Average Excellent Rating 2 3 5 9 1 Total 20 Frequency

Days Inn Quality Ratings Bar Graph Days Inn Quality Ratings 1 2 3 4 5 6 7 8 9 10 Frequency Rating Poor Below Average Average Above Average Excellent

Pie Chart The pie chart is another commonly used graphical device for presenting relative frequency distributions of 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. Example: As there are 360 degrees in a circle, a class with a relative frequency of .25 would take .25X360 = 90 degrees of the circle.

Recall the Relative Frequency for Days Inn Poor Below Average Average Above Average Excellent Rating 2 3 5 9 1 Total 20 .10 .15 .25 .45 .05 1.00

Days Inn Quality Ratings Pie Chart Days Inn Quality Ratings Excellent 5% Poor 10% Below Average 15% Above Average 45% Average 25%

Example: Days Inn Some Insights about Days Inn—from the Preceding Pie Chart One-half of the customers surveyed gave Days Inn 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.

Summarizing Quantitative Data Summarizing Data Summarizing Quantitative Data

2.2) Summarizing Quantitative Data Frequency Distribution Relative and Percent Frequency Distributions Dot Plot Histogram Cumulative Distributions Ogive Tabular Methods Graphical Methods

2.2.1) Frequency Distribution Three Important things when working on Frequency Distribution for Quantitative Data: Quantitative data do not naturally come in categories- So, you Determine the number of non-overlapping classes (categories)— Determining the Width of each class (Equal class widths are preferred) Specifying the Class Limits: (a given observation should belong to one and only one class)

2.2.1) Frequency Distribution In Determining the Number of CLASSES Use a thumb rule of 5 to 20 classes. Larger data sets usually require a larger number of classes. Smaller data sets usually require fewer classes

2.2.1) Frequency Distribution To Determine the WIDTH of Classes:

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

2.2.1) Frequency Distribution If we choose six classes: then the Approximate Class width would be determined as follows Class Width = (109 - 52)/6 = 9.5   10 50-59 60-69 70-79 80-89 90-99 100-109 Parts Cost ($) 2 13 16 7 5 Total 50 Frequency

2.2.2) Relative and Percent Frequency Distributions Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 2 13 16 7 5 Total 50 .04 .26 .32 .14 .10 1.00 4 26 32 14 10 100 2/50 .04(100)

2.2.2) Relative Frequency and Percent Frequency Distributions Some Insights ….on the Parts Cost Data 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.

2.2.3) Dot Plot Is one of the simplest graphical summaries of data. It has two components: A horizontal axis that shows the range of data values. And a dot placed above the axis, that represents each data value.

2.2.3) Dot Plot Tune-up Parts Cost 50 60 70 80 90 100 110 Cost ($)

2.2.4) Histogram Histogram is a common graphical presentation of quantitative data . In describing data using a Histogram, we place the variable of interest on the horizontal axis. Then we draw a rectangle 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.

2.2.4) Histogram Tune-up Parts Cost Frequency Parts Cost ($) 18 16 14 10 12 14 16 18 Frequency Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-110

Shapes of Histogram Depending upon the data set, we may see different shapes when we summarize a data using histogram.

Various Shapes of Histogram 1. Symmetric Left tail is the mirror image of the right tail Examples: heights and weights of people .05 .10 .15 .20 .25 .30 .35 Relative Frequency

Various Shapes of Histogram 2. Moderately Skewed Left A longer tail to the left Example: exam scores .05 .10 .15 .20 .25 .30 .35 Relative Frequency

Various Shapes of Histogram 3. Moderately Right Skewed A Longer tail to the right Example: housing values .05 .10 .15 .20 .25 .30 .35 Relative Frequency

Various Shapes of Histogram 4. Highly Skewed Right A very long tail to the right Example: executive salaries .05 .10 .15 .20 .25 .30 .35 Relative Frequency

2.2.5) Cumulative Distributions 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. 3. Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.

Example Recall the Hudson Auto Repair Data: 50-59 60-69 70-79 80-89 90-99 100-109 Parts Cost ($) 2 13 16 7 5 Total 50 Frequency

2.2.5) Cumulative Distributions Hudson Auto Repair Cumulative Relative Frequency Cumulative Percent Frequency Frequency Cumulative Frequency < 59 < 69 < 79 < 89 < 99 < 109 Cost ($) 2 13 16 7 5 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)

2.2.6) Ogive An ogive is a graph of a cumulative distribution. Can be constructed using one of the following Shown on the vertical axis are the: cumulative frequencies, or cumulative relative frequencies, or cumulative percent frequencies The plotted points are connected by straight lines.

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