Chapter 2 Descriptive Statistics: Tabular and Graphical Methods Summarizing the Qualitative Data Frequency Distribution Relative Frequency Percent Frequency Distribution Bar Graph Pie Chart

Frequency Distribution A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several classes.

Example: Marada Inn Guests staying at Marada 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 quests are shown below. Below Average Average Above Average Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor Above Average Excellent Above Average Average Above Average Average Above Average Average

Example: Marada Inn Frequency Distribution Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20

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

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

Example: Marada Inn Relative Frequency and Percent Frequency Distributions Relative Percent Rating Frequency Frequency Poor .10 10 Below Average .15 15 Average .25 25 Above Average .45 45 Excellent .05 5 Total 1.00 100

Bar Graph A bar graph is a graphical device for depicting qualitative data. On 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 vertical axis. The bars are separated to emphasize the fact that each class is a separate category.

Example: Marada Inn Bar Graph 1 2 3 4 5 6 7 8 9 Poor Below Average Above Excellent Frequency Rating

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.

Example: Marada Inn Pie Chart Average 25% Below 15% Poor 10% Above 45% Exc. 5% Quality Ratings

Example: Marada Inn Insights Gained from the Preceding Pie Chart One-half of the customers surveyed gave Marada 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 Frequency Distribution Relative Frequency Percent Frequency Distributions Cumulative Distributions Dot Plot Histogram Ogive/ Frequency Polygon

Example: Hudson Auto Repair The manager of Hudson Auto would like to get a better picture of the distribution of costs for engine tune-up parts. A sample of 50 customer invoices has been taken and the costs of parts, rounded to the nearest dollar, are listed below.

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.

Frequency Distribution Guidelines for Selecting Width of Classes Use classes of equal width. Approximate Class Width =

Example: Hudson Auto Repair Frequency Distribution If we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 10 Cost ($) Frequency 50-59 2 60-69 13 70-79 16 80-89 7 90-99 7 100-109 5 Total 50

Example: Hudson Auto Repair Relative Frequency and Percent Frequency Distributions Relative Percent Cost ($) Frequency Frequency 50-59 .04 4 60-69 .26 26 70-79 .32 32 80-89 .14 14 90-99 .14 14 100-109 .10 10 Total 1.00 100

Example: Hudson Auto Repair 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.

Dot Plot One of the simplest graphical summaries of quantitative 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.

Example: Hudson Auto Repair Dot Plot . . .. . . . 50 60 70 80 90 100 110 . . . ..... .......... .. . .. . . ... . .. . . .. .. .. .. . . Cost ($)

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’s frequency, relative frequency, or percent frequency. Unlike a bar graph, a histogram has no natural separation between rectangles of classes.

Example: Hudson Auto Repair Histogram 18 16 14 12 Frequency 10 8 6 4 2 Parts Cost ($) 50 60 70 80 90 100 110

Cumulative Distributions Cumulative frequency distribution -- shows the number of items with values less than or equal to the upper limit of each class. Cumulative relative frequency distribution -- shows the proportion of items with values less than or equal to the upper limit of each class. Cumulative percent frequency distribution -- shows the percentage of items with values less than or equal to the upper limit of each class.

Example: Hudson Auto Repair Cumulative Distributions Cumulative Cumulative Cumulative Relative Percent Cost ($) Frequency Frequency Frequency < 59 2 .04 4 < 69 15 .30 30 < 79 31 .62 62 < 89 38 .76 76 < 99 45 .90 90 < 109 50 1.00 100

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.

Example: Hudson Auto Repair Ogive 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.

Example: Hudson Auto Repair Ogive with Cumulative Percent Frequencies 100 80 60 Cumulative Percent Frequency 40 20 Parts Cost ($) 50 60 70 80 90 100 110