McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 2 Descriptive Statistics: Tabular and Graphical Methods

2-2 Descriptive Statistics 2.1Graphically Summarizing Qualitative Data 2.2Graphically Summarizing Quantitative Data 2.3Dot Plots 2.4Stem-and-Leaf Displays 2.5Crosstabulation Tables (Optional)

2-3 Descriptive Statistics Continued 2.6Scatter Plots (Optional) 2.7Misleading Graphs and Charts (Optional)

2-4 Graphically Summarizing Qualitative Data Frequency distribution: A table that summarizes the number of items in each of several non-overlapping classes. The purpose is to make the data easier to understand. With qualitative data, the possible values naturally identify the different categories. For example, in the dealership data, the vehicle models sold.

2-5 Example 2.1: Describing 2006 Jeep Purchasing Patterns Table 2.1 lists all 251 vehicles sold in 2006 by the greater Cincinnati Jeep dealers Table 2.1 does not reveal much useful information A frequency distribution is a useful summary –Simply count the number of times each model appears in Table 2.1

2-6 The Resulting Frequency Distribution Jeep ModelFrequency Commander71 Grand Cherokee70 Liberty80 Wrangler30 251

2-7 Relative Frequency and Percent Frequency Relative frequency summarizes the proportion of items in each class For each class, divide the frequency of the class by the total number of observations Multiply times 100 to obtain the percent frequency

2-8 The Resulting Relative Frequency and Percent Frequency Distribution Jeep Model Relative Frequency Percent Frequency Commander % Grand Cherokee70/251 Liberty80/251 Wrangler30/ %

2-9 Bar Charts and Pie Charts Bar chart: A vertical or horizontal rectangle represents the frequency for each category –Height can be frequency, relative frequency, or percent frequency Pie chart: A circle divided into slices where the size of each slice represents its relative frequency or percent frequency

2-10 Excel Bar Chart of the Jeep Sales Data

2-11 Excel Pie Chart of the Jeep Sales Data

2-12 Pareto Chart Pareto chart: “In many economies, most of the wealth is held by a small minority of the population.” “Most of the defects are caused by a small portion of reasons” law –Bar height represents the frequency of occurrence –Bars are arranged in decreasing height from left to right

2-13 Pareto charts are typically used to prioritize competing or conflicting "problems," so that resources are allocated to the most significant areas. In general, though, they can be used to determine which of several classifications have the most "count" or cost associated with them. For instance, the number of people using the various ATM's vs. each of the indoor teller locations, or the profit generated from each of twenty product lines. The important limitations are that the data must be in terms of either counts or costs. The data can not be in terms that can't be added, such as percent yields or error rates.

2-14 Excel Frequency Table and Pareto Chart of Labeling Defects

2-15 Graphically Summarizing Quantitative Data Often need to summarize and describe the shape of the distribution One way is to group the measurements into classes of a frequency distribution and then displaying the data in the form of a histogram

2-16 Frequency Distribution A frequency distribution is a list of data classes, non-overlapping intervals, with the count of values that belong to each class. The frequency distribution is organized as a table. –“tally and count” Show the frequency distribution in a histogram –The histogram is a picture of the frequency distribution, a special bar chart for quantitative data, with no gaps between the bars.

2-17 Constructing a Frequency Distribution Steps in making a frequency distribution: 1.Find the number of classes 2.Find the class length 3.Form non-overlapping classes of equal width 4.Tally and count 5.Graph the histogram Given the non-overlapping classes you should know how to tally and count and make the histogram.

2-18 Example 2.2 The Payment Time Case: A Sample of Payment Times Table 2.4

2-19 Make a histogram Find the number of classes Find the class length Constructing nonoverlapping classes of equal length Tally and count the number of entries in each class Graph the histogram.

2-20 Number of Classes Group all of the n data into K number of classes K is the smallest whole number for which 2 K  n In Examples 2.2 n = 65 –For K = 6, 2 6 = 64, < n –For K = 7, 2 7 = 128, > n –So use K = 7 classes

2-21 Class Length Find the length of each class as the largest measurement minus the smallest divided by the number of classes found earlier (K) Always round up to the same level of precision as the data For Example 2.2, (29-10)/7 = –Because payments measured in days, round to three days

2-22 Form Non-Overlapping Classes of Equal Width The classes start on the smallest value –This is the lower limit of the first class The upper limit of the first class is smallest value + class length –In the example, the first class starts at 10 days and goes up to 13 days The next class starts at this upper limit and goes up by class length And so on

2-23 Seven Non-Overlapping Classes Payment Time Example Class 110 days and less than 13 days Class 213 days and less than 16 days Class 316 days and less than 19 days Class 419 days and less than 22 days Class 522 days and less than 25 days Class 625 days and less than 28 days Class 728 days and less than 31 days

2-24 Tally and Count the Number of Measurements in Each Class Class FrequencyRelative Frequency 10 ≤ x< ≤ x< 16 16≤ x< ≤ x< ≤ x< ≤ x< ≤ x< 31

2-25 Histogram Rectangles represent the classes The base represents the class length and limits The height represents –the frequency in a frequency histogram, or –the relative frequency in a relative frequency histogram

2-26 Histograms Frequency Histogram Relative Frequency Histogram

2-27 Some Common Distribution Shapes Skewed to the right: The right tail of the histogram is longer than the left tail Skewed to the left: The left tail of the histogram is longer than the right tail Symmetrical: The right and left tails of the histogram appear to be mirror images of each other

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2-29 Mound-shaped or bell-shaped distribution vs non mound-shaped distribution. Symmetric Distribution

2-30 Cumulative Distributions Another way to summarize a distribution is to construct a cumulative distribution To do this, use the same number of classes, class lengths, and class boundaries used for the frequency distribution Rather than a count, we record the number of measurements that are less than the upper boundary of that class. The cumulative count is the sum of the count of current class and the counts of all previous classes. –In other words, a running total

2-31 Frequency, Cumulative Frequency, and Cumulative Relative Frequency Distribution ClassFrequency Cumulative Frequency Cumulative Relative Frequency Cumulative Percent Frequency 10 < 13333/65= % 13 < /65= % 16 < % 19 < % 22 < % 25 < % 28 < %

2-32 Stem-and-Leaf Display Purpose is to see the overall pattern of the data, by grouping the data into classes –the variation from class to class –the amount of data in each class –the distribution of the data within each class Best for small to moderately sized data distributions

2-33 Car Mileage Example Refer to the Car Mileage Case –Data in Table 2.14; all digits except the last one, leaf is the last digit The stem-and-leaf display: = = 29.8

2-34 Constructing a Stem-and-Leaf Display Can split the stems as needed For example you can divide one stem into the lower part, which only contains the leaves of ‘0’ ‘1’ ‘2’ ‘3’ ‘4’, and the upper part, which only contains the leaves of ‘5’ ‘6’ ‘7’ ‘8’ ‘9’.

2-35 Split Stems from Car Mileage Example Starred classes (*) extend from 0.0 to 0.4 Unstarred classes extend from 0.5 to * * * * 03

2-36 Comparing Two Distributions To compare two distributions, can construct a back-to-back stem-and-leaf display (or histogram) Uses the same stems for both One leaf is shown on the left side and the other on the right

2-37 Sample Back-to-Back Stem-and-Leaf Display

2-38 Scatter Plots (Optional) Used to study relationships between two variables Place one variable on the x-axis Place a second variable on the y-axis Place dot on pair coordinates

2-39 Types of Relationships Linear: A straight line relationship between the two variables Positive: When one variable goes up, the other variable goes up Negative: When one variable goes up, the other variable goes down No Linear Relationship: There is no coordinated linear movement between the two variables

2-40 A Scatter Plot Showing a Positive Linear Relationship

2-41 A Scatter Plot Showing a Little or No Linear Relationship

2-42 A Scatter Plot Showing a Negative Linear Relationship