Descriptive Statistics: Tabular and Graphical Methods Statistics for Business (Env) Chapter 2 Descriptive Statistics: Tabular and Graphical Methods

Descriptive Statistics 2.1 Graphically Summarizing Qualitative Data 2.2 Graphically Summarizing Quantitative Data 2.3 Dot Plots 2.4 Stem-and-Leaf Displays (Optional) 2.6 Scatter Plots 2.7 Misleading Graphs and Charts (Optional)

Types of Variables/data Interval Ratio Nominal Ordinal

Graphically Summarizing Qualitative Data With qualitative data, names identify the different and non-overlapping categories (classes) This data can be summarized using a frequency distribution Frequency distribution: A table that summarizes the number of items in each of several non-overlapping categories/classes.

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

Table 2.1 lists all 251 vehicles sold in 2006 by the greater Cincinnati Jeep dealers

The Resulting Frequency Distribution Jeep Model Frequency Commander 71 Grand Cherokee 70 Liberty 80 Wrangler 30 251

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

The Resulting Relative Frequency and Percent Frequency Distribution Jeep Model Relative Frequency Percent Frequency Commander 0.2829 28.29% Grand Cherokee 0.2789 27.89% Liberty 0.3187 31.78% Wrangler 0.1195 11.95% 1.0000 100.00%

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

Excel Bar Chart of the Jeep Sales Data

Excel Pie Chart of the Jeep Sales Data Pie chart is usually for percent Frequency Distribution.

Pareto Chart Pareto chart: a type of chart that contains both bars and a line graph A bar chart having the different kinds of categories listed on the horizontal scale Bar height represents the frequency of occurrence Bars are arranged in decreasing height from left to right Augmented by plotting a cumulative percentage point for each bar

Excel Frequency Table and Pareto Chart of Labeling Defects

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 A Histogram is a graph in which the class (numerical) midpoints or limits are marked on the horizontal axis and the class frequencies on the vertical axis.

Frequency Distribution A frequency distribution is a list of data classes with the count of values that belong to each class “Classify and count” The frequency distribution is a table Show the frequency distribution in a histogram picture of the frequency distribution (tabulated frequencies) shown as bars and the bars are drawn adjacent to each other.

Constructing a Frequency Distribution Steps in making a frequency distribution: Find the number of classes Find the class length Form non-overlapping classes of equal width Tally and count Graph the histogram

Example 2.2 The Payment Time Case: A Sample of Payment Times (in days after billing) 22 29 16 15 18 17 12 13 19 10 21 14 20 25 23 24 26 27 Table 2.4

Determine the number of Classes If the number of classes is too small lack of detail too large some classes will be empty Group all of the n data into K number of classes General rule: K is the smallest whole number for which 2K  n In Examples 2.2 n = 65 For K = 6, 26 = 64, < n For K = 7, 27 = 128, > n So use K = 7 classes

Number of Classes In General Size of Data Set 2 1≤n<4 3 4≤n<8 4 8≤n<16 5 16≤n<32 6 32≤n<64 7 64≤n<128 8 128≤n<256 9 256≤n<528 10 528≤n<1056

Determine the Class Length Find the length of each class as the largest measurement minus the smallest divided by the number of classes found earlier (K) For Example 2.2, (29-10)/7 = 2.7143 Because payments measured in days, round to three days

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

Seven Non-Overlapping Classes Payment Time Example 10 days and less than 13 days Class 2 13 days and less than 16 days Class 3 16 days and less than 19 days Class 4 19 days and less than 22 days Class 5 22 days and less than 25 days Class 6 25 days and less than 28 days Class 7 28 days and less than 31 days

Tally and Count the Number of Measurements in Each Class First 4 Tally Marks All 65 Tally Marks Frequency 10 < 13 ||| 3 13 < 16 |||| |||| |||| 14 16 < 19 || |||| |||| |||| |||| ||| 23 19 < 22 I |||| |||| || 12 22 < 25 | |||| ||| 8 25 < 28 |||| 4 28 < 31 1

Histogram Rectangles represent the classes The base represents the class length The height represents the frequency in a frequency histogram, or the relative frequency in a relative frequency histogram

Histograms Frequency Histogram Relative Frequency Histogram

Example of a frequency distribution histogram for discrete data The frequency distribution of quiz scores in a class. The score, X, is the number of problems that is answered correctly.

Example of a frequency distribution histogram for continuous data

CONTINUOUS VARIABLES AND REAL LIMITS

CONTINUOUS VARIABLES AND REAL LIMITS For a continuous variable, each score actually corresponds to an interval on the scale. The boundaries that separate these intervals are called real limits. The real limit separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits, one at the top of its interval called the upper real limit and one at the bottom of its interval called the lower real limit.

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

skewed to the left skewed to the right

A Right-Skewed Distribution

A Left-Skewed Distribution

Frequency Polygons Plot a point above each class midpoint at a height equal to the frequency of the class Useful when comparing two or more distributions

Example 2.3: Comparing The Grade Distribution for Two Statistics Exams Table 2.8 (in textbook) gives scores earned by 40 students on first statistics exam Table 2.9 gives the scores on the second exam after an attendance policy Due to the way exams are reported, used the classes: 30<40, 40<50, 50<60, 60<70, 70<80, 80<90, and 90<100

A Percent Frequency Polygon of the Exam Scores

A Percent Frequency Polygon Comparing the Two Exam Scores

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 In other words, a running total

Frequency, Cumulative Frequency, and Cumulative Relative Frequency Distribution Class Frequency Cumulative Frequency Cumulative Relative Frequency Cumulative Percent Frequency 10 < 13 3 3/65=0.0462 4.62% 13 < 16 14 17 17/65=0.2615 26.15% 16 < 19 23 40 0.6154 61.54% 19 < 22 12 52 0.8000 80.00% 22 < 25 8 60 0.9231 92.31% 25 < 28 4 64 0.9846 98.46% 28 < 31 1 65 1.0000 100.00%

Ogive (Cumulative Frequency distribution) Ogive: A graph of a cumulative distribution Plot a point above each upper class boundary at height of cumulative frequency Connect points with line segments Can also be drawn using Cumulative relative frequencies Cumulative percent frequencies

A Percent Frequency Ogive of the Payment Times

Frequency Distribution For Hours Studying

Cumulative Frequency Distribution For Hours Studying

Dot Plots On a number line, each data value is represented by a dot placed above the corresponding scale value Dot plots are useful for detecting outliers Unusually large or small observations that are well separated from the remaining observations

Dot Plots Example

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

A set of 24 quiz scores presented as raw data and organized in a Stem-and-Leaf Display

Advantage of the Stem-and-Leaf Display

Car Mileage Example Refer to the Car Mileage Case Data in Table 2.14 The stem-and-leaf display: 29 8 30 13455677888 31 0012334444455667778899 32 01112334455778 33 03 29 + 0.8 = 29.8 33 + 0.3 = 33.3

Constructing a Stem-and-Leaf Display There are no rules that dictate the number of stem values Can split the stems as needed

Split Stems from Car Mileage: Example Starred classes (*) extend from 0.0 to 0.4 Unstarred classes extend from 0.5 to 09 29 8 30* 134 30 55677888 31* 00123344444 31 55667778899 32* 011123344 32 55778 33* 03

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

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

Comparing Two Distributions with back-to-back bar charts Back-to-Back histogram Display Comparing Two Distributions with back-to-back bar charts

Comparing Two Distributions with back-to-back bar charts Back-to-Back histogram Display Comparing Two Distributions with back-to-back bar charts

Scatter Plots Used to study relationships between two quantitative variables Place one variable on the x-axis Place a second variable on the y-axis Place dot on pair coordinates

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

A Scatter Plot Showing a Positive Linear Relationship

A Scatter Plot Showing a Little or No Linear Relationship

A Scatter Plot Showing a Negative Linear Relationship

Misleading Graphs and Charts: Scale Break Break the vertical scale to exaggerate effect Mean Salaries at a Major University, 2004 - 2007

Misleading Graphs and Charts: Horizontal Scale Effects

You can use simple mathematical operations (like averages) to create nonsensical “facts” that can drive whatever agenda you’d like. Example: the average wealth of the citizens of a particular town is $100,000, therefore they don’t need any government assistance. (The town consists of 1 stingy millionaire and 9 homeless people.)