Chapter 2 Graphs, Charts, and Tables – Describing Your Data Business Statistics Department of Quantitative Methods & Information Systems Dr. Mohammad Zainal QMIS 120

Chapter Goals After completing this chapter, you should be able to: Construct a frequency distribution both manually and with a computer Construct and interpret a histogram Create and interpret bar charts, pie charts, and stem-and-leaf diagrams Present and interpret data in line charts and scatter diagrams QMIS 120, by Dr. M. Zainal Chap 2-2

Raw data Ages (in years) of 20 students selected from CBA are reported in the way they are collected. The data values are recorded in the following table. Ages of 20 Students QMIS 120, by Dr. M. Zainal Chap 2-3

Raw data The same students were asked about their status. The responses of the sample are recorded in the following table Status of 20 Students SSJSFJFSFJ SSFJFJSJJF QMIS 120, by Dr. M. Zainal Chap 2-4

Frequency Distributions What is a Frequency Distribution? A frequency distribution is a list or a table … containing the values of a variable (or a set of ranges within which the data fall)... and the corresponding frequencies with which each value occurs (or frequencies with which data fall within each range) QMIS 120, by Dr. M. Zainal Chap 2-5

Frequency Distributions Weekly Earnings of 100 Employees of a company Number of employees f Weekly Earnings (dollars) 9401 to to to to to to 1600 Variable First class Frequency Of first class Frequency Column Lower limit of the sixth class Upper limit of the sixth class QMIS 120, by Dr. M. Zainal Chap 2-6

Why Use Frequency Distributions? A frequency distribution is a way to summarize data The distribution condenses the raw data into a more useful form... and allows for a quick visual interpretation of the data QMIS 120, by Dr. M. Zainal Chap 2-7

Frequency Distribution: Discrete Data Discrete data: possible values are countable Example: An advertiser asks 200 customers how many days per week they read the daily newspaper. Row Data 5,6,1,2,4,5,7,2,3,5,1,3,2,5,0,2,2,0,7,7,1,2,4,3,5,6,7,1,1,1,1,2,5,0,0,0,1,20,7,5,3,6,2,1,6,2,1,4,2,4,5,3,1,0,2,3,6,5,7,4,1,2,3,5,6,1,0, 0,0,0,0,1,1,1,2,3,5,1,4…………………….. QMIS 120, by Dr. M. Zainal Chap 2-8

Frequency Distribution: Discrete Data Number of days read Frequency Total200 It is called “Single-Value” approach QMIS 120, by Dr. M. Zainal Chap 2-9

Relative Frequency Relative Frequency: What proportion is in each category? Number of days read Frequency Relative Frequency Total % of the people in the sample report that they read the newspaper 0 days per week QMIS 120, by Dr. M. Zainal Chap 2-10

Frequency Distribution: Discrete Data Example: Construct a frequency distribution table for the following data Home RunsTeamHome RunsTeam 139Milwaukee152Anaheim 167Minnesota165Arizona 162Montreal164Atlanta 160New York Mets165Baltimore 223New York Yankees177Boston 205Oakland200Chicago Cubs 165Philadelphia217Chicago White Sox 142Pittsburgh169Cincinnati 175St. Louis192Cleveland 136San Diego152Colorado 198San Francisco124Detroit 152Seattle146Florida 133Tampa Bay167Houston 230Texas140Kansas City 187Toronto155Los Angeles Home Runs Hit by Major League Baseball Teams During the 2002 Season QMIS 120, by Dr. M. Zainal Chap 2-11

Frequency Distribution: Discrete Data Number of classes: C = log 30 = 5.87 ≈ 5 or 6 Class width: W = (230 – 124) / 5 = 21.2 ≈ 22 Starting point: x 22 = 234 checks! Home Runs 124 – – – – – 233 Total f QMIS 120, by Dr. M. Zainal Chap 2-12

Frequency Distribution: Continuous Data Continuous Data: may take on any value in some interval Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27 Temperature is a continuous variable because it could be measured to any degree of precision desired QMIS 120, by Dr. M. Zainal Chap 2-13

Grouping Data by Classes Sort raw data from low to high: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: = 46 Select number of classes: 5 (usually between 5 and 20) Compute class width: 10 (46/5 then round off) Determine class boundaries:10, 20, 30, 40, 50 (Sometimes class midpoints are reported: 15, 25, 35, 45, 55) Count the number of values in each class QMIS 120, by Dr. M. Zainal Chap 2-14

Frequency Distribution Example Data from low to high: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class Frequency 10 but under but under but under but under but under Total Relative Frequency Frequency Distribution QMIS 120, by Dr. M. Zainal Chap 2-15

Frequency Histograms The classes or intervals are shown on the horizontal axis frequency is measured on the vertical axis Bars of the appropriate heights can be used to represent the number of observations within each class Such a graph is called a histogram QMIS 120, by Dr. M. Zainal Chap 2-16

Class Midpoints Histogram Example Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 No gaps between bars, since continuous data Class Endpoints Chap 2-17

Questions for Grouping Data into Classes 1.How wide should each interval be? (How many classes should be used?) 2.How should the endpoints of the intervals be determined? Often answered by trial and error, subject to user judgment The goal is to create a distribution that is neither too "jagged" nor too "blocky” Goal is to appropriately show the pattern of variation in the data QMIS 120, by Dr. M. Zainal Chap 2-18

How Many Class Intervals? Many (Narrow class intervals) may yield a very jagged distribution with gaps from empty classes Can give a poor indication of how frequency varies across classes Few (Wide class intervals) may compress variation too much and yield a blocky distribution can obscure important patterns of variation. (X axis labels are upper class endpoints) QMIS 120, by Dr. M. Zainal Chap 2-19 (X axis labels are upper class endpoints)

General Guidelines Number of Data Points Number of Classes under – – over Class widths can typically be reduced as the number of observations increases Distributions with numerous observations are more likely to be smooth and have gaps filled since data are plentiful QMIS 120, by Dr. M. Zainal Chap 2-20

Class Width The class width is the distance between the lowest possible value and the highest possible value for a frequency class The class width is Largest Value - Smallest Value Number of Classes W = QMIS 120, by Dr. M. Zainal Chap 2-21

Histograms in Excel Select “Data” Tab 1 Choose Histogram 3 2 Data Analysis QMIS 120, by Dr. M. Zainal Chap 2-22

Histograms in Excel (continued) 4 Input data and bin ranges Select Chart Output QMIS 120, by Dr. M. Zainal Chap 2-23

Ogives An Ogive is a graph of the cumulative relative frequencies from a relative frequency distribution Ogives are sometime shown in the same graph as a relative frequency histogram QMIS 120, by Dr. M. Zainal Chap 2-24

Ogives 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Add a cumulative relative frequency column: Class Frequency 10 but under but under but under but under but under Total Relative Frequency Frequency Distribution (continued) Cumulative Relative Frequency QMIS 120, by Dr. M. Zainal Chap 2-25

Class Midpoints Ogive Example Cumulative Frequency (%) / Ogive Class Endpoints Chap 2-26

Excel will show the Ogive graphically if the “Cumulative Percentage” option is selected in the Histogram dialog box Ogives in Excel QMIS 120, by Dr. M. Zainal Chap 2-27

Other Graphical Presentation Tools Categorical Data Bar Chart Stem and Leaf Diagram Pie Charts Quantitative Data QMIS 120, by Dr. M. Zainal Chap 2-28

Bar and Pie Charts Bar charts and Pie charts are often used for qualitative (category) data Height of bar or size of pie slice shows the frequency or percentage for each category QMIS 120, by Dr. M. Zainal Chap 2-29

Bar Chart Example 1 (Note that bar charts can also be displayed with vertical bars) QMIS 120, by Dr. M. Zainal Chap 2-30

Bar Chart Example 2 Number of days read Frequency Total200 QMIS 120, by Dr. M. Zainal Chap 2-31

Pie Chart Example Percentages are rounded to the nearest percent Current Investment Portfolio Savings 15% CD 14% Bonds 29% Stocks 42% Investment Amount Percentage Type (in thousands $) Stocks Bonds CD Savings Total (Variables are Qualitative) QMIS 120, by Dr. M. Zainal Chap 2-32

Tabulating and Graphing Multivariate Categorical Data Investment in thousands of dollars Investment Investor A Investor B Investor C Total Category Stocks Bonds CD Savings Total QMIS 120, by Dr. M. Zainal Chap 2-33

Tabulating and Graphing Multivariate Categorical Data Side by side charts (continued) QMIS 120, by Dr. M. Zainal Chap 2-34

Side-by-Side Chart Example Sales by quarter for three sales territories: QMIS 120, by Dr. M. Zainal Chap 2-35

Dot Plot A One of the simplest methods for graphing and understanding quantitative data is to create a dot plot. A horizontal axis shows the range of values for the observations. Each data point is represented by a dot placed above the axis. QMIS 120, by Dr. M. Zainal Chap 2-36

Dot Plot Dot plots can help us detect outliers (also called extreme values) in a data set. Outliers are the values that are extremely large or extremely small with respect to the rest of the data values. QMIS 120, by Dr. M. Zainal Chap 2-37

Dot Plot Example : The following table lists the number of runs batted in (RBIs) during the 2004 Major League Baseball playoffs by members of the Boston Red Sox team with at least one at-bat. Create a dot plot for these data. QMIS 120, by Dr. M. Zainal Chap 2-38

Dot Plot Step 1. First we draw a horizontal line that includes the minimum and the maximum values in this data set. Step 2. Place a dot above the value on the numbers line that represents each RBI listed in the table Outlier QMIS 120, by Dr. M. Zainal Chap 2-39

Stem and Leaf Diagram Another simple way to see distribution details from qualitative data METHOD 1. Separate the sorted data series into leading digits (the stem) and the trailing digits (the leaves) 2. List all stems in a column from low to high 3. For each stem, list all associated leaves QMIS 120, by Dr. M. Zainal Chap 2-40

Example: Here, use the 10’s digit for the stem unit: Data sorted from low to high: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, is shown as 35 is shown as Stem Leaf QMIS 120, by Dr. M. Zainal Chap 2-41

Example: Completed Stem-and-leaf diagram: Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 28, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 StemLeaves QMIS 120, by Dr. M. Zainal Chap 2-42

Using other stem units Using the 100’s digit as the stem: Round off the 10’s digit to form the leaves 613 would become would become becomes 12 2 Stem Leaf QMIS 120, by Dr. M. Zainal Chap 2-43

Line charts show values of one variable vs. time Time is traditionally shown on the horizontal axis Scatter Diagrams show points for bivariate data one variable is measured on the vertical axis and the other variable is measured on the horizontal axis A trend line is a line that provides an approximation of that relationship. Line Charts and Scatter Diagrams QMIS 120, by Dr. M. Zainal Chap 2-44

Line Chart Example Year Inflation Rate QMIS 120, by Dr. M. Zainal Chap 2-45

Scatter Diagram Example Volume per day Cost per day Chap 2-46

Types of Relationships Linear Relationships QMIS 120, by Dr. M. Zainal Chap 2-47

Curvilinear Relationships Types of Relationships (continued) QMIS 120, by Dr. M. Zainal Chap 2-48

No Relationship Types of Relationships (continued) QMIS 120, by Dr. M. Zainal Chap 2-49

Cross-tabulation Example: Draw a scatter diagram for the following data which lists the total amount spent in KD by costumers in a restaurant. x (person) y (KD) QMIS 120, by Dr. M. Zainal Chap 2-50

Cross-tabulation A cross tab is a tabular summary of data of two variables. They are usually presented in a matrix format. Not like a frequency distribution (one variable). A contingency table describes the distribution of two or more variables simultaneously. Each cell shows the number of respondents that gave a specific combination of responses It can be used with any level of data (What are they?) QMIS 120, by Dr. M. Zainal Chap 2-51

Cross-tabulation example Example: In a survey of the quality rating and the meal price conducted by a consumer restaurant review agency, the following table was produced: RestaurantQuality ratingMeal Price 1Good18 2Very Good22 3Good28 4Excellent38 5Very Good33 6Good28... QMIS 120, by Dr. M. Zainal Chap 2-52

Cross-tabulation example Quality rating is a qualitative variable with the rating categories of good, very good and excellent Meal price Quality rating Total Good Very Good Excellent Total QMIS 120, by Dr. M. Zainal Chap 2-53

Cross-tabulation example Also, we can find the row percentage Meal price Quality rating Total Good Very Good Excellent QMIS 120, by Dr. M. Zainal Chap 2-54

Cross-tabulation example Dividing the totals in the right margin of the cross tab by the grand total provides relative and percentage frequency distribution for the quality rating variable. Quality ratingRelative frequencyPercentage frequency Good Very Good Excellent Total QMIS 120, by Dr. M. Zainal Chap 2-55

Cross-tabulation example Try it for the meal price (column totals) Meal price Quality rating Total Good Very Good Excellent QMIS 120, by Dr. M. Zainal Chap 2-56

Cross-tabulation example Example: The following data are for 30 observations involving two qualitative variables x (A, B and C) and y (1 and 2). Obs.xy xy xy 1A111A121C2 2B112B122B1 3B113C223C2 4C214C224A1 5B115C225B1 6C216B226C2 7B117C127C2 8C218B128A1 9A119C129B1 10B120B130B2 1- Construct a cross tabulation for the data 2- Calculate the row percentages QMIS 120, by Dr. M. Zainal Chap 2-57

Cross-tabulation example y x12Total A505 B11213 C21012 Total y x12Total A1000 B C QMIS 120, by Dr. M. Zainal Chap 2-58

Cross-tabulation example XRelative frequency Percentage frequency A5/30= B13/30= C12/30= Total YRelative frequency Percentage frequency 118/30= /30=.4040 Total QMIS 120, by Dr. M. Zainal Chap 2-59

Chapter Summary Data in raw form are usually not easy to use for decision making -- Some type of organization is needed:  Table  Graph Techniques reviewed in this chapter: Frequency Distributions, Histograms, and Ogives Bar Charts and Pie Charts Stem and Leaf Diagrams Line Charts and Scatter Diagrams QMIS 120, by Dr. M. Zainal Chap 2-60

Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. QMIS 120, by Dr. M. Zainal Chap 2-61