Chapter 2 Describing Data Frequency Distributions PDF

Describing Data: Frequency Tables, Frequency Distributions, and Graphic Presentation CHAPTER 2 N A T I O N A L L Y R A N K E D SPU is the only pri...

Describing Data: Frequency Tables, Frequency Distributions, and Graphic Presentation CHAPTER 2 N A T I O N A L L Y R A N K E D SPU is the only private university in the Pacific Northwest to make U.S. News & World Report’s 2018 “Best National Universities” list. 1-1  Learning Objectives LO2-1 Summarize qualitative variables with frequency and relative frequency tables LO2-2 Display a frequency table using a bar or pie chart LO2-3 Summarize quantitative variables with frequency and relative frequency distributions LO2-4 Display a frequency distribution using a histogram or frequency polygon 2-2  Getting started Techniques used to describe a set of data are called Descriptive Statistics Descriptive statistics are used to: Organize and show the general pattern of data Identify where values tend to concentrate Expose extreme or unusual data One of the techniques used to describe data is called: Frequency table. 2-3  Constructing Frequency Tables FREQUENCY TABLE A grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class. Mutually exclusive means the data can be divided into distinct categories. Collectively exhaustive means there is a class for each value. 2-4  Constructing Frequency Tables Let’s construct a frequency table of the location of cars! 2-5  Constructing Frequency Tables To construct a frequency table First sort the data into classes Count the number in each class and report as the class frequency 2-6  Constructing Frequency Tables Convert each frequency to a relative frequency Each of the class frequencies is divided by the total number of observations Shows the fraction of the total number observations in each class 2-7 2-8 Graphic Presentation of Qualitative Data BAR CHART A graph that shows the qualitative classes on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are proportional to the heights of the bars. Use a bar chart when you wish to compare the number of observations for each class of a qualitative variable. 2-9 Graphic Presentation of Qualitative Data PIE CHART A chart that shows the proportion or percentage that each class represents of the total number of frequencies. Use a pie chart when you wish to compare relative differences in the percentage of observations for each class of a qualitative variable.  Constructing Frequency Distributions FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class. This is a four-step process 1. Decide on the number of classes 2. Determine the class interval 3. Set the individual class limits 4. Tally the data into classes and determine the number of the observations in each class 2-10  Frequency Distributions Step 1: Decide on the number of classes (k). Use the 2k > n rule, where n=180 k is the number of classes n is the number of values in the data set 2k > 180, let k = 8 So use 8 classes 2-11  Frequency Distributions (2 of 4) Step 2 Determine the class interval, i Round up to some convenient number So decide to use an interval of $400 The interval is also referred to as the class width 2-12  Frequency Distributions Step 3 Set the individual class limits Lower limit should be rounded to an easy-to-read number when possible; It should also be lower than the lowest values in the data. 2-13  Frequency Distributions Step 4: Tally the individual data into the classes and determine the number of observations in each class The number of observations is the class frequency 2-14  Relative Frequency Distributions To find the relative frequencies, simply take the class frequency and divide by the total number of observations 2-15  Practice Problem Wachesaw Manufacturing Inc. produced the following number of units in the last 16 days. The information is to be organized into a frequency distribution. a. How many classes would you recommend? b. What class interval would you suggest? c. What lower limit would you recommend for the first class? d. Organize the information into a frequency distribution and determine the relative frequency distribution. 2-16  Solution a. 2^4 = 16 suggests 5 classes. b. i ≥ (31 − 25)/5 = 1.2. Use interval of 1.5. c. 24 d.Units f Relative Frequency 24.0 up to 25.5 2 0.125 25.5 up to 27.0 4 0.250 27.0 up to 28.5 8 0.500 28.5 up to 30.0 0 0.000 30.0 up to 31.5 2 0.125 Total 16 1.000 2-17 2-18 Graphic Presentation of a Frequency Distribution HISTOGRAM A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.  A histogram shows the shape of a distribution for a continuous variable.  Each class is depicted as a rectangle, with the height of the bar representing the number in each class. 2-19 Graphical Presentation of a Frequency Distribution  A frequency polygon, similar to a histogram, also shows the shape of a distribution  These are good to use when comparing two or more distributions  Comparing two Distributions  Typical vehicle profit at Fowler (~ $2400) is more expensive than at Applewood (~$2000).  There is less variation or dispersion in profits at Fowler Motors than at Applewood. 2-20 2-21 Cumulative Frequency Distributions  To construct a cumulative frequency distribution, add each frequency to the frequencies before it  This shows how many values have accumulated as you move from one class down to the next class  Cumulative Relative Frequency Distribution To construct a cumulative relative frequency distribution, we divide the cumulative frequencies by the total number of observations 2-22 2-23 Cumulative Frequency Polygon  Data Visualization: Best Practices in Creating Effective Graphical Displays Data visualization is the use of graphical displays to summarize and present information about a data set. The goal is to communicate, as effectively and clearly as possible, the key information about the data. 2-24  Creating Effective Graphical Displays Creating effective graphical displays is as much art as it is science. Here are some guidelines: 1. Give the display a clear and concise title. 2. Keep the display simple. 3. Clearly label each axis and provide the units of measure. 4. If colors are used, make sure they are distinct. 5. Provide a legend if multiple colors or line types are used. 2-25  Creating Effective Graphical Displays Example Figure 3: School Completion Rates in Kasai and Kasai Central by Sex in 2017 Source: INS 2021 2-26  Practical Applications – Case Study of Marketing Analyst Scenario: Assume a Marketing Analyst at a retail company is asked to categorize customer purchasing behavior for targeted marketing campaigns. The analyst could use frequency tables and create bar graphs. This could allow them to identify that a large segment of customers frequently buys eco-friendly products, leading to a successful targeted promotion for sustainable goods.  Practical Applications – Student Business Scenario: Assume you have a business of performing as a violinist at different events. You could use bar graphs to check the trends in sales data. This could allow you to identify periods with the highest sales, leading to a successful targeted advertisement.