CHAPTER 2: Visual Description of Data to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group

Chapter 2 - Learning Objectives Convert raw data into a data array. Construct: –a frequency distribution. –a relative frequency distribution. –a cumulative relative frequency distribution. Construct a stem-and-leaf diagram. Visually represent data by using graphs and charts. © 2002 The Wadsworth Group

Chapter 2 - Key Terms Data array –An orderly presentation of data in either ascending or descending numerical order. Frequency Distribution –A table that represents the data in classes and that shows the number of observations in each class. © 2002 The Wadsworth Group

Chapter 2 - Key Terms Frequency Distribution – Class - The category – Frequency - Number in each class – Class limits - Boundaries for each class – Class interval - Width of each class – Class mark - Midpoint of each class © 2002 The Wadsworth Group

Sturges’ rule How to set the approximate number of classes to begin constructing a frequency distribution. where k = approximate number of classes to use and n = the number of observations in the data set. © 2002 The Wadsworth Group

How to Construct a Frequency Distribution 1. Number of classes Choose an approximate number of classes for your data. Sturges’ rule can help. 2. Estimate the class interval Divide the approximate number of classes (from Step 1) into the range of your data to find the approximate class interval, where the range is defined as the largest data value minus the smallest data value. 3. Determine the class interval Round the estimate (from Step 2) to a convenient value. © 2002 The Wadsworth Group

How to Construct a Frequency Distribution, cont. 4. Lower Class Limit Determine the lower class limit for the first class by selecting a convenient number that is smaller than the lowest data value. 5. Class Limits Determine the other class limits by repeatedly adding the class width (from Step 2) to the prior class limit, starting with the lower class limit (from Step 3). 6. Define the classes Use the sequence of class limits to define the classes. © 2002 The Wadsworth Group

Converting to a Relative Frequency Distribution 1. Retain the same classes defined in the frequency distribution. 2. Sum the total number of observations across all classes of the frequency distribution. 3. Divide the frequency for each class by the total number of observations, forming the percentage of data values in each class. © 2002 The Wadsworth Group

Forming a Cumulative Relative Frequency Distribution 1. List the number of observations in the lowest class. 2. Add the frequency of the lowest class to the frequency of the second class. Record that cumulative sum for the second class. 3. Continue to add the prior cumulative sum to the frequency for that class, so that the cumulative sum for the final class is the total number of observations in the data set. © 2002 The Wadsworth Group

Forming a Cumulative Relative Frequency Distribution, cont. 4. Divide the accumulated frequencies for each class by the total number of observations -- giving you the percent of all observations that occurred up to an including that class. An Alternative: Accrue the relative frequencies for each class instead of the raw frequencies. Then you don’t have to divide by the total to get percentages. © 2002 The Wadsworth Group

Example: Problem 2.53 The average daily cost to community hospitals for patient stays during 1993 for each of the 50 U.S. states was given in the next table. –a) Arrange these into a data array. –b) Construct a stem-and-leaf display. –*) Approximately how many classes would be appropriate for these data? [* not in textbook ] –c & d) Construct a frequency distribution. State interval width and class mark. –e) Construct a histogram, a relative frequency distribution, and a cumulative relative frequency distribution. © 2002 The Wadsworth Group

Problem The Data AL $775HI 823MA 1,036NM 1,046SD 506 AK 1,136ID 659MI 902NY 784TN 859 AZ 1,091IL 917MN 652NC 763TX 1,010 AR 678IN 898MS 555ND 507UT 1,081 CA 1,221IA 612MO 863OH 940VT 676 CO 961KS 666MT 482OK 797VA 830 CT 1,058KY 703NE 626OR 1,052WA 1,143 DE 1,024LA 875NV 900PA 861WV 701 FL 960ME 738NH 976RI 885WI 744 GA 775MD 889NJ 829SC 838WY 537 © 2002 The Wadsworth Group

Problem (a) Data Array CA 1,221TX 1,010RI 885NY 784KS 666 WA 1,143NH 976LA 875AL 775ID 659 AK 1,136CO 961MO 863GA 775MN 652 AZ 1,091FL 960PA 861NC 763NE 626 UT 1,081CH 940TN 859WI 744IA 612 CT 1,058IL 917SC 838ME 738MS 555 OR 1,052MI 902VA 830KY 703WY 537 NM 1,046NV 900NJ 829WV 701ND 507 MA 1,036IN 898HI 823AR 678SD 506 DE 1,024MD 889OK 797VT 676MT 482 © 2002 The Wadsworth Group

Problem (b) The Stem-and-Leaf Display Stem-and-Leaf DisplayN = 50 Leaf Unit: , , 81, 58, 52, 46, 36, 24, , 61, 60, 40, 17, 02, 00 (11) 898, 89, 85, 75, 63, 61, 59, 38, 30, 29, , 84, 75, 75, 63, 44, 38, 03, , 76, 66, 59, 52, 26, , 37, 07, Range: $482 - $1,221 © 2002 The Wadsworth Group

Problem Continued To approximate the number of classes we should use in creating the frequency distribution, use Sturges’ Rule, n = 50: Sturges’ rule suggests we use approximately 7 classes. © 2002 The Wadsworth Group

Constructing the Frequency Distribution Step 1. Number of classes –Sturges’ Rule: approximately 7 classes. The range is: $1,221 – $482 = $739 $739/7 ­ $106 and $739/8 ­ $92 Steps 2 & 3. The Class Interval –So, if we use 8 classes, we can make each class $100 wide. © 2002 The Wadsworth Group

Constructing the Frequency Distribution Step 4. The Lower Class Limit –If we start at $450, we can cover the range in 8 classes, each class $100 in width. The first class : $450 up to $550 Steps 5 & 6. Setting Class Limits $450 up to $550$850 up to $950 $550 up to $650$950 up to $1,050 $650 up to $750 $1,050 up to $1,150 $750 up to $850 $1,150 up to $1,250 © 2002 The Wadsworth Group

Problem (c) & (d) Average daily cost NumberMark $450 – under $5504$500 $550 – under $650 3$600 $650 – under $750 9$700 $750 – under $850 9$800 $850 – under $950 11$900 $950 – under $1,050 7 $1,000 $1,050 – under $1,150 6 $1,100 $1,150 – under $1,250 1 $1,200 Interval width: $100 © 2002 The Wadsworth Group

Problem (e) The Histogram © 2002 The Wadsworth Group

Problem The Relative Frequency Distribution Average daily cost Number Rel. Freq. $450 – under $550 44/50 =.08 $550 – under $650 33/50 =.06 $650 – under $750 99/50 =.18 $750 – under $850 99/50 =.18 $850 – under $ /50 =.22 $950 – under $1,050 77/50 =.14 $1,050 – under $1,150 66/50 =.12 $1,150 – under $1,250 11/50 =.02 © 2002 The Wadsworth Group

Problem (e) The Percentage Polygon © 2002 The Wadsworth Group

Problem The Cumulative Frequency Distribution Average daily cost Number Cum. Freq. $450 – under $ $550 – under $ $650 – under $ $750 – under $ $850 – under $ $950 – under $1, $1,050 – under $1, $1,150 – under $1, © 2002 The Wadsworth Group

Problem The Cumulative Relative Frequency Distribution Average daily cost Cum.Freq. Cum.Rel.Freq. $450 – under $550 44/50 =.02 $550 – under $650 77/50 =.14 $650 – under $ /50 =.32 $750 – under $ /50 =.50 $850 – under $ /50 =.72 $950 – under $1, /50 =.86 $1,050 – under $1, /50 =.98 $1,150 – under $1, /50 = 1.00 © 2002 The Wadsworth Group

Problem (e) The Percentage Ogive (Less Than) © 2002 The Wadsworth Group

The Scatter Diagram A scatter diagram is a two-dimensional plot of data representing values of two quantitative variables. x, the independent variable, on the horizontal axis y, the dependent variable, on the vertical axis Four ways in which two variables can be related: 1. Direct2. Inverse3. Curvilinear 4. No relationship © 2002 The Wadsworth Group

An Example: Problem 2.38 For 6 local offices of a large tax preparation firm, the following data describe x = service revenues and y = expenses for supplies, freight, postage, etc. Draw a scatter diagram representing the data. Does there appear to be any relationship between the variables? If so, is the relationship direct or inverse? © 2002 The Wadsworth Group

Problem 2.38, continued © 2002 The Wadsworth Group There appears to be a direct relationship between the service revenue and the office expenses incurred.