Chapter 4-Describing Data: Displaying and Exploring Data Jie Zhang, Ph.D. Student Account and Information Systems Department College of Business Administration The University of Texas at El Paso jzhang6@utep.edu Spring, 2014 Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 1
Learning Objectives LO1 Construct a dot plot. LO2 Construct and describe a stem-and-leaf display. LO3 Identify and compute measures of position. LO4 Construct and analyze a box plot. LO5 Compute and describe the coefficient of skewness. LO6 Create and interpret a scatter diagram LO7 Develop and explain a contingency table Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 2
Dot Plots A dot plot groups the data as little as possible and the identity of an individual observation is not lost. To develop a dot plot, each observation is simply displayed as a dot along a horizontal number line indicating the possible values of the data. If there are identical observations or the observations are too close to be shown individually, the dots are piled on top of each other. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 3
Example: Dots plots for random numbers hist(x1,main="norm") Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 4
Stem-and-Leaf In Chapter 2, frequency distribution was used to organize data into a meaningful form. A major advantage to organizing the data into a frequency distribution is that we get a quick visual picture of the shape of the distribution. There are two disadvantages, however, to organizing the data into a frequency distribution: 1)The exact identity of each value is lost 2)Difficult to tell how the values within each class are distributed. One technique that is used to display quantitative information in a condensed form is the stem-and-leaf display. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 5
Stem-and-leaf display is a statistical technique to present a set of data. Each numerical value is divided into two parts. The leading digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis. Advantage of the stem-and-leaf display over a frequency distribution - the identity of each observation is not lost. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 6
Stem-and-leaf Plot Example Listed in Table 4 1 is the number of 30-second radio advertising spots purchased by each of the 45 members of the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf display. Around what values do the number of advertising spots tend to cluster? What is the fewest number of spots purchased by a dealer? The largest number purchased? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 7
The usual procedure is to sort the leaf values from the smallest to largest. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 8
Self-Review 4-1 2. The rate of return for 21 stocks is; 8.3 9.6 9.5 9.1 8.8 11.2 7.7 10.1 9.9 10.8 10.2 8.0 8.4 8.1 11.6 9.6 8.8 8.0 10.4 9.8 9.2 Organize this information into a stem-and-leaf display. (a)how many rates are less than 9.0? (b)list this rates in the 10.0 up to 11.0 category (c)what is the median? (d)what are the maximum and the minimum rates of return? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 9
Measures of Position-Lecture two The standard deviation is the most widely used measure of dispersion. Alternative ways of describing spread of data include determining the location of values that divide a set of observations into equal parts. These measures include : quartiles, (divide into four parts) deciles, (10 equal parts) and percentiles. (100 equal parts) Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 10
To formalize the computational procedure, let L p refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L 33 and if we wanted the median, the 50th percentile, then L 50. The number of observations is n, so if we want to locate the median, its position is at (n + 1)/2, or we could write this as (n + 1)(P/100), where P is the desired percentile. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 11
Percentiles - Example Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney s Oakland, California, office. $2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287 $1,940 $2,311 $2,054 $2,406 $1,471 $1,460 Locate the median, the first quartile, and the third quartile for the commissions earned. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 12
Step 1: Organize the data from lowest to largest value $1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038 $2,047 $2,054 $2,097 $2,205 $2,287 $2,311 $2,406 Step 2: Compute the first and third quartiles. Locate L25 and L75 using 25 75 L25 (15 1) 4 L75 (15 1) 12 100 100 Therefore, the first and third quartiles are located at the 4th and12th positions, respectively L L 25 75 $1,721 $2,205 Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 13
In the previous example the location formula yielded a whole number. What if there were 6 observations in the sample with the following ordered observations: 43, 61, 75, 91, 101, and 104, that is n=6, and we wanted to locate the first quartile? L 25 (6 1) 100 25 1.75 Locate the first value in the ordered array and then move.75 of the distance between the first and second values and report that as the first quartile. Like the median, the quartile does not need to be one of the actual values in the data set. The 1st and 2nd values are 43 and 61. Moving 0.75 of the distance between these numbers, the 25 th percentile is 56.5, obtained as 43 + 0.75*(61-43) Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 14
Box Plot A box plot is a graphical display, based on quartiles, that helps us picture a set of data. To construct a box plot, we need only five statistics: 1. the minimum value, 2. Q1(the first quartile), 3. the median, 4. Q3 (the third quartile), and 5. the maximum value. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 15
Boxplot - Example Alexander s Pizza offers free delivery of its pizza within 15 miles. Alex, the owner, wants some information on the time it takes for delivery. For a sample of 20 deliveries, he determined the following information: 1. Minimum value = 13 minutes 2. Q1 = 15 minutes 3. Median = 18 minutes 4. Q3 = 22 minutes 5. Maximum value = 30 minutes Develop a box plot for the delivery times. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 16
Step1: Create an appropriate scale along the horizontal axis. Step 2: Draw a box that starts at Q1 (15 minutes) and ends at Q3 (22 minutes). Inside the box we place a vertical line to represent the median (18 minutes). Step 3: Extend horizontal lines from the box out to the minimum value (13 minutes) and the maximum value (30 minutes). Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 17
Box plot in R The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973 74 models). Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 18
Skewness In Chapter 3, measures of central location (the mean, median, and mode) for a set of observations and measures of data dispersion (e.g. range and the standard deviation) were introduced Another characteristic of a set of data is the shape. There are four shapes commonly observed: 1. symmetric, 2. positively skewed, 3. negatively skewed, 4. bimodal. Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 19
Skewness - Formulas for Computing The coefficient of skewness can range from -3 up to 3. A value near -3, indicates considerable negative skewness. A value such as 1.63 indicates moderate positive skewness. A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present. Professor Karl Pearson(1857-1936) Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 20
Commonly Observed Shapes Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 21
Skewness An Example Following are the earnings per share for a sample of 15 software companies for the year 2010. The earnings per share are arranged from smallest to largest. Compute the mean, median, and standard deviation. Find the coefficient of skewness using Pearson s estimate. What is your conclusion regarding the shape of the distribution? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 22
Step 1: Compute the Mean X $ 74. 26 X n 15 $ 4. 95 Step 2 : Compute the StandardDeviation s X X n 1 2 ($ 0. 09 $ 4. 95) 2... ($ 16. 40 $ 4. 95) 15 1 2 ) $ 5. 22 Step 3 :Find the Median The middlevalue in the setof data, arrangedfrom smallestto largestis3.18 Step 4 : Compute the Skewness sk 3( X Median ) s 3($ 4. 95 $ 3. 18) $ 5. 22 1. 017 Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 23
What s does the value of skewness mean? Can you get any idea from the graph below? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 24
Self-Review 4-4 P123 A sample of five data entry clerks employed in the Horry County Tax Office revised the following number of tax records last hour: 73, 98, 60, 92, and 84 (a) Find the mean median, and the standard deviation (b) Compute the coefficient of skewness using Person s method (c) calculate the coefficient of skewness using the software method, then compare its value with Person s method (d) what is your conclusion regarding the skewness of the data? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 25
Describing the relationship between two variables-scatter Diagram Questions: Can you get some intuitive idea from the left tree graphs? Or more precisely, what the relationship between the two variable? Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 26