Chapter 4-Describing Data: Displaying and Exploring...

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

[email protected]

Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP 1

mailto:[email protected]

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


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.


Dot Plots - Examples• The Service Departments at Tionesta Ford Lincoln Mercury and Sheffield Motors,

Inc., two of the four Applewood Auto Group Dealerships, were both open 24 days last month. Listed below is the number of vehicles serviced during the 24 working at the two Dealerships. Construct dot plots and report summary statistics to compare the two dealerships.



0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30

compare Tinesta and Sheffield

Column1 Column2

Mean 31.291667 Mean 35.791667

Standard Error 0.8394252 Standard Error 0.8866546

Median 32 Median 35.5

Mode 32 Mode 36

Standard Deviation 4.1123268 Standard Deviation 4.3437028

Sample Variance 16.911232 Sample Variance 18.867754

Kurtosis -0.634428 Kurtosis -0.410937

Skewness -0.133631 Skewness 0.6333923

Range 16 Range 14

Minimum 23 Minimum 30

Maximum 39 Maximum 44

Sum 751 Sum 859

Count 24 Count 24

• How to Make a Dotplot

• Descriptive Statistics in Excel

• How to Make Scatterplot with Groups


https://www.youtube.com/watch?v=N7HHmTpccZI

https://www.youtube.com/watch?v=Czgl_VZH370

https://www.youtube.com/watch?v=Czgl_VZH370

https://www.youtube.com/watch?v=PTRTva-assc

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.


• 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.


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?



The usual procedure is to sort the leaf values from the smallest to largest.

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?

Measures of Position

• 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)



To formalize the computational procedure, let Lp refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L33 and if we wanted the median, the 50th percentile, then L50.

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.

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.



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

205,2$

721,1$

lyrespective positions,

12th and4th at the located are quartiles thirdandfirst theTherefore,

12100

75)115( 4

100

25)115(

75

25

7525

L

L

LL

• 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?


75.1100

25)16(25 L

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 25th percentile is 56.5, obtained as 43 + 0.75*(61- 43)

Example: percentiles with Excel


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.


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.


• 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).


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).


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.


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)


Commonly Observed Shapes


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?


0171225

18395433

225115

9544016954090

1

95415

2674

222

..$

).$.($)(

Skewness the Compute :4 Step

3.18 is largest to smallest from arranged data, of set the in value middle The

Median the Find :3 Step

.$)).$.($...).$.($

Deviation Standard the Compute :2 Step

.$.$

Mean the Compute :1 Step

s

MedianXsk

n

XXs

n

XX


What’s does the value of skewness mean? Can you get any idea from the graph below?

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?


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?

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