Doing Statistics for Business Data, Inferences, and Decision

Doing Statistics for Business

Data, Inference, and Decision Making

Marilyn K. Pelosi

Theresa M. Sandifer

Chapter 4

Numerical

Descriptors of

Data

1


Chapter 4 Objectives

 Numerical Measures of Center: The Mean, the Median, and the Mode

 Numerical Measures of Variability: The

Range & the Standard Deviation

 Describing a Set of Data: The Empirical

Rule & Boxplots

2


Chapter 4 Objectives

(con’t)

 Measures of Relative Standing:

Percentiles

Percentile Rank

 Identifying Outliers: z -scores

Boxplots

3


A Statistic is a numerical descriptor that is calculated from sample data and is used to describe the sample. Statistics are usually represented by Roman letters.

4


A Parameter is a numerical descriptor that is used to describe a population.

Parameters are usually represented by

Greek letters.

5


The Sample Mean is the center of balance of a set of data, and is found by adding up all of the data values and dividing by the number of observations.

6


The Population Mean is represented by the Greek letter



(mu).

7


TRY IT NOW!

Restaurant Table Times

Calculating the Sample Mean

A restaurant is trying to decide whether it has an adequate number of tables available. The restaurant owner decides that she would like some information on the amount of time a table is occupied by a customer. She collects data on the length of time a customer occupies a table for a random sample of 10 customers and obtains the following data.

8


.

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Restaurant Table Times

Calculating the Sample Mean (con’t)

Customer 1 2 3 4 5 6 7 8 9 10

Time (min.) 59.3 58.6 62.7 65.4 59.0 67.3 62.8 68.1 59.4 63.7

Calculate the sample mean for the length of time a table is occupied.

9


The Sample Median is the value of the middle observation in an ordered set of data.

10


TRY IT NOW!

Town Hall Traffic

Calculating the Sample Median

In the past few years the town council of a small town has received complaints that it has become increasingly difficult to cross the main street in town near the library. The council decides to look at traffic flow on the street. It selects a site directly in front of the library where most people try to cross the road and records the number of cars that pass the point in a two-minute period.

11


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Calculating the Sample Median (con’t)

This is done for 10 two-minute periods at 3:00 p.m. over several weeks and the following data are obtained.

Number of cars 20 27 29 28 37 23 21 28 29 28

Find the median number of cars that pass the site in two minutes.

Remember to SORT the data before you locate the median!

12


Figure 4.2

Mean and

Median for a Symmetric

Distribution

Mean

Median

13


Figure 4.3

Mean and

Median for Skewed

Distributions

Left skew Right skew

14


TRY IT NOW!

Airline Cancellations

Comparing the Mean and the Median

An airline company is wondering about the number of cancellations that it receives for a particular business commuter flight. The airline takes a random sample of 15 days from the first quarter of the year and obtains the following data:

# of cancellations 4 9 9 12 12 13 14 14 15 15 16 16 17 17 24

15


TRY IT NOW!

Airline Cancellations

Comparing the Mean and the Median

(con’t)

Find the mean and median for the # of cancellations for the commuter f light.

When compared, do the data appear symmetric or skewed?

Make a dotplot of the data.

From the dotplot, do the data appear symmetric or skewed?

Note: the data have been sorted for you.

16


Discovery Exercise 4.1

The Trimmed Mean

Part I. Investigating the Data

In a report to the administration of a large university, the Psychology Department states that the average class size is greater than the 35 students per class allowed by the university charter.

The report indicates that the mean class size is 39.4.

17




Part I. Investigating the Data (con’t)

No data are appended to the report, but you can obtain the current enrollments easily. The data you find are:

3 14 22 26 42

3 15 23 27 45

5 15 24 28 45

9 17 24 28 190

11 21 25 36 193

13 22 26 38 193

18




Part I. Investigating the Data (con’t)

A. Do you think that the mean is a god measure of center for these data? Why or why not?

B. By simply studying the data, what do you think a typical class size for the Psychology Department is?

C. What is the median of the data? Is this closer to what you thought?

D. Compare the mean and median. What doe the comparison lead you to believe about the data?

E. Display the data graphically. Do you still think the same thing?

19


The Sample Mode is the data value that has the highest frequency of occurrence in the sample.

20


The Modal Class is the class interval in a frequency distribution or histogram that has the highest frequency.

21


Figure 4.4

Histogram of Bimodal Data

20

Frequency 10

0

3 5 7 9 11 13

X

15 17 19 21 23

22



Investigating Variability

The table contains air-quality data collected by the

Environmental Protection Agency. The data show the number of days in which the ozone level was dangerous for 14 major U.S. cities in 2000.

City

Atlanta

Boston

Chicago

Dallas

Denver

Houston

Kansas City

Number of unhealthy days

18

0

0

5

0

94

0

23




(con’t)

City

Los Angeles

New York

Philadelphia

Pittsburgh

San Francisco

Seattle

Washington, DC

Number of unhealthy days

1

13

2

3

0

0

0

24




(con’t)

A. Display these data using a dotplot.

B. Find the typical number of unhealthy days by calculating the average value.

C. Can you expect every observation to be typical? Why not?

25


A Sample Range, R, is the difference between the maximum and minimum observations in the sample.

26


TRY IT NOW!

Restaurant Table Time

Calculating the Sample Range

The restaurant looking at the turnaround time for its tables, wonders how variable the occupation time for a table really is. The data the restaurant had collected are:

Time (min) 59.3 58.6 62.7 65.4 59.0 67.3 62.8 68.1 59.4 63.7

27


TRY IT NOW!

Restaurant Table Time

Calculating the Sample Range (con’t)

What is the range of turnaround times?

Previously you calculated the mean turnaround time to be 62.6 minutes.

Using this information and the value for the range, what would the restaurant expect as its lowest turnaround time? Its highest turnaround time?

28


The Sample Variance, s 2 , is the average of the squared deviations of the data values from the sample mean.

29


The Sample Standard Deviation, s, is the positive square root of the sample variance.

30


The population variance and standard deviation are represented by the Greek letter,



(sigma), where



2 is the population variance and

 is the population standard deviation

31


The Empirical Rule says that for a mound-shaped, symmetric distribution:





 about 68% of all data values are within one standard deviation of the mean about 95% of all observations are within two standard deviations of the mean almost all (more than 99%) of the observations are within three standard deviations of the mean.

32


TRY IT NOW!

Town Hall Traffic Flow

Calculating the Sample Variance and Standard Deviation

The town council looking at the traffic flow problem has seen reports that use the standard deviation, and wants to use it to describe the variability of traffic flow. The data are:

Number of Cars 20 27 29 28 37 23 21 28 29 28

What is the sample standard deviation of the traffic flow?

Use whatever method you feel most comfortable with. If you have a stat. calc. learn how to use it now

33








68%

95%

99%







x

Figure 4.5

The Empirical Rule

34


TRY IT NOW!

Loan Processing

The Empirical Rule

Errors in filling out loan applications can lead to delays in having the loans approved. Bank employees must contact the applicants to correct the errors. This sometimes requires multiple contacts. To understand the extent to which the errors affect the application process a bank collected data on the number of follow-up contacts required before a loan could be processed.

35


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Loan Processing

The Empirical Rule (con’t)

The bank looked at 25 different applications and found:

1

1

0 1

0

1

2

2

2

2

Make a dotplot of the data.

2

2

3

3

3

3

4

4

4

4

4

4

5

5

7

36


TRY IT NOW!

Loan Processing

The Empirical Rule (con’t)

From the dotplot, do you think that the assumption that the data have a symmetric, bell-shaped distribution is a reasonable one?

Find the mean and standard deviation of the data.

According to the empirical rule, between what two values should 68% of the observations fall?

Between what two values should 95% of the observations fall?

Between what two values should more than 99% of the observations fall?

37


A z-score measures the number of standard deviations that a data value is from the mean.

38


TRY IT NOW!


Calculating z-Scores

The town that was looking at traffic flow in front of the town hall wonders if the observation of 37 cars is unusual. Although the town officials know that their sample size of 10 cars is not large enough to ensure accuracy; they want to use zscores to look at the data:

Number of Cars 20 27 29 28 37 23 21 28 29 28

What is the z -score for the observation of 37 cars?

39


TRY IT NOW!


Calculating z-Scores

Comparing the z -score to the empirical rule, do you think that the value is unusual?

40


The Pth Percentile of a data set is the value that has p% of the data at or below it.

41


The Percentile Rank of a value is the percentage of the data in the sample that are at or below the value of interest.

42


TRY IT NOW!

Aptitude Test Scores

Calculating the Percentile Rank

A group of employees at a manufacturing facility take a test to determine their aptitude for training. The tests are scored on a

400-point scale and are shown here in increasing order:

185 227 241 257 281 299 314 329

195 228 243

196 234 248

199 238 250

223 241 253

261

269

271

272

283

283

291

297

304

307

309

310

318

319

322

328

333

335

349

353

43


TRY IT NOW!

Aptitude Test Scores

Calculating the Percentile Rank

One of the employees who scored 283 wants to know how he stands relative to the other employees who took the exam.

What is the percentile rank for the employee’s score?

What is the percentile rank of the employee that scored 319?

44


The first quartile, Q

1

, is the value in the sample that has 25% of the data at or below it.

45


The third quartile, Q

3

, is the value in the sample that has 75% of the data at or below it.

46


TRY IT NOW!

Training Aptitude

Finding the Quartiles

The company looking at training aptitude wants to give employees who scored in the top 25% on the test the opportunity to attend a seminar on training. The test scores are:

185 227 241 257 281 299 314 329

195 228 243

196 234 248

261

269

283

283

304

307

318

319

333

335

199 238 250

223 241 253

271

272

291

297

309

310

322

328

349

353

47


TRY IT NOW!



(con’t)

In the sample, what is the cutoff score for those people who will be able to attend the seminar?

Hint: the value that defines the top 25% is the same as the value that defines the bottom 75%.

Suppose that the company decides that the employees who scored in the bottom 25% need some additional classes on team building. What is the cutoff score for those employees who need the classes on team building?

48


A Boxplot or Box and Whisker diagram is a graphical display that uses summary statistics to display the distribution of a set of data.

49


A Interquartile Range (IQR) is the difference between the third and first quartiles Q

3

Q

1.

50


Figure 4.6

Box Portion of Boxplot

Figure 4.7

Boxplot with Whiskers

51


The Inner Fences of a boxplot are located at Q

1

1.5

(IQR) and Q

3

+ 1.5

(IQR).

52


The Outer Fences of a boxplot are located at Q

1

3 (IQR) and Q

3

+ 3 (IQR).

53


Figure 4.8

Boxplots for Skewed Data

54


TRY IT NOW!



The company that administered the training aptitude test to its employees would like a better picture of how the employees performed on the test. The data are:

185 227 241 257 281 299 314 329

195 228 243

196 234 248

199 238 250

223 241 253

261

269

271

272

283

283

291

297

304

307

309

310

318

319

322

328

333

335

349

353

55


TRY IT NOW!



(con’t)

In the previous exercise, you found the first and third quartiles of the data set. Use these values to complete the calculations needed for a boxplot.

Draw a complete boxplot of the data.

Were there any outliers? If so, which data values were they?

56


The basics of creating a chart in Excel, using the Chart Wizard.

1. Highlight the data (Frequency table) that you want to graph.

2. Invoke the Chart Wizard by clicking on the icon on the toolbar.

3. Follow the directions and hints from the Chart Wizard.

4. Edit the graph to include any other features or changes you want.

57


Calculating Summary Statistics in Excel

1. Position the cursor in the textbox labeled Input Range and highlight the range of data for which you want to calculate summary statistics.

2. Specify location for the output, either a section of the current worksheet, or a new worksheet or workbook. Click

on the radio button for your choice. If you select Output

Range, you must specify a location on the worksheet.

58


Calculating Summary Statistics in Excel

(con’t)

2. Position the cursor in the textbox for Output Range and click on the cell where you want the upper left corner of the results to appear. If you want to put the results in a new worksheet, you have the option of giving the sheet a name in the textbox or just letting Excel create a new, numbered sheet.

3. Click on the box labeled Summary statistics and finally

click on OK. The output does not include the quartiles.

When a statistic cannot be computed, the output will read

59

N/A.


Figure 4.10

Descriptive Statistics Dialog Box

60


Figure 4.11

Output from Tools>Data

Analysis>Descriptive Statistics

61


Making a Boxplot in KaddStat (note Excel does not include boxplots as part of the graphs it can create)

Be sure you have enabled the KaddStat add-in!

1. From the KADD menu select Boxplots. The Boxplot

Dialog Box will open.

2. Position the cursor in the textbox labeled Input Range and highlight the cells that contain the data.

3. Indicate where you want the boxplot to appear.

4. Click OK.

62


Figure 4.12

KADDSTAT Menu Selection

63


Figure 4.13

The Boxplot Dialog Box

64


Figure 4.14

Finished Boxplot for Golf Ball Data

65


Chapter 4 Summary

In this chapter you have learned:

 There are many ways to describe a set of data using sample statistics. No single number will do the job, nor is there any standard way to proceed.

 The measures that you choose must reflect the characteristics of the data itself.

66


Chapter 4 Summary (con’t)

 Often the best descriptions come from the use of multiple measure and conclusions that can be reached by comparing them.

 It is useful to create images of data using combinations of different statistics.

 The Empirical Rule and Boxplots are examples of using summary statistics to get a picture of the distribution of data.

67

Doing Statistics for Business Data, Inferences, and Decision

Data, Inference, and Decision Making

Chapter 4

Numerical

Descriptors of

Data

Chapter 4 Objectives

Chapter 4 Objectives

Chapter 4 Summary

Chapter 4 Summary (con’t)

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Doing Statistics for Business Data, Inferences, and Decision

Data, Inference, and Decision Making

Chapter 4

Numerical

Descriptors of

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Chapter 4 Objectives

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Chapter 4 Summary

Chapter 4 Summary (con’t)

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