Chapter 3, Part B Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Exploratory

Chapter 3, Part B Descriptive Statistics: Numerical

Measures Measures of Distribution Shape, Relative

Location, and Detecting Outliers Exploratory Data Analysis Measures of Association Between Two

Variables The Weighted Mean and Working with Grouped Data

Measures of Distribution Shape,Relative Location, and Detecting Outliers

Distribution Shape z-Scores Chebyshev’s

Theorem Empirical Rule Detecting Outliers

Distribution Shape: Skewness

An important measure of the shape of a distribution is called skewness.

The formula for the skewness of sample data is

Skewness can be easily computed using statistical software.

3

)2)(1(Skewness

s

xx

nn

n i


Symmetric (not skewed)

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = 0

• Skewness is zero.• Mean and median are equal.

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00


Moderately Skewed Left

Skewness = - .31

• Skewness is negative.• Mean will usually be less than the median.


Moderately Skewed Right

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = .31

• Skewness is positive.• Mean will usually be more than the median.


Highly Skewed RightR

ela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = 1.25

• Skewness is positive (often above 1.0).• Mean will usually be more than the median.

Seventy apartments were randomlysampled in Suva. The monthly rent pricesfor the apartments are listed below in ascending

order.


Example: Apartment Rents

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = .92



The z-score is often called the standardized value. The z-score is often called the standardized value.

It denotes the number of standard deviations a data value xi is from the mean. It denotes the number of standard deviations a data value xi is from the mean.

z-Scores

zx xsii

Excel’s STANDARDIZE function can be used to compute the z-score. Excel’s STANDARDIZE function can be used to compute the z-score.

z-Scores

A data value less than the sample mean will have a z-score less than zero. A data value greater than the sample mean will have a z-score greater than zero. A data value equal to the sample mean will have a z-score of zero.

An observation’s z-score is a measure of the relative location of the observation in a data set.

• z-Score of Smallest Value (425)

425 490.80 1.20

54.74ix x

zs

z-Scores

Standardized Values for Apartment Rents-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27


Chebyshev’s Theorem

At least (1 - 1/z2) of the items in any data set will be within z standard deviations of the mean, where z is any value greater than 1.

At least (1 - 1/z2) of the items in any data set will be within z standard deviations of the mean, where z is any value greater than 1.

Chebyshev’s theorem requires z > 1, but z need not be an integer. Chebyshev’s theorem requires z > 1, but z need not be an integer.

At least of the data values must be

within of the mean.


within of the mean.

75%75%

z = 2 standard deviations z = 2 standard deviations



within of the mean.


within of the mean.

89%89%



within of the mean.


within of the mean.

94%94%



Let z = 1.5 with = 490.80 and s = 54.74x

At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%

of the rent values must be between

x - z(s) = 490.80 - 1.5(54.74) = 409

andx + z(s) = 490.80 + 1.5(54.74) = 573

(Actually, 86% of the rent values are between 409 and 573.)


Empirical Rule

When the data are believed to approximate a bell-shaped distribution …

The empirical rule is based on the normal distribution, which is covered in Chapter 6. The empirical rule is based on the normal distribution, which is covered in Chapter 6.

The empirical rule can be used to determine the percentage of data values that must be within a specified number of standard deviations of the mean.

The empirical rule can be used to determine the percentage of data values that must be within a specified number of standard deviations of the mean.

Empirical Rule

For data having a bell-shaped distribution:

of the values of a normal random variable are within of its mean.

of the values of a normal random variable are within of its mean.68.26%68.26%

+/- 1 standard deviation+/- 1 standard deviation



+/- 2 standard deviations+/- 2 standard deviations



+/- 3 standard deviations+/- 3 standard deviations

Empirical Rule

xm – 3s m – 1s

m – 2sm + 1s

m + 2sm + 3sm

68.26%

95.44%99.72%

Detecting Outliers

An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier.

It might be:• an incorrectly recorded data value• a data value that was incorrectly included in the

data set• a correctly recorded data value that belongs in

the data set

Detecting Outliers

• The most extreme z-scores are -1.20 and 2.27• Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set.

-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

Standardized Values for Apartment Rents


Exploratory Data Analysis

Exploratory data analysis procedures enable us to use simple arithmetic and easy-to-draw pictures to summarize data.

Exploratory data analysis procedures enable us to use simple arithmetic and easy-to-draw pictures to summarize data.

We simply sort the data values into ascending order and identify the five-number summary and then construct a box plot.

We simply sort the data values into ascending order and identify the five-number summary and then construct a box plot.

Five-Number Summary

11 Smallest Value Smallest Value

First Quartile First Quartile

Median Median

Third Quartile Third Quartile

Largest Value Largest Value

22

33

44

55

Five-Number Summary

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Lowest Value = 425 First Quartile = 445Median = 475

Third Quartile = 525Largest Value = 615


Box Plot

A box plot is a graphical summary of data that is based on a five-number summary. A box plot is a graphical summary of data that is based on a five-number summary.

A key to the development of a box plot is the computation of the median and the quartiles Q1 and Q3.

A key to the development of a box plot is the computation of the median and the quartiles Q1 and Q3.

Box plots provide another way to identify outliers. Box plots provide another way to identify outliers.

400400

425425

450450

475475

500500

525525

550550

575575

600600

625625

• A box is drawn with its ends located at the first and third quartiles.

Box Plot

• A vertical line is drawn in the box at the location of the median (second quartile).

Q1 = 445 Q3 = 525Q2 = 475


Box Plot

Limits are located (not drawn) using the interquartile range (IQR).

Data outside these limits are considered outliers. The locations of each outlier is shown with the

symbol * .continued

Box Plot

Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325

Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645

• The lower limit is located 1.5(IQR) below Q1.

• The upper limit is located 1.5(IQR) above Q3.

• There are no outliers (values less than 325 or greater than 645) in the apartment rent data.


Box Plot

• Whiskers (dashed lines) are drawn from the ends

of the box to the smallest and largest data values

inside the limits.

400400

425425

450450

475475

500500

525525

550550

575575

600600

625625

Smallest valueinside limits = 425

Largest valueinside limits = 615


Box Plot

An excellent graphical technique for making

comparisons among two or more groups.

Measures of Association Between Two Variables

Thus far we have examined numerical methods used to summarize the data for one variable at a time. Thus far we have examined numerical methods used to summarize the data for one variable at a time.

Often a manager or decision maker is interested in the relationship between two variables. Often a manager or decision maker is interested in the relationship between two variables.

Two descriptive measures of the relationship between two variables are covariance and correlation coefficient.

Two descriptive measures of the relationship between two variables are covariance and correlation coefficient.

Covariance

Positive values indicate a positive relationship. Positive values indicate a positive relationship.

Negative values indicate a negative relationship. Negative values indicate a negative relationship.

The covariance is a measure of the linear association between two variables. The covariance is a measure of the linear association between two variables.

Covariance

The covariance is computed as follows:

The covariance is computed as follows:

forsamples

forpopulations

sx x y ynxy

i i

( )( )

1

xyi x i yx y

N

( )( )

Correlation Coefficient

Just because two variables are highly correlated, it does not mean that one variable is the cause of the other.

Just because two variables are highly correlated, it does not mean that one variable is the cause of the other.

Correlation is a measure of linear association and not necessarily causation. Correlation is a measure of linear association and not necessarily causation.

The correlation coefficient is computed as follows:

The correlation coefficient is computed as follows:

forsamples

forpopulations

rs

s sxyxy

x y

xyxy

x y



Values near +1 indicate a strong positive linear relationship. Values near +1 indicate a strong positive linear relationship.

Values near -1 indicate a strong negative linear relationship. Values near -1 indicate a strong negative linear relationship.

The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1.

The closer the correlation is to zero, the weaker the relationship. The closer the correlation is to zero, the weaker the relationship.

A golfer is interested in investigating therelationship, if any, between driving distance

and 18-hole score.

277.6259.5269.1267.0255.6272.9

697170707169

Average DrivingDistance (yds.)

Average18-Hole Score

Covariance and Correlation Coefficient

Example: Golfing Study


277.6259.5269.1267.0255.6272.9

697170707169

x y

10.65 -7.45 2.15 0.05-11.35 5.95

-1.0 1.0 0 0 1.0-1.0

-10.65 -7.45 0 0-11.35 -5.95

( )ix x ( )( )i ix x y y ( )iy y

AverageStd. Dev.

267.0 70.0 -35.408.2192.8944

Total


• Sample Covariance

• Sample Correlation Coefficient


7.08 -.9631

(8.2192)(.8944)xy

xyx y

sr

s s

( )( ) 35.40 7.08

1 6 1i i

xy

x x y ys

n


The Weighted Mean andWorking with Grouped Data

Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data

Weighted Mean

When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

Weighted Mean

i i

i

wxx

w

where: xi = value of observation i

wi = weight for observation i

Grouped Data

The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.

Mean for Grouped Data

i if Mx

n

N

Mf ii

where: fi = frequency of class i Mi = midpoint of class i

Sample Data

Population Data

The previously presented sample of apartment

rents is shown here as grouped data in the form of

a frequency distribution.

Sample Mean for Grouped Data

Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6


Sample Mean for Grouped Data

This approximationdiffers by $2.41 fromthe actual samplemean of $490.80.

34,525 493.21

70x

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

f iM i

3436.07641.55634.03916.03566.52118.01099.02278.01179.03657.034525.0


Variance for Grouped Data

sf M xn

i i22

1

( )

22

f M

Ni i( )

For sample data

For population data

Sample Variance for Grouped Data

continued

Rent ($) f i

420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6

Total 70

M i

429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5

M i - x

-63.7-43.7-23.7-3.716.336.356.376.396.3116.3

(M i - x )2

4058.961910.56562.1613.76

265.361316.963168.565820.169271.76

13523.36

f i(M i - x )2

32471.7132479.596745.97110.11

1857.555267.866337.13

23280.6618543.5381140.18

208234.29


3,017.89 54.94s

s2 = 208,234.29/(70 – 1) = 3,017.89

This approximation differs by only $.20 from the actual standard deviation of $54.74.

• Sample Variance

• Sample Standard Deviation


Sample Variance for Grouped Data

End of Chapter 3, Part B

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Chapter 3, Part B Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Exploratory