1 Slide

Descriptive Statistics: Numerical Measures

Exploratory Data Analysis

Chapter 3BA 201

2 Slide

EXPLORATORY ANALYSIS

3 Slide

Five-Number Summary

1 Smallest Value

First Quartile Median Third Quartile Largest Value

2345

4 Slide

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

Apartment Rents

5 Slide

Box Plot

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.

Box plots provide another way to identify outliers.

6 Slide

400

425

450

475

500

525

550

575

600

625

• 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

Apartment Rents

7 Slide

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 * .

8 Slide

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.

Apartment Rents

9 Slide

Box Plot

• Whiskers (dashed lines) are drawn from the ends

of the box to the smallest and largest data values

inside the limits.

400

425

450

475

500

525

550

575

600

625

Smallest valueinside limits = 425

Largest valueinside limits = 615

Apartment Rents

10 Slide

Box Plot

400

425

450

475

500

525

550

575

600

625

Apartment Rents

11 Slide

Box Plot

An excellent graphical technique for making

comparisons among two or more groups.

12 Slide

PRACTICEEXPLORATORY DATA ANALYSIS

13 Slide

Practice – Draw a Box Plot for this Data

716113

2318

ix

14 Slide

Practice – Box Plot

Minimum MaximumMedian

ix

15 Slide

Practice – Box Plot

Minimum MaximumMedian1st Quartile 3rd Quartile

3 7 11 16 18 23ix

16 Slide

Practice – Box Plot3 7 11 16 18 23ix

Minimum MaximumMedian1st Quartile 3rd Quartile

0 5 10 15 20 25

17 Slide

Practice – Box Plot1st Quartile 3rd Quartile

0 5 10 15 20 25

Lower Limit Upper Limit

18 Slide

COVARIANCE AND CORRELATION COEFFICIENT

19 Slide

Measures of Association Between Two Variables

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.

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

20 Slide

Covariance

Positive values indicate a positive relationship.

Negative values indicate a negative relationship.

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

21 Slide

Covariance

The covariance is computed as follows:

forsamples

forpopulations

s x x y ynxy

i i

( )( )

1

xyi x i yx y

N

( )( )

22 Slide

Correlation Coefficient

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.

23 Slide

The correlation coefficient is computed as follows:

forsamples

forpopulations

rss sxyxy

x y

xyxy

x y

Correlation Coefficient

24 Slide

Correlation Coefficient

Values near +1 indicate a strong positive linear relationship.

Values near -1 indicate a strong negative linear relationship.

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

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

25 Slide

A golfer is interested in investigating the relationship, 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

Golfing Study

26 Slide

Covariance and Correlation Coefficient

Golfing Study

x277.6 10.65 113.42259.5 -7.45 55.50269.1 2.15 4.62267.0 0.05 0.00255.6 -11.35 128.82272.9 5.95 35.40

Sum 1602 337.78Mean 266.95

Variance 67.56Std. Dev. 8.22

xxi 2)( xxi

27 Slide

Covariance and Correlation Coefficient

Golfing Study

y69 -1.00 1.0071 1.00 1.0070 0.00 0.0070 0.00 0.0071 1.00 1.0069 -1.00 1.00

Sum 420 4.00Mean 70.00

Variance 0.80Std. Dev. 0.89

yyi 2)( yyi

28 Slide

Covariance and Correlation Coefficient

Golfing Study

10.65 -1.00 -10.65-7.45 1.00 -7.452.15 0.00 0.000.05 0.00 0.00

-11.35 1.00 -11.355.95 -1.00 -5.95

Total -35.40

yyi xxi ))(( yyxx ii

29 Slide

• Sample Covariance

• Sample Correlation Coefficient

Covariance and Correlation Coefficient

7.08 -.9631(8.2192)(.8944)xy

xyx y

sr

s s

( )( ) 35.40 7.081 6 1i i

xyx x y y

sn

Golfing Study

30 Slide

Clarification (Day Class)

If x and y are positively correlated• x and y move in the same direction.• As x increases, y increases.

(Also, as x decreases, y decreases.) If x and y are negatively correlated

• x and y move in opposite directions.• As x increases, y decreases.

(Also, ax x decreases, y increases.)

31 Slide

PRACTICECOVARIANCE AND CORRELATION COEFFICIENT

32 Slide

Practice - Covariance andCorrelation Coefficient

x y65 14271 16154 12867 14994 20693 194

Do the following:1. Compute the mean and

standard deviation for x and y.

2. Compute the Covariance and Correlation Coefficient.

33 Slide

Practice - Covariance andCorrelation Coefficient

x657154679493

MeanStd. Dev.

xxi 2)( xxi

34 Slide

Practice - Covariance andCorrelation Coefficient

y142161128149206194

MeanStd. Dev.

yyi 2)( yyi

35 Slide

Practice - Covariance andCorrelation Coefficient

-9.00 -21.33-3.00 -2.33

-20.00 -35.33-7.00 -14.3320.00 42.6719.00 30.67

Total

))(( yyxx ii yyi xxi

36 Slide

Practice - Covariance andCorrelation Coefficient

Covariance

1))((

n

yyxxs iixy

Correlation Coefficient

yx

xyxy ss

sr

37 Slide

WEIGHTED MEAN AND GROUPED DATA

38 Slide

The Weighted Mean andWorking with Grouped Data

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

39 Slide

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.

40 Slide

Weighted Mean

i i

i

wxx

w

where:

xi = value of observation i wi = weight for observation i

41 Slide

PRACTICEWEIGHTED MEAN

42 Slide

Practice – Weighted Mean

CourseHoursW

PointsX

HxPWxX

Botany 4 4

Astrology 3 2

Calculus 5 3

Geeimatree 4 1

Advanced Comic Books 6 3

Weighted Mean

43 Slide

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.

44 Slide

Mean for Grouped Data

i if Mx

n

NMf ii

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

Sample Data

Population Data

45 Slide

The previously presented sample of apartment rents is shown here as grouped data in the form ofa 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

Apartment Rents

46 Slide

Sample Mean for Grouped Data

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

34,525 493.2170x

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

Apartment Rents

47 Slide

Variance for Grouped Data

s f M xn

i i22

1

( )

22

f MN

i i( )

For sample data

For population data

48 Slide

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

Apartment Rents

49 Slide

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

Apartment Rents

Sample Variance for Grouped Data

50 Slide