describing data: summary measures

© 1999 Prentice-Hall, Inc. Chap. 3 - 1

Measures of Central Location Mean, Median, Mode

Measures of Variation Range, Variance and Standard Deviation

Measures of Association Covariance and Correlation

Describing Data: Summary Measures

© 1999 Prentice-Hall, Inc. Chap. 3 - 2

•It is the Arithmetic Average of data values:

•The Most Common Measure of Central Tendency

•Affected by Extreme Values (Outliers)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Sample Mean

Mean

nxxx n2i

n

xn

1ii

x

© 1999 Prentice-Hall, Inc. Chap. 3 - 3

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

•Important Measure of Central Tendency

•In an ordered array, the median is the “middle” number.

•If n is odd, the median is the middle number.•If n is even, the median is the average of the 2

middle numbers.•Not Affected by Extreme Values

Median

© 1999 Prentice-Hall, Inc. Chap. 3 - 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

•A Measure of Central Tendency•Value that Occurs Most Often•Not Affected by Extreme Values•There May Not be a Mode•There May be Several Modes•Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6

No Mode

Mode

© 1999 Prentice-Hall, Inc. Chap. 3 - 5

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

100%

X

SCV

Measures Of Variability

© 1999 Prentice-Hall, Inc. Chap. 3 - 6

• Measure of Variation

• Difference Between Largest & Smallest Observations:

Range =

• Ignores How Data Are Distributed:

The Range

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

SmallestrgestLa xx

© 1999 Prentice-Hall, Inc. Chap. 3 - 7

•Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Variance

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

N

(Xi 2

2

1

22

n

XXs i

© 1999 Prentice-Hall, Inc. Chap. 3 - 8

•Most Important Measure of Variation

•Shows Variation About the Mean:

•For the Population:

•For the Sample:

Standard Deviation

For the Population: use N in the denominator.

For the Sample : use n - 1 in the denominator.

N

Xi

2

1

2

n

XXs i

© 1999 Prentice-Hall, Inc. Chap. 3 - 9

Sample Standard Deviation

For the Sample : use n - 1 in the denominator.

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

1624161816171615161416121610 2222222

)()()()()()()(

Sample Standard Deviation= 4.2426

s

:X i

1

2

n

XX i

© 1999 Prentice-Hall, Inc. Chap. 3 - 10

Comparing Standard Deviations

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5 s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Data C

© 1999 Prentice-Hall, Inc. Chap. 3 - 11

Coefficient of Variation

•Measure of Relative Variation

•Always a %

•Shows Variation Relative to Mean

•Used to Compare 2 or More Groups

•Formula ( for Sample):

100%

X

SCV

© 1999 Prentice-Hall, Inc. Chap. 3 - 12

Comparing Coefficient of Variation

Stock A: Average Price last year = $50

Standard Deviation = $5

Stock B: Average Price last year = $100

Standard Deviation = $5

100%

X

SCV

Coefficient of Variation:

Stock A: CV = 10%

Stock B: CV = 5%

© 1999 Prentice-Hall, Inc. Chap. 3 - 13

Shape

• Describes How Data Are Distributed

• Measures of Shape: Symmetric or skewed

Right-SkewedLeft-Skewed SymmetricMean = Median = ModeMean Median Mode Median MeanMode