© 2002 Prentice-Hall, Inc. Chap 3-1

Basic Business Statistics(8th Edition)

Chapter 3Numerical Descriptive

Measures

© 2002 Prentice-Hall, Inc. Chap 3-2

Chapter Topics

Measures of central tendency Mean, median, mode, geometric mean,

midrange

Quartile Measure of variation

Range, Interquartile range, variance and standard deviation, coefficient of variation

Shape Symmetric, skewed, using box-and-whisker plots

© 2002 Prentice-Hall, Inc. Chap 3-3

Chapter Topics

Coefficient of correlation Pitfalls in numerical descriptive

measures and ethical considerations

(continued)

© 2002 Prentice-Hall, Inc. Chap 3-4

Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Geometric Mean

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

© 2002 Prentice-Hall, Inc. Chap 3-5

Measures of Central Tendency

Central Tendency

Average Median Mode

Geometric Mean1

1

n

ii

N

ii

XX

n

X

N

1/

12

n

Gn XXXX

© 2002 Prentice-Hall, Inc. Chap 3-6

Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values Sample mean

Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

© 2002 Prentice-Hall, Inc. Chap 3-7

Mean (Arithmetic Mean)

The most common measure of central tendency

Affected by extreme values (outliers)

(continued)

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

© 2002 Prentice-Hall, Inc. Chap 3-8

Median Robust measure of central tendency Not affected by extreme values

In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle

number If n or N is even, the median is the average of

the two middle numbers

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

© 2002 Prentice-Hall, Inc. Chap 3-9

Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical

data There may may be no mode There may be several modes

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

Mode = 9

0 1 2 3 4 5 6

No Mode

© 2002 Prentice-Hall, Inc. Chap 3-10

Geometric Mean

Useful in the measure of rate of change of a variable over time

Geometric mean rate of return Measures the status of an investment over

time

1/

1 2

n

G nX X X X

1/

1 21 1 1 1n

G nR R R R

© 2002 Prentice-Hall, Inc. Chap 3-11

ExampleAn investment of $100,000 declined to $50,000 at

the end of year one and rebounded to $100,000 at end of year two:

1 2 3$100,000 $50,000 $100,000X X X

1/ 2

1/ 2 1/ 2

Average rate of return:

( 50%) (100%)25%

2Geometric rate of return:

1 50% 1 100% 1

0.50 2 1 1 1 0%

G

X

R

© 2002 Prentice-Hall, Inc. Chap 3-12

Quartiles Split Ordered Data into 4 Quarters

Position of i-th Quartile

and Are Measures of Noncentral Location = Median, A Measure of Central Tendency

25% 25% 25% 25%

1Q 2Q 3Q

Data in Ordered Array: 11 12 13 16 16 17 18 21 22

1 1

1 9 1 12 13Position of 2.5 12.5

4 2Q Q

1Q 3Q

2Q

1

4i

i nQ

© 2002 Prentice-Hall, Inc. Chap 3-13

Measures of Variation

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance

Sample

Variance

PopulationStandardDeviationSample

Standard

Deviation

Range

Interquartile Range

© 2002 Prentice-Hall, Inc. Chap 3-14

Range

Measure of variation Difference between the largest and the

smallest observations:

Ignores the way in which data are distributed

Largest SmallestRange X X

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

© 2002 Prentice-Hall, Inc. Chap 3-15

Measure of variation Also known as midspread

Spread in the middle 50% Difference between the first and third

quartiles

Not affected by extreme values

3 1Interquartile Range 17.5 12.5 5Q Q

Interquartile Range

Data in Ordered Array: 11 12 13 16 16 17 17 18 21

© 2002 Prentice-Hall, Inc. Chap 3-16

2

2 1

N

ii

X

N

Important measure of variation Shows variation about the mean

Sample variance:

Population variance:

2

2 1

1

n

ii

X XS

n

Variance

© 2002 Prentice-Hall, Inc. Chap 3-17

Standard Deviation Most important measure of variation Shows variation about the mean Has the same units as the original data

Sample standard deviation:

Population standard deviation:

2

1

1

n

ii

X XS

n

2

1

N

ii

X

N

© 2002 Prentice-Hall, Inc. Chap 3-18

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

© 2002 Prentice-Hall, Inc. Chap 3-19

Coefficient of Variation

Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of

data measured in different units 100%

SCV

X

© 2002 Prentice-Hall, Inc. Chap 3-20

Comparing Coefficient of Variation

Stock A: Average price last year = $50 Standard deviation = $5

Stock B: Average price last year = $100 Standard deviation = $5

Coefficient of variation: Stock A:

Stock B:

$5100% 100% 10%

$50

SCV

X

$5100% 100% 5%

$100

SCV

X

© 2002 Prentice-Hall, Inc. Chap 3-21

Shape of a Distribution

Describes how data is distributed Measures of shape

Symmetric or skewed

Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean

Right-SkewedLeft-Skewed Symmetric

© 2002 Prentice-Hall, Inc. Chap 3-22

Exploratory Data Analysis

Box-and-whisker plot Graphical display of data using 5-number

summary

Median( )

4 6 8 10 12

XlargestXsmallest1Q 3Q

2Q

© 2002 Prentice-Hall, Inc. Chap 3-23

Distribution Shape and Box-and-Whisker Plot

Right-SkewedLeft-Skewed Symmetric

1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q

© 2002 Prentice-Hall, Inc. Chap 3-24

Coefficient of Correlation

Measures the strength of the linear relationship between two quantitative variables

1

2 2

1 1

n

i ii

n n

i ii i

X X Y Yr

X X Y Y

© 2002 Prentice-Hall, Inc. Chap 3-25

Features of Correlation Coefficient

Unit free Ranges between –1 and 1 The closer to –1, the stronger the negative

linear relationship The closer to 1, the stronger the positive linear

relationship The closer to 0, the weaker any positive linear

relationship

© 2002 Prentice-Hall, Inc. Chap 3-26

Scatter Plots of Data with Various Correlation

CoefficientsY

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0

r = .6 r = 1

© 2002 Prentice-Hall, Inc. Chap 3-27

Pitfalls in Numerical Descriptive

Measures Data analysis is objective

Should report the summary measures that best meet the assumptions about the data set

Data interpretation is subjective Should be done in fair, neutral and clear

manner

© 2002 Prentice-Hall, Inc. Chap 3-28

Ethical Considerations

Numerical descriptive measures:

Should document both good and bad results

Should be presented in a fair, objective and neutral manner

Should not use inappropriate summary measures to distort facts

© 2002 Prentice-Hall, Inc. Chap 3-29

Chapter Summary

Described measures of central tendency Mean, median, mode, geometric mean,

midrange

Discussed quartile Described measure of variation

Range, interquartile range, variance and standard deviation, coefficient of variation

Illustrated shape of distribution Symmetric, skewed, box-and-whisker plots

© 2002 Prentice-Hall, Inc. Chap 3-30

Chapter Summary

Discussed correlation coefficient Addressed pitfalls in numerical

descriptive measures and ethical considerations

(continued)