1

Pre University-Mathematics for Business

Session 2

Numerical Descriptive Measures

2

Learning Objectives

• Explain Numerical Data Properties• Describe Summary Measures• Analyze Numerical Data Using Summary

Measures

3

Summary Measures

Arithmetic Mean

Median

Mode

Describing Data Numerically

Variance

Standard Deviation

Coefficient of Variation

Range

Interquartile Range

Geometric Mean

Skewness

Central Tendency Variation ShapeQuartiles

4

Measures of Central Tendency

Central Tendency

Arithmetic Mean Median Mode Geometric Mean

n

XX

n

ii

1

n/1n21G )XXX(X

Overview

Midpoint of ranked values

Most frequently observed value

5

Arithmetic Mean

• The arithmetic mean (mean) is the most common measure of central tendency

– For a sample of size n:

Sample size

n

XXX

n

XX n21

n

1ii

Observed values

6

Arithmetic Mean

• The most common measure of central tendency• Mean = sum of values divided by the number of values• Affected by extreme values (outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

35

15

5

54321

4

5

20

5

104321

7

Median

• In an ordered array, the median is the “middle” number (50% above, 50% below)

• Not affected by extreme values

0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

Median = 3

8

Finding the Median

• The location of the median:

– If the number of values is odd, the median is the middle number– If the number of values is even, the median is the average of the

two middle numbers

• Note that is not the value of the median, only the

position of the median in the ranked data

dataorderedtheinposition2

1npositionMedian

2

1n

9

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

(nominal) data• There 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

Chap 3-10

• Five houses on a hill by the beach

Review Example

$2,000 K

$500 K

$300 K

$100 K

$100 K

House Prices:

$2,000,000 500,000 300,000 100,000 100,000

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Review Example:Summary Statistics

• Mean: ($3,000,000/5)

= $600,000

• Median: middle value of ranked data

= $300,000

• Mode: most frequent value = $100,000

House Prices:

$2,000,000

500,000 300,000 100,000 100,000

Sum $3,000,000

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• Mean is generally used, unless extreme values (outliers) exist

• Then median is often used, since the median is not sensitive to extreme values.– Example: Median home prices may be

reported for a region – less sensitive to outliers

Which measure of location is the “best”?

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Quartiles

• Quartiles split the ranked data into 4 segments with an equal number of values per segment

25% 25% 25% 25%

The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger

Q2 is the same as the median (50% are smaller, 50% are larger)

Only 25% of the observations are greater than the third quartile

Q1 Q2 Q3

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Quartile Formulas

Find a quartile by determining the value in the appropriate position in the ranked data, where

First quartile position: Q1 = (n+1)/4

Second quartile position: Q2 = (n+1)/2 (the median position)

Third quartile position: Q3 = 3(n+1)/4

where n is the number of observed values

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(n = 9)

Q1 is in the (9+1)/4 = 2.5 position of the ranked data

so use the value half way between the 2nd and 3rd values,

so Q1 = 12.5

Quartiles

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

Example: Find the first quartile

Q1 and Q3 are measures of noncentral location

Q2 = median, a measure of central tendency

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(n = 9)

Q1 is in the (9+1)/4 = 2.5 position of the ranked data,

so Q1 = 12.5

Q2 is in the (9+1)/2 = 5th position of the ranked data,

so Q2 = median = 16

Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data,

so Q3 = 19.5

Quartiles

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

Example:(continued)

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Geometric Mean• Geometric mean

– Used to measure the rate of change of a variable over time

• Geometric mean rate of return– Measures the status of an investment over time

– Where Ri is the rate of return in time period i

n/1n21G )XXX(X

1)]R1()R1()R1[(R n/1n21G

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Example

An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:

000,100$X000,50$X000,100$X 321

50% decrease 100% increase

The overall two-year return is zero, since it started and ended at the same level.

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Example

Use the 1-year returns to compute the arithmetic mean and the geometric mean:

%0111)]2()50[(.

1%))]100(1(%))50(1[(

1)]R1()R1()R1[(R

2/12/1

2/1

n/1n21G

%252

%)100(%)50(X

Arithmetic mean rate of return:

Geometric mean rate of return:

Misleading result

More accurate result

(continued)

20Same center,

different variation

Measures of Variation

Variation

Variance Standard Deviation

Coefficient of Variation

Range Interquartile Range

Measures of variation give information on the spread or variability of the data values.

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Range• Simplest measure of variation• Difference between the largest and the

smallest values in a set of data:

Range = Xlargest – Xsmallest

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

Range = 14 - 1 = 13

Example:

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• Ignores the way in which data are distributed

• Sensitive to outliers

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

Disadvantages of the Range

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120

Range = 5 - 1 = 4

Range = 120 - 1 = 119

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Interquartile Range• Can eliminate some outlier problems by

using the interquartile range

• Eliminate some high- and low-valued observations and calculate the range from the remaining values

• Interquartile range = 3rd quartile – 1st quartile

= Q3 – Q1

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Interquartile Range

Median

(Q2)

XmaximumX

minimum Q1 Q3

Example:

25% 25% 25% 25%

12 30 45 57 70

Interquartile range

= 57 – 30 = 27

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• Average (approximately) of squared deviations of values from the mean

– Sample variance:

Variance

1-n

)X(XS

n

1i

2i

2

Where = mean

n = sample size

Xi = ith value of the variable X

X

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Standard Deviation

• Most commonly used measure of variation• Shows variation about the mean• Is the square root of the variance• Has the same units as the original data

– Sample standard deviation:

1-n

)X(XS

n

1i

2i

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Calculation Example:Sample Standard Deviation

Sample Data (Xi) : 10 12 14 15 17 18 18 24

n = 8 Mean = X = 16

4.30957

130

18

16)(2416)(1416)(1216)(10

1n

)X(24)X(14)X(12)X(10S

2222

2222

A measure of the “average” scatter around the mean

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Measuring variation

Small standard deviation

Large standard deviation

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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 = 0.926

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 S = 4.567

Data C

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Advantages of Variance and Standard Deviation

• Each value in the data set is used in the calculation

• Values far from the mean are given extra weight (because deviations from the mean are squared)

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Coefficient of Variation

• Measures relative variation• Always in percentage (%)• Shows variation relative to mean• Can be used to compare two or more

sets of data measured in different units

100%X

SCV

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Comparing Coefficient of Variation

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

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

Both stocks have the same standard deviation, but stock B is less variable relative to its price

10%100%$50

$5100%

X

SCVA

5%100%$100

$5100%

X

SCVB

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Shape of a Distribution

• Describes how data are distributed• Measures of shape

– Symmetric or skewed

Mean = Median Mean < Median Median < Mean

Right-SkewedLeft-Skewed Symmetric

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Numerical Measures for a Population

• Population summary measures are called parameters

• The population mean is the sum of the values in the

population divided by the population size, N

N

XXX

N

XN21

N

1ii

μ = population mean

N = population size

Xi = ith value of the variable X

Where

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• Average of squared deviations of values from the mean

– Population variance:

Population Variance

N

μ)(Xσ

N

1i

2i

2

Where μ = population mean

N = population size

Xi = ith value of the variable X

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Population Standard Deviation• Most commonly used measure of variation• Shows variation about the mean• Is the square root of the population

variance• Has the same units as the original data

– Population standard deviation:

N

μ)(Xσ

N

1i

2i

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Summary of Variation Measures

Measure Formula Description

Range X largest – X smallest Total Spread

Standard Deviation(Sample)

X X

ni

2

1

Dispersion aboutSample Mean

Standard Deviation(Population)

X

N

i X 2 Dispersion about

Population Mean

Variance(Sample)

(Xi X )2

n – 1Squared Dispersionabout Sample Mean

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• If the data distribution is approximately bell-shaped, then the interval:

• contains about 68% of the values in the population or the sample

The Empirical Rule

1σμ

μ

68%

1σμ

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• contains about 95% of the values in the population or the sample

• contains about 99.7% of the values in the population or the sample

The Empirical Rule

2σμ

3σμ

3σμ

99.7%95%

2σμ

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Interpreting Standard Deviation

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• Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)

– Examples:

(1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ)

(1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ)

(1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ)

Chebyshev Rule

withinAt least

42

Interpreting Standard Deviation: Chebyshev’s Theorem

• Applies to any shape data set

No useful information about the fraction of data in the interval x – s to x + s

At least 3/4 of the data lies in the interval x – 2s to x + 2s

At least 8/9 of the data lies in the interval x – 3s to x + 3s

In general, for k > 1, at least 1 – 1/k2 of the data lies in the interval x – ks to x + ks

43

Interpreting Standard Deviation: Chebyshev’s Theorem

sx 3 sx 3sx 2 sx 2sx xsx

No useful information

At least 3/4 of the data

At least 8/9 of the data

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Thinking Challenge

• You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11.

• What are the variance and standard deviation of the stock prices?

45

Variation Solution*

Sample Variance

Raw Data: 17 16 21 18 13 16 1211

S

X X

nX

X

n

S

ii

n

ii

n

2

2

1 1

2

2 2 21

15 5

17 15 5 16 15 5 11 15 5

8 11114

( )

( ) ( ) ( )where .

. . .

.

…

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Variation Solution*

Sample Standard Deviation

S S

X X

n

ii

n

2

2

1

11114 3 34

( ). .

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Chebyshev’s Theorem Example

• Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is 3.34.

• Use this information to form an interval that will contain at least 75% of the closing stock prices of new stock issues.

48

Chebyshev’s Theorem Example

At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean.

x = 15.5 s = 3.34

(x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34)

= (8.82, 22.18)

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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

50

Chapter Summary

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

• Discussed quartiles• Described measures of variation

– Range, interquartile range, variance and standard deviation, coefficient of variation, Z-scores

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