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10th Edition
Chapter 3 Numerical Descriptive Measures

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc..

Chap 3-1

Learning Objectives
In this chapter, you learn:
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?

?

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To describe the properties of central tendency, variation, and shape in numerical data To calculate descriptive summary measures for a population To calculate the coefficient of variation and Zscores To construct and interpret a box-and-whisker plot To calculate the covariance and the coefficient of correlation
Chap 3-2

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chapter Topics
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Measures of central tendency, variation, and shape
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Mean, median, mode, geometric mean Quartiles Range, interquartile range, variance and standard deviation, coefficient of variation, Z-scores Symmetric and skewed distributions Mean, variance, and standard deviation The empirical rule and Chebyshev rule
Chap 3-3

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Population summary measures
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Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chapter Topics
(continued)
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Five number summary and box-and-whisker plot

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Covariance and coefficient of correlation
Pitfalls in numerical descriptive measures and ethical issues

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-4

Summary Measures
Describing Data Numerically

Central Tendency Arithmetic Mean Median Mode Geometric Mean

Quartiles

Variation

Shape

Range
Interquartile Range Variance Standard Deviation

Skewness

Coefficient of Variation

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-5

Measures of Central Tendency
Overview Central Tendency

Arithmetic Mean

Median

Mode

Geometric Mean

X?

?X
i ?1

n

i

XG ? ( X1 ? X2 ??? Xn )1/ n
Midpoint of ranked values Most frequently observed value
Chap 3-6

n

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Arithmetic Mean
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The arithmetic mean (mean) is the most common measure of central tendency
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For a sample of size n:

X?

?X
i ?1

n

i

n

X1 ? X 2 ? ? ? Xn ? n
Observed values
Chap 3-7

Sample size
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Arithmetic Mean
(continued)
?
? ?

The most common measure of central tendency Mean = sum of values divided by the number of values 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

Mean = 3
1 ? 2 ? 3 ? 4 ? 5 15 ? ?3 5 5
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Mean = 4
1 ? 2 ? 3 ? 4 ? 10 20 ? ?4 5 5
Chap 3-8

Median
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In an ordered array, the median is the “middle” number (50% above, 50% below)
0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Median = 3

Median = 3

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Not affected by extreme values

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-9

Finding the Median
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The location of the median:
n ?1 Median position ? position in the ordered data 2
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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

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n ?1 is not the value of the median, only the 2 position of the median in the ranked data

Note that

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-10

Mode
? ? ? ?

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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 may be no mode There may be several modes
0 1 2 3 4 5 6

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

Mode = 9
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

No Mode
Chap 3-11

Review Example
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Five houses on a hill by the beach
\$2,000 K

House Prices: \$2,000,000 500,000 300,000 100,000 100,000

\$500 K \$300 K

\$100 K \$100 K
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-12

Review Example: Summary Statistics
House Prices: \$2,000,000 500,000 300,000 100,000 100,000 Sum \$3,000,000
? ?

Mean:

(\$3,000,000/5) = \$600,000

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Median: middle value of ranked data = \$300,000 Mode: most frequent value = \$100,000
Chap 3-13

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Which measure of location is the “best”?
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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.
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Example: Median home prices may be reported for a region – less sensitive to outliers

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-14

Quartiles
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Quartiles split the ranked data into 4 segments with an equal number of values per segment 25%
Q1

25%
Q2

25%
Q3

25%

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?

?

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
Chap 3-15

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-16

Quartiles
?

Example: Find the first quartile

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

(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

Q1 and Q3 are measures of noncentral location Q2 = median, a measure of central tendency
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-17

Quartiles
(continued)
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Example:

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

(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
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-18

Geometric Mean
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Geometric mean
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Used to measure the rate of change of a variable over time

XG ? ( X1 ? X2 ??? Xn )
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1/ n

Geometric mean rate of return
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Measures the status of an investment over time
1/ n

RG ? [(1? R1) ? (1? R2 ) ? ?? (1? Rn )]
?

?1

Where Ri is the rate of return in time period i
Chap 3-19

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

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:

X1 ? \$100,000

X2 ? \$50,000

X3 ? \$100,000

50% decrease

100% increase

The overall two-year return is zero, since it started and ended at the same level.
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-20

Example
(continued)

Use the 1-year returns to compute the arithmetic mean and the geometric mean:
Arithmetic mean rate of return:
Geometric mean rate of return:

X?

( ?50%) ? (100%) ? 25% 2

R G ? [(1 ? R1 ) ? (1 ? R 2 ) ? ? ? (1 ? Rn )]1/ n ? 1 ? [(1 ? ( ?50%))? (1 ? (100%))]1/ 2 ? 1 ? [(.50) ? (2)]1/ 2 ? 1 ? 11/ 2 ? 1 ? 0%
More accurate result
Chap 3-21

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Measures of Variation
Variation
Range Interquartile Range Variance Standard Deviation Coefficient of Variation

?

Measures of variation give information on the spread or variability of the data values.
Same center, different variation

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-22

Range
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Simplest measure of variation Difference between the largest and the smallest values in a set of data: Range = Xlargest – Xsmallest

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

Range = 14 - 1 = 13
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-23

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Ignores the way in which data are distributed
7 8 9 10 11 12 7 8 9 10 11 12

Range = 12 - 7 = 5
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Range = 12 - 7 = 5

Sensitive to outliers
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
Range = 5 - 1 = 4

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 = 120 - 1 = 119
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-24

Interquartile Range
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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
Chap 3-25

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?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Interquartile Range
Example: X
minimum
25%

Q1
25%

Median (Q2)
25%

Q3
25%

X

maximum

12

30

45

57

70

Interquartile range = 57 – 30 = 27

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-26

Variance
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Average (approximately) of squared deviations of values from the mean
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Sample variance:

S ?
2
Where

? (X ? X)
i?1 i

n

2

n -1

X = mean
n = sample size

Xi = ith value of the variable X
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-27

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

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Sample standard deviation:

S?

? (X ? X)
i i?1

n

2

n -1
Chap 3-28

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Calculation Example: Sample Standard Deviation
Sample Data (Xi) :

10

12
n=8

14

15

17

18

18

24

Mean = X = 16

S?

(10 ? X)2 ? (12 ? X)2 ? (14 ? X)2 ? ? ? (24 ? X)2 n ?1 (10 ? 16)2 ? (12 ? 16)2 ? (14 ? 16)2 ? ? ? (24 ? 16)2 8 ?1 130 7 ?

?

?

4.3095

A measure of the “average” scatter around the mean
Chap 3-29

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Measuring variation

Small standard deviation

Large standard deviation

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-30

Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 S = 3.338

Data B
11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 S = 0.926 Mean = 15.5 S = 4.567
Chap 3-31

Data C
11 12 13 14 15 16 17 18 19 20 21

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Advantages of Variance and Standard Deviation
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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)

?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-32

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

?S? ? ? 100% CV ? ? ?X? ? ?
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-33

Comparing Coefficient of Variation
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Stock A: ? Average price last year = \$50 ? Standard deviation = \$5

?

Stock B:
? ?

?S? \$5 ? ? ? 100% ? CVA ? ? ? ? 100% ? 10% \$50 ?X?

Average price last year = \$100 Standard deviation = \$5

?S? \$5 CVB ? ? ? ? 100% ? ? 100% ? 5% ?X? \$100 ? ?
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

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

Chap 3-34

Z Scores
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A measure of distance from the mean (for example, a Z-score of 2.0 means that a value is 2.0 standard deviations from the mean) The difference between a value and the mean, divided by the standard deviation A Z score above 3.0 or below -3.0 is considered an outlier

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X?X Z? S
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-35

Z Scores
(continued)

Example:
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If the mean is 14.0 and the standard deviation is 3.0, what is the Z score for the value 18.5?

X ? X 18.5 ? 14.0 Z? ? ? 1.5 S 3.0
?

The value 18.5 is 1.5 standard deviations above the mean
(A negative Z-score would mean that a value is less than the mean)
Chap 3-36

?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Shape of a Distribution
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Describes how data are distributed Measures of shape
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Symmetric or skewed

Left-Skewed
Mean < Median

Symmetric
Mean = Median

Right-Skewed
Median < Mean

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-37

Using Microsoft Excel
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Descriptive Statistics can be obtained from Microsoft? Excel
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Use menu choice: tools / data analysis / descriptive statistics

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Enter details in dialog box

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-38

Using Excel
?

tools / data analysis / descriptive statistics

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-39

Using Excel
(continued)

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Enter dialog box details

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Check box for summary statistics Click OK
Chap 3-40

?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Excel output
Microsoft Excel descriptive statistics output, using the house price data:
House Prices: \$2,000,000 500,000 300,000 100,000 100,000

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-41

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

??
Where

?X
i?1

N

i

N

X1 ? X2 ? ? ? XN ? N

μ = population mean N = population size

Xi = ith value of the variable X
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-42

Population Variance
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Average of squared deviations of values from the mean
?

Population variance:

σ2 ?
Where μ = population mean N = population size

? (X ? μ)
i?1 i

N

2

N

Xi = ith value of the variable X
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-43

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:

σ?

(Xi ? μ)2 ?
i?1

N

N
Chap 3-44

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

The Empirical Rule
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If the data distribution is approximately bell-shaped, then the interval:
μ ? 1σ contains about 68% of the values in the population or the sample

?

68%

μ

μ ? 1σ
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-45

The Empirical Rule
?

?

μ ? 2σ contains about 95% of the values in the population or the sample μ ? 3σ contains about 99.7% of the values in the population or the sample

95%

99.7%

μ ? 2σ
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

μ ? 3σ
Chap 3-46

Chebyshev Rule
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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: At least

within

(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)
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 3-47

Approximating the Mean from a Frequency Distribution
?

?

Sometimes only a frequency distribution is available, not the raw data Use the midpoint of a class interval to approximate the values in that class

X?
?

?m f
j?1

c

j j

n

Where

n = number of values or sample size
c = number of classes in the frequency distribution mj = midpoint of the jth class fj = number of values in the jth class

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-48

Approximating the Standard Deviation from a Frequency Distribution
?

Assume that all values within each class interval are located at the midpoint of the class
?

Approximation for the standard deviation from a frequency distribution:

S?
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

? (m ? X)
j?1 j

c

2

fj

n -1
Chap 3-49

Exploratory Data Analysis
?

Box-and-Whisker Plot: A Graphical display of data using 5-number summary:
Minimum -- Q1 -- Median -- Q3 -- Maximum

Example:
25% 25% 25% 25%

Minimum
Minimum

1st 1st Quartile
Quartile

Median
Median

3rd 3rd Quartile
Quartile

Maximum
Maximum

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-50

Shape of Box-and-Whisker Plots
?

The Box and central line are centered between the endpoints if data are symmetric around the median

Min
?

Q1

Median

Q3

Max

A Box-and-Whisker plot can be shown in either vertical or horizontal format

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-51

Distribution Shape and Box-and-Whisker Plot
Left-Skewed Symmetric Right-Skewed

Q1

Q2 Q3

Q1 Q2 Q3

Q1 Q2 Q3

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-52

Box-and-Whisker Plot Example
?

Below is a Box-and-Whisker plot for the following data:
Min Q1 Q2 Q3 Max

0

2

2

2

3

3

4

5

5

10

27

0 23 5 0 2 3 5
?

27 27

The data are right skewed, as the plot depicts
Chap 3-53

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

The Sample Covariance
?

The sample covariance measures the strength of the linear relationship between two variables (called bivariate data)
The sample covariance:

?

cov ( X , Y ) ?
?

? ( X ? X)(Y ? Y )
i ?1 i i

n

n ?1

Only concerned with the strength of the relationship

?

No causal effect is implied
Chap 3-54

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Interpreting Covariance
?

Covariance between two random variables:
X and Y tend to move in the same direction
X and Y tend to move in opposite directions X and Y are independent

cov(X,Y) > 0
cov(X,Y) < 0 cov(X,Y) = 0

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-55

Coefficient of Correlation
?

?

Measures the relative strength of the linear relationship between two variables Sample coefficient of correlation:

cov(X, Y) r? SX SY
where
cov (X, Y) ?

? (X ? X)(Y ? Y)
i?1 i i

n

n ?1

SX ?

? (X ? X)
i?1 i

n

2

n ?1

SY ?

? (Y ? Y)
i?1 i

n

2

n ?1
Chap 3-56

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Features of Correlation Coefficient, r
? ? ?

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 the linear relationship

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-57

Scatter Plots of Data with Various Correlation Coefficients
Y Y Y

X r = -1 Y Y r = -.6

X r=0 Y

X

r = +1

X
r = +.3

X
r=0

X
Chap 3-58

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Using Excel to Find the Correlation Coefficient
?

Select Tools/Data Analysis Choose Correlation from the selection menu Click OK . . .

?

?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-59

Using Excel to Find the Correlation Coefficient

(continued)

?

?

Input data range and select appropriate options Click OK to get output

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-60

Interpreting the Result
?

r = .733
100 95

Scatter Plot of Test Scores

Test #2 Score

?

There is a relatively strong positive linear relationship between test score #1 and test score #2

90 85 80 75 70 70 75 80 85 90 95 100

Test #1 Score

?

Students who scored high on the first test tended to score high on second test, and students who scored low on the first test tended to score low on the second test
Chap 3-61

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-62

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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-63

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 Symmetric, skewed, box-and-whisker plots

?

Illustrated shape of distribution
?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-64

Chapter Summary
(continued)
?

Discussed covariance and correlation coefficient

?

Addressed pitfalls in numerical descriptive measures and ethical considerations

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 3-65