Describing Data: Summary Measures

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# Describing Data: Summary Measures - PowerPoint PPT Presentation

Describing Data: Summary Measures. Measures of Central Location Mean, Median, Mode Measures of Variation Range, Variance and Standard Deviation Measures of Association Covariance and Correlation. Mean. It is the Arithmetic Average of data values:

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Presentation Transcript
Describing Data: Summary Measures

Measures of Central Location

Mean, Median, Mode

Measures of Variation

Range, Variance and Standard Deviation

Measures of Association

Covariance and Correlation

Mean
• It is the Arithmetic Average of data values:
• The Most Common Measure of Central Tendency
• Affected by Extreme Values (Outliers)

+

+

·

·

·

+

x

x

x

n

=

x

å

x

=

i

2

n

i

=

Sample Mean

i

1

n

n

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

Median
• 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

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

Mode
• 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

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

No Mode

Mode = 9

Measures Of Variability

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Population

Variance

Population

Standard

Deviation

Sample

Variance

Sample

Standard

Deviation

The Range

• Measure of Variation
• Difference Between Largest & Smallest
• Observations:
• Range =
• Ignores How Data Are Distributed:

-

x

x

La

rgest

Smallest

Range = 12 - 7 = 5

Range = 12 - 7 = 5

7 8 9 10 11 12

7 8 9 10 11 12

Variance

• Important Measure of Variation
• Shows Variation About the Mean:
• For the Population:
• For the Sample:

)

2

-

m

å

(X

2

s

=

i

N

(

)

2

-

å

X

X

2

=

i

s

-

n

1

For the Population: use N in the denominator.

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

Standard Deviation

• Most Important Measure of Variation
• Shows Variation About the Mean:
• For the Population:
• For the Sample:

(

)

2

-

m

å

X

s

=

i

N

(

)

2

-

å

X

X

=

i

s

-

n

1

For the Population: use N in the denominator.

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

Sample Standard Deviation

(

)

2

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

-

å

X

X

s

=

i

-

n

1

Data:10 12 14 15 17 18 18 24

n = 8 Mean =16

s =

Sample Standard Deviation= 4.2426

Comparing Standard Deviations

Data A

Mean = 15.5

s = 3.338

11 12 13 14 15 16 17 18 19 20 21

Data B

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Data C

Mean = 15.5

s = 4.57

11 12 13 14 15 16 17 18 19 20 21

Coefficient of Variation
• Measure of Relative Variation
• Always a %
• Shows Variation Relative to Mean
• Used to Compare 2 or More Groups
• Formula ( for Sample):
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: CV = 10%

Stock B: CV = 5%

Shape
• Describes How Data Are Distributed
• Measures of Shape:
• Symmetric or skewed

Right-Skewed

Left-Skewed

Symmetric

Mean

Median

Mode

Mean

=

Median

=

Mode

Mode

Median

Mean