Measures of Variation

Training Course on Basic Statistics for Research

August 24-28, 2009

STATISTICAL RESEARCH AND TRAINING CENTERJ and S Building, 104 Kalayaan Avenue, Diliman, Quezon City

Measures of Variation

Prepared by:Josefina V. AlmedaProfessor and College SecretarySchool of StatisticsUniversity of the Philippines, DilimanAugust 2009

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Statistical Research and Training Center Training Course on Basic Statistics for ResearchAugust 24 - 28, 2009

Learning Objectives

After the session, participants should be able to:

Gain skills in the computation of the different quantitative measures of dispersion;

Describe and compare groups and individuals within groups using the measures of dispersion;

Interpret results obtained from each measure

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Measures of Dispersion indicate the extent to which individual items in a

series are scattered about an average.

1. Measures of Absolute Dispersion

Use to compare two or more data sets with the

same means and the same units of measurement.

2. Measures of Relative Dispersion

Used to compare two or more data sets with

different means and different units of

measurement.

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Measures of VariationMeasures of Variation

Coefficient of Variation

Range

Variation

Variance Standard Deviation

PopulationVariance

Sample

Variance

PopulationStandardDeviation

Sample

Standard

Deviation

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Measures of Absolute Dispersion:

Range and Standard Deviation

Range highest observation – lowest observation

Standard deviation is the positive square root of the variance and

measures on the average the dispersion of each

observation from the mean.

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• Difference Between Largest & Smallest Observations:

Range = X Largest - X Smallest

• Simple to compute and easy to understand

• Quick measure of spread

• Ignores How Data Are Distributed

RangeRange

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Range = 12 - 7 = 5

7 8 9 10 11 12

7 8 9 10 11 12

Range = 12 - 7 = 5

RangeRange

What is the range of the monthly salary of people working in Y company?

3050, 3273, 3552, 4009, 5118, 6370, 8950, 10835

R = 10835 – 3050 = 7785

Q:

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Important measure of variation Shows variation about the mean

• Sample variance:

• Population variance: 2

2 1

N

ii

X

N

2

2 1

1

n

ii

X XS

n

Variance

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Standard Deviation

Most important measure of variation Shows variation about the mean Has the same units as the original data It is always positive

• Sample standard deviation:

• Population standard deviation:

2

1

1

n

ii

X XS

n

2

1

N

ii

X

N

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1

2

n

XX i

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

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

= 4.2426

S =

:X i

Calculating the Sample SDCalculating the Sample SD

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1

2

n

XX is =

= 4.2426

N

X i

2 = 3.9686

Value for the Standard Deviation is larger for data considered as a Sample.

Data : Xj 10 12 14 15 17 18 18 24

N= 8 Mean =16

Sample vs Pop’n SDSample vs Pop’n SD

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Remarks:1. If there is a large amount of variation in the data set, then on the average, the data values will be far from the mean. Hence, the standard deviation will be large.

2. If there is only a small amount of variation in the data set, then on the average, the data values will be close to the mean. Hence, the standard deviation will be small.

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 = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5 s = 4.57

Data C

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Example: Team A - Heights of five marathon players in inches

65 “ 65 “ 65 “ 65 “ 65 “

Mean = 65 s = 0

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Example: Team B - Heights of five marathon players in inches

62 “ 67 “ 66 “ 70 “ 60 “

Mean = 65” s = 3.6”

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Standard Deviation

11. It is the most widely used measure of dispersion. It is based on all the items and is rigidly defined.

2. It is of great significance for testing the reliability of measures calculated from samples, the difference between such measures, and in comparing the extent of fluctuation in two or more samples.

Advantages

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Standard Deviation

D

1. The standard deviation is sensitive to the presence of extreme values.

2. It is not easy to calculate by hand.

Disadvantages

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Measure of relative dispersion

are unitless and are used to compare the

scatter of one distribution with the scatter of

another distribution.

Coefficient of Variation

utilizes two measures and these are the mean and

the standard deviation.

is a percentage

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The formula of the coefficient of variation is given as,

population CV = %100x

where is the population standard deviation is the population mean

sample CV = %100x

sx

where s is the sample standard deviation x is the sample mean

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ComparingComparing CV’sCV’s

• Stock A: Average Price last year = P50

Standard Deviation = P5• Stock B: Average Price last year = P100

Standard Deviation = P5

100%

X

SCV

Coefficient of Variation:

Stock A: CV = 10%

Stock B: CV = 5%

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Example: To illustrate, you want to buy a stock and you have the option to select one out of the two. The given information is that Stock 1 is priced at P2000 per share and stock 2 is priced at P550 per share. In buying stocks, we lessen the risk by selecting a stock that has less variable price. On the other hand, if we want to take a chance that the price of the stock will go up, then we would want the stock that has more varied price. Let’s say a sample of price of Stock 1 and Stock 2 was collected at the close of trading for the past months and the following statistics were obtained: Stock Mean Price Standard Deviation 1 P1975 P578 2 P 565 P 85

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To determine which of the two stocks have a more variable price, we compute for the coefficient of variation.

CVstock1 = 229 x1001975

578. % CVstock2 = 100x

565

85 15.04%

Stock 1 price is more variable than stock 2 price. As a matter of fact, stock 1 price is almost twice as variable as stock 2 price.

Training Course on Basic Statistics for Research

August 24-28, 2009

STATISTICAL RESEARCH AND TRAINING CENTERJ and S Building, 104 Kalayaan Avenue, Diliman, Quezon City

Thank you.

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Measures of Variation

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