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Measures of Dispersion

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

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

Quartile Formulas

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

First quartile position: Q1 = n/4

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

Third quartile position: Q3 = 3n/4

where n is the number of observed values

(n = 9)

Q1 is in the 9/4 = 2.25 position of the ranked data

so Q1 = 12.25

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

Quartiles

Example:(continued)

Overtime Hours

10-15 15-20 20-25 25-30 30-35 35-40 Total

No. of employees

11 20 35 20 8 6 100

Calculate Median, First Quartile, 7th Decile & Interquartile Range

Same 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.

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:

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

Coefficient of Range

Coefficient of Range = [L-S]/[L+S]

Where,

L = Largest value in the data set

S = Smallest value in the data set

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

Interquartile Range

Median(Q2)

XmaximumX

minimum Q1 Q3

Example:

25% 25% 25% 25%

12 30 45 57 70

Interquartile range = 57 – 30 = 27

Coefficient of QD

Coefficient of QD= [Q3-Q1]/[Q3+Q1]

Average Deviation

Coefficient of AD = (Average Deviation)/(Mean or Median)

Calculate the Coefficient of AD (mean)

Sales (in thousand Rs) No. of days

10-20 3

20-30 6

30-40 11

40-50 3

50-60 2

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

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∑

=

−=

Calculation Example:Sample Standard Deviation

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

n = 8 Mean = X = 16

4.3087

130

7

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

1-n

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

2222

2222

==

−++−+−+−=

−++−+−+−=

A measure of the “average” scatter around the mean

Population vs. Sample Standard Deviation

Population vs. Sample Standard Deviation (Grouped Data)

Calculate the Standard Deviation for the following sample

Number of pins knocked down in ten-pin bowling matches

Solution

Number of pins knocked down in ten-pin bowling matches

Calculate the Standard Deviation for the following sample

Heights of students

Solution

Calculate the Standard Deviation for the following sample

Number of typing errors

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

Measuring variation

Small standard deviation

Large standard deviation

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)

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 ⋅

=

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 =⋅=⋅

=

Sample vs. Population CV

Calculate!

Calculate the coefficient of variation for the following data set. The price, in cents, of a stock over five trading days

was 52, 58, 55, 57, 59.

Solution

Pooled Standard Deviation

Pooled Standard Deviation

Z Scores

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

S

XXZ

−=

Z Scores

Example: If the mean is 14.0 and the standard deviation is 3.0,

what is the Z score for the value 18.5?

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)

1.53.0

14.018.5

S

XXZ =−=−=

(continued)

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

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 ±

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 ±

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

Calculate!

Calculate the mean & the standard dev. Calculate the percentage of

observations between the mean & 2σ±

Variable Frequency

44-46 3

46-48 24

48-50 27

50-52 21

52-54 5

Find the correct Mean & Std. dev.

The Mean & Standard Deviation of a set of 100 observations were worked out as 40 & 5 respectively. It was later learnt that a mistake had been made in data entry – in place of 40 (the correct value), 50 was entered.

Skewness

A fundamental task in many statistical analyses is to characterize the location and variability of a data set (Measures of central tendency vs. measures of dispersion)

Both measures tell us nothing about the shape of the distribution

It is possible to have frequency distributions which differ widely in their nature and composition and yet may have same central tendency and dispersion.

Therefore, a further characterization of the data includes skewness

Positive & Negative Skew

Positive skewness There are more observations below the mean than

above it When the mean is greater than the median

Negative skewness There are a small number of low observations and a

large number of high ones When the median is greater than the mean

Measures of Skew

Skew is a measure of symmetry in the distribution of scores

Positive Skew

Negative Skew

Normal (skew = 0)

Measures of Skew

The Rules

I. Rule One. If the mean is less than the median, the data are skewed to the left.

II. Rule Two. If the mean is greater than the median, the data are skewed to the right.

Karl Pearson Coefficient of Skewness

Karl Pearson Coefficient - Properties

Advantage – Uses the data completelyDisadvantage – Is sensitive to extreme values

Calculate!

Calculate the coefficient of skewness & comment on its value

Calculate!

Calculate the coefficient of skewness & comment on its value

Profits (lakhs) No. of Companies

100-120 17

120-140 53

140-160 199

160-180 194

180-200 327

200-220 208

220-240 2

Bowley’s Coefficient of Skewness

Bowley’s Coefficient - Properties

Advantage – Is not sensitive to extreme valuesDisadvantage – Does not use the data completely

Calculate!

Kelly’s Coefficient of Skewness

Calculate!

Obtain the limits of daily wages of the central 50% of the workers

Calculate Bowley’s & Kelly’s Coefficients of Skewness

Wages No. of Workers

Below 200 10

200-250 25

250-300 145

300-350 220

350-400 70

Above 400 30

Moments about the Mean