Statistics-Measures of dispersions

MEASURES OF DISPERSIONS

STATISTICS

MEASURES OF DISPERSIONS

• A quantity that measures the variability among the data, or how the data one dispersed about the average, known as Measures of dispersion, scatter, or variations.

2. Common Measures of Dispersion

• The main measures of dispersion1. Range

2. Mean deviation or the average deviation

3. The variance & the standard deviation

1. RANGE

• It is the difference between the largest and the smallest observation in a set of data.

• Range = xm – xo

• Its relative measure known as coefficient of dispersion.

• Coefficient of dispersion =

• It is used in daily temperature recording stick prices rate• It ignores all the information available in middle of data.• It might give a misleading picture of the spread of data.

om

om

xx

xx

1. RANGE• Example:

1. Find the range in the following data.

31,26,15,43,19,10,12,37

Range = xm – xo 33 = 43 – 10

2. Find the range in the following F.D. (Ungrouped)

5 = 8 – 3

Range 5 = 8 – 3

3. Find the range in the following data.

Range = 60 – 10 = 50

X 3 4 5 6 7 8

f 5 8 12 10 4 2

X 10 - 20 20 - 30 30- 40 40 – 50 50 - 60

f 5 8 12 10 4

MEAN (OR AVERAGE) DEVIATION

• It is defined as the “Arithmetic mean of the absolute deviation measured either from the mean or median.

• or for ungroup.

• or for grouped.

n

xxDM

..

N

xxf

N

medianx

N

medianxf


• Example:

1. Calculate mean deviation from the FD (Ungrouped Data).

MD (x) = 33.6 / 20 = 1.68

X f f.x I x – 4.9 I f I x - 4.9 I

2 3 6 2.9 8.7

4 9 36 0.9 8.1

6 5 30 1.1 5.5

8 2 16 3.1 6.2

10 1 10 5.1 5.1

Total Σf =20 Σf.x =98 Σ f I x - 4.9 I = 33.6


• Exp: Calculate mean deviation from the FD (Grouped Data).

MD (x) = 33.6 / 20 = 1.68

M.D = 23.72 / 14 = 1.69

X f Class Mark ( x )

f.x I x – 6.57 I f I x – 6.57 I

2 – 4 2 3 6 3.57 7.14

4 - 6 3 5 15 1.57 4.71

6 – 8 6 7 42 0.43 2.58

8 – 10 2 9 18 2.43 4.86

10 – 12 1 11 11 4.43 4.43

Total Σf =14 Σ f.x =92 Σ f I x – 6.57 I = 23.72

ẋ=92/14=6.57

• It is an absolute measure.

• It is relative measure is coefficient of M.D.

• Coefficient of M.D. =

• It is based on all the observed values.


median

DMor

mean

DM ....

• EXAMPLES


THE VARIANCE ANDSTANDARD DEVIATION

• It is defined as “The mean of the squares of deviations of all the observation from their mean.” It’s square root is called “standard deviation”.

• Usually it is denoted by (for population of statistics) S2 (for sample)

• = for ungrouped

2

2n

xx 2)(

• = for grouped

• It is an absolute measure;

• It is relative measure is coefficient of variation.

•

• Shortcut method

N

xxf 2)(2

100.

VC 100..

.. x

DSVC

222

N

x

N

x

222 .

N

fx

N

xf


• EXAMPLES


VARIANCE AND STANDARD DEVIATION

• Example:

1. Calculate Variance and SD from the FD (Ungrouped Data).

Using Short cut method

var = (564 / 20) - (98 / 20) ^ 2 = 28.2 – 24.01 = 4.09

Sd = √ σ^2 = √ 4.09 = 2.02

X f f.x X^2 f.x^2

2 3 6 4 12

4 9 36 16 144

6 5 30 36 180

8 2 16 64 128

10 1 10 100 100

Total Σf =20 Σf.x = 98 Σ f.x^2=564

222 .

N

fx

N

xf

VARIANCE AND STANDARD DEVIATION

• Exp: Calculate Variance and Standard deviation from the FD (Grouped Data).

Using Short cut method:

var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67

Sd = √ σ^2 = √ 4.67 = 2.16

X f Class Mark ( x )

f.x x^2 f.x^2

2 – 4 2 3 6 9 18

4 - 6 3 5 15 25 75

6 – 8 6 7 42 49 294

8 – 10 2 9 18 81 162

10 – 12 1 11 11 121 121

Total Σf =14 Σ f.x =92 Σ f.x^2 =670

222 .

N

fx

N

xf

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Relative Measures of Relative Measures of DispersionDispersion

Coefficient of Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation (CV)

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Relative Measures of Variation Relative Measures of Variation

Largest Smallest

Largest Smallest

Coefficient of RangeX X

X X

3 1

3 1

Coefficient of Quartile DeviationQ Q

Q Q

Coefficient of Mean DeviationMD

Mean

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Coefficient of Variation (CV)Coefficient of Variation (CV)

Can be used to compare two or more sets of data measured in

different units or same units but different average size.

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100%X

SCV

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Use of Coefficient of VariationUse of Coefficient of VariationStock A:

Average price last year = $50Standard deviation = $5

Stock B:Average price last year = $100Standard deviation = $5 but stock B is

less variable relative to its price

10%100%$50

$5100%

X

SCVA

5%100%$100

$5100%

X

SCVB

Both stocks have the same standard deviation

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Five Number SummaryFive Number SummaryThe five number summary of a data set consists of the minimum value, the first quartile, the second quartile, the third quartile and the maximum value written in that order: Min, Q1, Q2, Q3, Max.

From the three quartiles we can obtain a measure of central tendency (the median, Q2) and measures of variation of the two middle quarters of the distribution, Q2-Q1 for the second quarter and Q3-Q2 for the third quarter.

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The weekly TV viewing times (in hours).

25 41 27 32 43 66 35 31 15 5 34 26 32 38 16 30 38 30 20 21

The array of the above data is given below:

5 15 16 20 21 25 26 27 30 30 31 32 32 34 35 37 38 41 43 66

Five Number SummaryFive Number Summary

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Hrs 22.021}-0.25{2521obs.}5th -obs.0.25{6th obs.5th ; Q1 of VALUE

obs.5.25th data in the obs.th 4

1)1(20 ; Q1 of LOCATION

Five Number SummaryFive Number Summary

Hrs 30.530}-0.50{3103obs.}10th -obs.0.50{11th obs.th 10; Q2 of VALUE

obs.th 50.10data in the obs.th 4

1)2(20;2Q of LOCATION

Minimum value=5.0 Maximum value=66.0

Hrs 36.535}-0.75{37 35 obs}15th -obs{16th 75.0 obs15th ;3Q of VALUE

obs.15.75th data in the obs.th 4

1)3(20 ;3Q of LOCATION

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Box and Whisker DiagramBox and Whisker DiagramA box and whisker diagram or box-plot is a graphical mean for displaying the five number summary of a set of data. In a box-plot the first quartile is placed at the lower hinge and the third quartile is placed at the upper hinge. The median is placed in between these two hinges. The two lines emanating from the box are called whiskers. The box and whisker diagram was introduced by Professor Jhon W. Tukey.

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Construction of Box-PlotConstruction of Box-Plot

1. Start the box from Q1 and end at Q3

2. Within the box draw a line to represent Q2

3. Draw lower whisker to Min. Value up to Q1

4. Draw upper Whisker from Q3 up to Max. Value

Q1

Q3

Q2

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MaxValue

MinValue

25

Construction of Box-PlotConstruction of Box-Plot

1. Q1=22.0 Q3=36.52. Q2=30.53. Minimum Value=5.04. Maximum Value=66.0

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20

10

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Interpretation of Box-PlotInterpretation of Box-Plot

70

60

50

40

30

20

10

0

Box-Whisker Plot is useful to identify

•Maximum and Minimum Values in the data

•Median of the data

•IQR=Q3-Q1,

Lengthy box indicates more variability in the data

•Shape of the data From Position of line within box

Line At the center of the box----Symmetrical

Line above center of the box----Negatively skewed

Line below center of the box----Positively Skewed

•Detection of Outliers in the data12:24 AM

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OutliersOutliersAn outlier is the values that falls well outside the overall

pattern of the data. It might be

• the result of a measurement or recording error,• a member from a different population,• simply an unusual extreme value.

An extreme value needs not to be an outliers; it might, instead, be an indication of skewness.

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Inner and Outer FencesInner and Outer FencesIf Q1=22.0 Q2=30.5 Q3=36.5

25.58IQR1.5QFenceInner Upper

25.0IQR1.5QFenceInner Lower :FencesInner

3

1

0.80IQR3QFenceOuter Upper

5.21IQR3QFenceOuter Lower :FencesOuter

3

1

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Lower Inner Fence 22-1.5(36.5-22) = 0.25Upper Inner Fence 36+1.5(36.5-22) = 58.25

Lower Outer Fence 22-3(36.5-22) = -21.5Upper Outer Fence 36+3(36.5-22) = 80.0

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Identification of the OutliersIdentification of the Outliers

1. The values that lie within inner fences are normal values

2. The values that lie outside inner fences but inside outer fences are possible/suspected/mild outliers

3. The values that lie outside outer fences are sure outliers

80

70

60

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40

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20

10

0

Plot each suspected outliers with an asterisk and each sure outliers with an hollow dot.

*

Only 66 is a mild outlier

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Box plots are especially suitable for

comparing two or more data sets. In such a

situation the box plots are constructed on the

same scale.

Uses of Box and Whisker DiagramUses of Box and Whisker Diagram

MaleFemale

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Standardized VariableStandardized VariableA variable that has mean “0” and

Variance “1” is called standardized variable

Values of standardized variable are called standard scores

Values of standard variable i.e standard scores are unit-less

Construction

VariableofDeviation Standard

VariableofMeanVariableZ

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X Z

3 25 -1.3624 1.85611.8561

6 4 -0.5450 0.29700.2970

11 9 0.81741 0.66820.6682

12 16 1.0899 1.18791.1879

32 54 0 4.009

5.134

54

84

32

2

xS

n

XX

2)( XX

67.3

8

X

Sx

XXZ

14

009.4

0

2

zS

n

ZZ

2)( ZZ

Variable Z has mean “0” and

variance “1” so Z is a standard variable.

Standard Score at X=11 is8174.0

67.3

811

Sx

XXZ

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Standardized VariableStandardized Variable

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The industry in which sales rep Mr. Atif works has mean annual sales=$2,500

standard deviation=$500.

The industry in which sales rep Mr. Asad works has mean annual sales=$4,800

standard deviation=$600.

Last year Mr. Atif’s sales were $4,000 and Mr. Asad’s sales were $6,000.

Performance evaluation by z-scoresPerformance evaluation by z-scores

Which of the representatives would you hire if you have one sales position to fill?

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Performance evaluation by z-scoresPerformance evaluation by z-scores

3500

500,2000,4

B

B

BBB

Z

S

XXZ

Sales rep. Atif

XB= $2,500

S= $500

XB= $4,000

Sales rep. Asad

XP =$4,800

SP = $600

XP= $6,000

2600

800,4000,6

P

P

PPP

Z

S

XXZ

Mr. Atif is the best choice12:24 AM

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valuesof 68%about contains1SX

The Empirical RuleThe Empirical Rule

X

68%

1SX

valuesof 99.7%about contains3SX

valuesof 95%about contains2SX 95%

X 2S

X 3S

99.7%

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Chebysev’s TheoremChebysev’s Theorem

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A distribution in which the values equidistant from

the centre have equal frequencies is defined to be

symmetrical and any departure from symmetry is

called skewness.

1. Length of Right Tail = Length of Left

Tail

2. Mean = Median = Mode

3. Sk=0

a) Sk=(Mean-Mode)/SD

b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)

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Measures of Skewness

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A distribution is positively skewed, if the observations tend to concentrate more at the lower end of the possible values of the variable than the upper end. A positively skewed frequency curve has a longer tail on the right hand side

1. Length of Right Tail > Length of Left

Tail

2. Mean > Median > Mode

3. SK>0


b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)

MeasuresMeasures ofof SkewnessSkewness

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A distribution is negatively skewed, if the

observations tend to concentrate more at the upper

end of the possible values of the variable than the

lower end. A negatively skewed frequency curve

has a longer tail on the left side.

1. Length of Right Tail < Length of Left

Tail

2. Mean < Median < Mode

3. SK< 0


b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)

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Measures of Skewness

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The Kurtosis is the degree of peakedness or flatness of a unimodal (single humped) distribution,

• When the values of a variable are highly concentrated around the mode, the peak of the curve becomes relatively high; the curve is Leptokurtic.

• When the values of a variable have low concentration around the mode, the peak of the curve becomes relatively flat;curve is Platykurtic.

• A curve, which is neither very peaked nor very flat-toped, it is taken as a basis for comparison, is called Mesokurtic/Normal.

Measures of Kurtosis

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1. If Coefficient of Kurtosis > 3 ----------------- Leptokurtic.

2. If Coefficient of Kurtosis = 3 ----------------- Mesokurtic.

3. If Coefficient of Kurtosis < 3 ----------------- is Platykurtic.

4

22

n X-XCoefficient of Kurtosis=

X-X

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Moments about originC.I f x f.x f. x^2 f.x^3 f.x^4

2.5 2.9 2 2.7 5.4 14.58 39.366 106.2882

3 3.4 7 3.2 22.4 71.68 229.376 734.0032

3.5 3.9 17 3.7 62.9 232.73 861.101 3186.074

4 4.4 25 4.2 105 441 1852.2 7779.24

4.5 4.9 20 4.7 94 441.8 2076.46 9759.362

5 5.4 12 5.2 62.4 324.48 1687.296 8773.939

5.5 5.9 9 5.7 51.3 292.41 1666.737 9500.401

6 6.4 8 6.2 49.6 307.52 1906.624 11821.07

Total (Σ) 100 453 2126.2 10319.16 51660.38

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Moments about origin

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Regression

X Price

Y (Quantity

Demanded) x*y x^215 440 6600 22520 430 8600 40025 450 11250 62530 370 11100 90040 340 13600 160050 370 18500 2500

Σx=180 Σy=2400 Σx.y=69650 Σx^2=6250

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Documents

Statistics-Measures of dispersions