Stat 09,MOVariation

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Jahangirnagar University © Islam, M.T. http://sites.google.com/site/kjatbd/

Measures of Variation

Md. Tarikul Islam Jahangirnagar University, Bangladesh


The Course: Topics

No Topics

01 Basics of statistics and recap!

02 Collection of data

03 Presentation of data

04 Measures of central tendency

05 Measures of variation

06 Skewness, moments, and kurtosis

07 Correlation analysis

08 Regression analysis

09 Forecasting and time series analysis

10 Probability

11 Sampling

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Recap-1/2

q So far we have seen

ü  Recapping o  Research, types of research, and research methodology

Ø In core of all there is DATA

ü  Basics of statistics o  Data, types of data o  Place of data in statistics with the definition and

characteristics of statistics

ü Data collection and sampling o  What data to collect? From where? How? o  Sampling methods


Recap-2/2

q So far we have seen

ü Data presentation o  Classification, tabulation, and graphs

ü Central Tendency o  Mean, median, and mode, geometric mean, and harmonic

mean

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Today’s Topic

q  Today’s Topic

o  Today we shall see how the values in a data set are deviated from the central value


Which income stream will you like?

q Your preference

o  Which cash flow will you like and why from the following table?

Project –A Project-‐B Project-‐C

April 2011 Tk. 5000 Tk. 5000 Tk. 5000

May 2011 Tk. 5000 Tk. 5200 Tk. 4500

June 2011 Tk. 5000 Tk. 4700 Tk. 7000

July 2011 Tk. 5000 Tk. 5100 Tk.3500

Total Tk. 20000 Tk. 20000 Tk. 20000

Average Tk. 5000 Tk. 5000 Tk. 5000

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Why to know the deviations?

q Why to know the deviations?

o  Because Ø One wants to get as close idea as possible about the real

data set

o  Also Ø To determine the reliability of an average Ø To serve as a basis for the control of variability Ø To compare two or more series with regard to their

variability Ø To facilitate the use of other statistical measures

»  Followed is the explanation of them!


Why to know the deviations?

q Reasons cont’d

o  Reliability of average Ø Whether the average is representative or not

»  High or low variation

o  Basis for control Ø Once one knows the variation and amount of variation

then one can take the step to control

o  Enabling comparison Ø So that one can select the option with more consistency

o  Facilitator of other means Ø Helps in use of some other statistical techniques

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Variations: Types

q  Types

ü Absolute o  It tells us about amount of dispersion under some set of

observations. They give the result in the same units as the original observations Ø Usually calculated with range, average deviation, and

standard deviation

ü  Relative o  When two sets of data are expressed in two different units or

the average value is very much different relative measure is used normally through coefficient of variation Ø This is also measured with the purpose of comparison


Deviations: How to measure?

q Common measures

o  Range o  The interquartile range or quartile deviation o  The average deviation o  The standard deviation o  The Lorenz Curve

o  First four are mathematical while the last one is graphical

o  Remember Ø With all these we measure the amount of variation or its

degree NOT the direction of variation

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Range: Ungrouped data

q Ungrouped data

o  For ungrouped data the formula is Ø Range = L – S

»  L is the largest value and S is the smallest value

o  Coefficient of range Ø COR = (L-S) ÷ (L+S)

ü  In this example, o  Range = L-S = 32-4 = 28 o  COR = (L-s)÷ (L+S)

»  (32-4) ÷ (32+4) »  0.78

5 10 12 14 19

21 26 32 4 23


Range: Grouped data

q Grouped data o  For ungrouped data the formula is

Ø Range = L – S »  L = Upper limit of the highest class »  S = Lower limit of the lowest class

o  Coefficient of range Ø COR = (L-S) ÷ (L+S)

ü  In this example, o  Range = L-S = 50-10 = 40 o  COR = (L-s) ÷ (L+S) = (50-10) ÷ (50+10) = 0.67

Class f

10-20 12

20-30 34

30-40 45

40-50 40

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Range: Pros & Cons

q Range: Pros & Cons

ü  Pros o  Simplest one to calculate o  Simple to understand

ü Cons o  Not based on all observations o  Don’t represent the true characteristics of data


Range: Uses

q Range: Uses

ü  Use of range o  In quality control o  Share price fluctuations o  Weather forecasts

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Average deviation

q Average deviation

o  Absolute one takes the absolute values Ø Therefore AD is obtained by calculating the absolute

deviations of each observation from median or mean and the averaging these deviations by taking their arithmetic mean

o  Can be both absolute and relative


Average deviation: Ungrouped data

q Ungrouped data

ü Here o  Median is 4400 o  ADmedian = Σ ⎜X-median ⎜÷ N

»  N = 5 »  Σ ⎜X-median ⎜ = 1200

Ø So AD = 1200 ÷ 5 = 240

ü Co. of AD o  Co. of AD = AD ÷ median

»  240 ÷ 4400 = 0.054

o  Calculate with mean with the same data!

Av.!Dev.!=!

X!!!X_

"

N!!OR!!

X!!!Median"N

Income ⎜X-median⎜

4000 400

4200 200

4400 0

4600 200

4800 400

1200

C!of!AD!=!Av.!Dev.Mean

!!OR!!Av.!Dev.Median

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Average deviation: Grouped data

q Grouped data

o  Here Ø Mean = 33; using the mean formula for AM Ø ADmean = Σ f ⎜X-mean ⎜÷ N = 204 ÷ 25 = 8.16

o  Go for the median based calculation with the same data!

Av.!Dev.!=!

f! X!!!X_

"

N!!OR!!

f! X!!!Median"N

Sales X f fX ⎜X-µ⎜ f ⎜X-µ⎜

10-20 15 3 45 18 54

20-30 25 6 150 8 48

30-40 35 11 385 2 22

40-50 45 3 135 12 36

50-60 55 2 110 22 44

N=25 fX=825 204


AD: Pros & Cons

q AD: Pros & Cons

ü  Pros o  Simple to understand and calculate o  Considers all data in the set

ü Cons o  Does not consider algebraic sign o  Measured from median which itself is not a good measure o  Rarely used in business

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Standard Deviation

q Standard Deviation

o  Also known as the root mean square deviation Ø Denoted as sigma ‘σ’ Ø Square of SD is variance

o  Calculates the absolute variations of distributions Ø A small standard deviation means high degree of

uniformity of observations and homogeneity of a series; reverse is true too!


SD: Calculation

q SD: Calculation

ü Can be calculated in two ways o  With the actual mean o  With the assumed mean

Ø Assumed mean is used when its difficult to get the mean in absolute form; may be fractions!

o  Remember Ø First you calculate mean Ø Then you see the deviation Ø Finally you square all the deviations and divide by N

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SD: Ungrouped data

q Ungrouped data

o  SD = √ (622 ÷ 10) = 7.89

!!=!(x1"µ)2 +(x

2"µ)2 +(x

3"µ)2 + .........(x

N"µ)

N

2

!=!(x "µ)#

2

N

Workers Wages-X (X-µ) (X-µ)2

A 320 -3 9

B 310 -13 169

C 315 -8 64

D 322 -1 1

E 326 +3 9

F 340 +17 289

G 325 +2 4

H 321 -2 4

I 320 -3 9

J 331 +8 64

N = 10 3230 0 622


SD: Grouped data

q Grouped data

o  We just add ‘f’ for frequency and ‘X’ as the mid point o  These all are related to the mean which is calculated o  Practice SD calculation with the assumed mean!!

!!=!f!(x "µ)#

2

f#

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SD: Some interpretations

q  Some interpretations

ü  For a symmetrical distribution the following relationship holds true

ü  Mean ± 1σ covers 68.27% observations




SD: Relationships

q SD: Relationships

o  AD = 4/5 σ

o  Mean ± AD: includes 57.51% of the observations o  Mean ± σ: includes 68.27% of the observations

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SD: An interpretation

q SD: An interpretation

ü  Three distributions having o  More or less identical means o  Three different SD

Ø What could be the conclusion based on standard deviation?

Ø How much representative the means are?


SD: Pros & Cons

q SD: Pros & Cons

ü  ‘+’ points o  So far the best measure to calculate deviations o  Its possible to calculate combined standard deviations o  CV can be used to compare two or more distributions o  Most prominently used one in real world for further statistical

works

ü  ‘-’ points o  Comparatively difficult to calculate o  Gives more weights to extreme values because of squaring!

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Coefficient of variation

q Coefficient of variation

ü  To measure the relative variation its used ü  If

o  Two observations are different o  Two measurement units are also different o  CV is the right measure to use o  How?

Ø Usually expressed in percentage


CV: Interpretation

q CV: Interpretation

ü Higher CV means o  More variable data & o  Less consistency o  Less uniformity o  Less stability o  Less homogeneity

o  The reverse is true also!

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q Thank You!

q Any Question?!

Stat 09,MOVariation

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