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Measures of variation

Variability measures• In addition to locating the center of the observed

values of the variable in the data, another important aspect of a descriptive study of the variable is numerically measuring the extent of variation around the center. Two data sets of the same variable may exhibit similar positions of center but may be remarkably different with respect to variability.

• The variability measures should have the following characteristics:

- be minimum if all the value of the distribution are the same

-increase as increase the difference among the values of the distribution

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Shops

Revenues Costs employee

place Director gender

ShopOn-line

R.O

1 350 205 5 city male yes 145

2 200 100 3 suburbs male yes 100

3 600 350 10 Near the city

female no 250

4 500 270 10 suburbs female no 230

5 270 200 6 city male no 70

6 180 120 3 city male no 60

7 205 105 3 suburbs male no 100

8 340 210 5 Near the city

female no 120

9 280 140 4 city female yes 140

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Variability

revenue

350

200

600

500

270

180

205

340

280

revenue

(A)

revenue

(B)

revenue (C)

325 300 140

325 350 270

325 400 830

325 200 605

325 300 120

325 325 200

325 300 190

325 400 200

325 350 370

Observed distribution

Possible distribution

All the 3 possible distribution have the same mean of the observed one

325x

BUT the distribution are very different!!!

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Some measures of variability

Range It is the width of the interval that contain all

the values of the distribution.

Interquartile rangeIt is the width of the interval that contain 50%

the values of the distribution.(central ones).

minmax xxrange

13 QQdQ

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ExampleRevenue

350

200

600

500

270

180

205

340

280

Revenue

(A)Revenue

(B)Revenue

(C)

325 300 140

325 350 270

325 400 830

325 200 605

325 300 120

325 325 200

325 300 190

325 400 200

325 350 370

xmin180 325 200 120

xmax600 325 400 830

Range=xmax-xmin420 0 200 710

ANo VariabilityAll values are the same

From A to B and from B to C, the variability increasaes, the range is higher.

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Deviation from the mean

The variance σ2 is function of the differences among each value xi and the mean

The sum of squared deviation is

n

11

2

i2 xx

n1

n

1i

2

i xx)X(Dev

02

x

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The standard is the squared root of the variance

The coefficient of variation CV is the ratio between the standard dev. and the mean, multiplied 100

n

1i

2

i xxn1

100x

CV 0x

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Example

Revenue

xj

Differences from mean

(xj-μ)

Squared differences

(xj-μ)2

350 25 625

200 -125 15625

600 275 75625

500 175 30625

270 -55 3025

180 -145 21025

205 -120 14400

340 15 225

280 -45 2025

x 325mean

0xxn

1ii

163200)X(Devxxn

1i

2

i

3,181339

163200n

)X(Devxx

n1 2

n

1i

2

i

7,1343,18133

xxn1 n

1i

2

i

Mean property

s.s.dev.=163200

Variance=18133,3

Std.Dev.=134,7

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Variabilità dei ricavi dei punti vendita

• Un basso grado di variabilità indica che i punti vendita realizzano performance simili (i ricavi si discostano poco tra di loro)

• Viceversa un alto grado di variabilità fa capire che c’è una certa eterogeneità nei risultati delle vendite ottenuti nei diversi negozi

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Variance from a frequency distribution

10,6988,54

nxxn1

j

K

1j

2

j2

Employee(xj)

Shops(nj)

3 2

4 1

6 3

7 1

10 2

(xj-μ)2*nj

19,34

4,45

0,04

0,79

30,26

11,6x

47,210,6 %43,4010011,647,2

CV

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Standardised values

If a quantitative variable X as mean

and standard deviation σ, it is possible to obtain its standardised values

x

1...ni / xxy ii

The distribution of Y has zero mean and standard deviation equal to 1

Comparison among two founds (equal mean)

In last 5 years F1 and F2 had the same performance in mean, but variances are different Var(F1)>Var(F2)

F1 F2

2003 7,7 6,4

2004 6,1 5,9

2005 0,4 3,2

2006 9,8 7,1

2007 3,5 4,9

mean 5,5 5,5

var 10,7 1,8

Higher variability means that performance very different from the mean are more frequent. Higher volatility Higher risk

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Comparison among the performance of two founds (different mean)

F1 has a mean and a variance higher than F2.

Can we say that F1 is an higher risk found than F2?

F1 F2

2003 9,7 1,4

2004 7,1 1,9

2005 0,9 2,2

2006 9,9 2,1

2007 7,5 4,9

media 7,0 2,5

var 10,6 1,5

CV 46,5 49,3

We have to compare the CV F1 has less variability

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