Measures of Dispersion

&skewness

statistics topic-2

intro--

1.Measures of Dispersion

Dispersion is a measure of the variation of the items.

If value of items are same then there is no variance and

dispersion will be zero.

More variations more will be dispersion.

MethoDs of Measuring Dispersion

1.Range

2.Interquartile range

&

Quartile deviation

3.Means

deviation

4.Standard

deviation.

5.Coefficient

of variation

6.Lorenze

curve.

1.Range

It is simplest measure of dispersion.

It is define the difference b/w largest and smallest value.

R=L - S

15,22,17,32,28

Largest value = 32 smallest value = 15

R=L - S

R= 32-15= 17.

Coefficient of Range

L-S

L+S=

32-15

32+15=

17

470.36

2.Interquartile range & Quartile deviation

A. Interquartile range.

It also called as semi-interquartile.

IQ.Range = Q3- Q1

B. Quartile deviation.

It is the difference b/w upper upper quartile (Q3) and

lower quartile(Q3)

Q3-Q1

2Q.D =

C. Coefficient of Quartile deviation.

Q3+Q1Coeff. Of Q.D =

Q3-Q1

3.Means deviation

It is also known as average deviation.

M.D from median =

Mean deviation can be can be computed either from the

mean or median.

It is the difference b/w upper upper quartile (Q3) and

lower quartile(Q3)

∑(X-M)

N

M.D from Mean =∑(X-x ̄)

N

4.Standard deviation

Standard deviation is most widely used measure of

dispersion.

S.D first used by Karl Pearson. In 1893.

S.D is also called as root mean square deviation.

Nσ = ∑(X-x ̄̄)2

S.D is denoted as - σ

S.D is widely used in sampling theory and test of

significant.

Coefficient of S.D =x ̄

σ

Difference between mean deviation and S.D

1.In M.D sign + and – ignored.

In S.D +,- sign not ignored.

2.M.D can be calculated either from mean ,median and

mode.

S.D is always computed from mean only.

Variance

It is also a measure of dispersion.

The term variance is first used by R.A.Fisher in 1918.

Variance is the square of the standard deviation.

Variance= σ2

Variance

= N

∑f(X-x ̄̄)2

5.Coefficient of variance.

Coefficient of variation is an important relative measure of

dispersion.

It was developed by Karl Pearson.

It is widely used in comparing the variability of two or

more series.

It is denoted as C.V.

C.V.

=

σ

x ̄*100

C.V.

=

σ

x ̄*100

Variance= 324

324 = 18

18

180* 100

1800

180=10

6.Lorenze curve

It is graphical method of dispersion.

It studies the degree of inequality in the distribution of

income and wealth b/w countries.

1.Measures of skewness

1.Measures of skewness

Skewness means lack of symmetry in frequency

distribution.

It gives us idea about the shape of the frequency curve.

When a distribution is not symmetrical it is called a

skewed distribution.

Skewness tells us about the asymmetry of the frequency

distribution.

syMMetrical anD skeweD Distribution.

In a symmetrical dist. Skewness is not present.

1.Symmetrical distribution-

In this situation mean,median and mode are equal.

x ̄=M=Z

1.Perfectly symmetrical distribution.

In this case x ̄,M, and Z. are equal.

+VE-VE

x ̄=M=Z

In this case normal

distribution is bell shaped.

In normal distribution mean

is zero and variance is 1.

2.skewed distribution-

a. positively skewed

distribution.

b. Negatively

skewed distribution.

a. positively skewed.

In this case x ̄,>M> Z. .

+VE-VE x ̄

In this case normal

distribution is +vly skewed..

MZ

Tail right side.

b. negatively skewed.

In this case x ̄,<M< Z. .

+VE-VE x ̄

In this case normal

distribution is -vly skewed..

M Z

Tail left side.

Measures of skewness

1.Karl Pearson

method

2.Bowley Method

3.kelly’s method.

1.Karl Pearson method

Carl Pearson method based on 4 components.

Mean,median,mode,and S.D.

Absolute measure Relative measure/coef. Of skewness

Sk= x ̄-z

When mode is ill(eliminate)

Sk= 3(x ̄-M)

Coeff. Of Sk=x ̄-z

σ

When mode is ill(eliminate)

Coeff. Of Sk=3(x ̄-z)

σ

In Karl Pearson method , coeff. Of skewness usually

lies bw +1 or -1.

If mode(z) is ill then value of skewness lies b/w +3 or -3.

2.Bowley Method

Bowley method is based upon median(M).

Bowley used first quartile (Q1) and third quartile(Q3).

Bowley method is also called quartile method of

measuring skewness .

Absolute measure Relative measure/coef. Of skewness

Sk= Q3+Q1-2M Coeff. Of Sk=Q3+Q1-2M

sQ3-Q1

3.kelly’s method.

Kellys’s method is based on percentiles (%) and

deciles(.)

Absolute measure Relative measure/coef. Of skewness

Sk= P90+P10-2MCoeff. Of Sk=

P90+P10-2M

P90-P10

This method is not popular .

It is suitable when the skewness is based on percentiles

or deciles.

Measures of kurtosis

&correlation

statistics topic-3