Measures of dispersion

•Range•Mean absolute deviation•Variance•Standard deviation•Co-efficient of variation

Range

Difference between highest & lowest scores of distribution.• E.g. 3,5,7,9,12. Range = 12-3=9.

• Easy to compute and understand.• Quick impression of dispersion.• Useful for SD.• It is sensitive to extreme value• It dose not give you any information about the pattern of

distribution.• It is based on two variable.

Mean absolute deviation (MD)

• MD=∑│ │/n• ∑│ │ │= absolute deviation of each score from

the mean ignoring plus or minus sign.• n= Total number of scores.Procedure,• Calculate the difference between each item &

the mean. (X - ). • Sum up the values of the difference ignoring

plus and minus sign.

= 40/5=8

Score (X) X -

4 4 – 8 = -4 4

6 6 – 8 = - 2 2

8 8 – 8 = 0 0

10 10 – 8 = 2 2

12 12- 8 = 4 4

Total (X) = 40 ∑ = 12

• MD= 12/5 = 2.4• Evaluation –• It represents the overall dispersion.• MD value is not amenable to mathematical

manipulations.• Hence MD is not useful for advance statistical

analysis.

Variance

• These measures are based on the square deviation of all values from the mean.

• It eliminates the drawback of MD• The variance is the means of the squared

deviations from the mean of distribution.• Mean Variance SD CV• MD= σ 2 = σ = CV• σ 2 or S2 = ∑ 2 / n

= ∑X / n =48/6=8

Score (X) X - 2

3 3 – 8 = -5 25

5 5– 8 = - 3 9

7 7 – 8 = -1 1

9 9 – 8 = 1 1

10 10- 8 = 2 4

14 14 – 8 =6 36

Total (X) = 48 ∑ 2 = 76

• σ 2 or S2 = ∑ 2 / n =76/6 =12.6 Evaluation ,• It express the average dispersion not in the

original units of measurements but in squared units.

• The problem is solved by taking the square root of variance .

• This transform into SD

Standard deviation

• It is the square root of the means of the squared deviation from the mean of distribution.

• It express dispersion in the original scores.

• It is originally denoted by σ, the Greek letter Sigma or S .

SD

• S =3.5

• Grouped data

• = 3.9

Scores f mid point (m) fm Fm * m = fm2

4 – 6 1 5 5 25

7 -- 9 2 8 16 128

10 -- 12 4 11 44 484

13 -- 15 3 14 42 588

16 -- 18 1 17 17 289

19 -- 21 1 20 20 400

n=12 144 1914

2

-

• σ 2 or S2 = (∑fm2 / n) - 2 = 15.5

Evaluation,• SD is more stable from sample to sample.• It is possible to obtain SD for two or more

group combined.• It is more useful in more advanced analysis i.e.

to calculate co -efficient of variation.

Co-efficient of Variation

• SD can’t be compared in absolute magnitudes when the distribution compared have different means.

• E.g. mean of 7 than to mean of 75, it would convey different meaning.

• Therefore the degree of variability must be calculated in relation to the mean of the distribution.

• This is measured by coefficient of variation.• CV indicates the relative variation.

• CV= σ/ * 100

• Democratic participation in four co-operatives.

• There are no significant differences among the SD in the four cooperatives.

• However there are substantial differences between the means of indicating the varying degrees of democratic participation in each co operative.

Co operative A-160 B-150 C-190 D-170

Mean 4.7 5.4 2.9 5.6

SD 2.7 2.9 2.8 2.7

CV 57 % 54 % 95.5 % 48 %

• When the value of coefficient of variation is higher, it means that the data has high variability and less stability. When the value of coefficient of variation is lower, it means the data has less variability and high stability.

• CV shows that the relative deviation from the mean is higher in ‘c’ than in other co operatives, reflecting the given lower degree of Homogeneity in democratic participation i.e. higher degree of variability in democratic participation it.

• The lower the value of the coefficient of variation, the more precise the estimate.

• The advantage of the CV is that it is unit less.