Measures of Dispersion

Measures of Dispersion Dept of AGB Veterinary College, Hebbal, Bangalore

Central Tendency and Variation • Central tendency: mode, median, mean • Variation: range, average(mean) deviation, variance, and standard deviation The idea is the figure out what is happening with your group

Central Tendency • Statisticians need a way to indicate where, along a scale of measures, a given distribution is centered.

• They used three measures of central tendency – The Mode – The Median – The Mean

Measure of Dispersion Measures of dispersion are the measures of “spread” or “scatteredness” of observations about an average.

Measure of Dispersion • The term dispersion is used to indicate the fact that within a given group of data, the observations differ from one another in value or that there is lack of uniformity in their size.

Measure of Dispersion The degree to which numerical; data tend to spread about an average or central value is called the variation or dispersion of data.

Correlation visualized programmed by Erich Neuwirth target value

49

0.02 3

empirical (data) value

0.052226 2

1

Move slider to change correlation

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Measures of dispersion help to study the important characteristics of the dispersion, by measuring the extents to which there are differences between individual observations and the measure of central tendency.

The various measures of dispersion are 1.

2.

3.

4.

Range Variance Standard deviation Coefficient of variation.

Range It is defined as the difference between the highest and the lowest values for the variable in distribution denoted by ‘R’.

Range= Highest value- Lowest value.

In case of frequency distribution it is calculated as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Coefficient of range • It is a relative measure of the range and estimated by using the formula: Coefficient of Range=

L_S X100 L+S

L= Highest value in the series S= Smallest value in the series.

Variance R.A.Fisher first introduced the term in 1913.

• Variance is the mean of the squared deviations about the mean of a series.

Variance is the square of standard deviation • It is denoted by S2 for sample variance • σ2 for population.

Computation of variance Direct method: S2= ∑(X- X)2 n-1 Calculator of Machine formula: S2=∑X2-∑(X)2 n n-1

Standard Deviation • Standard deviation for population is denoted by ‘σ ’ and for sample as ‘s’.

• This was first suggested by Karl Pearson in 1893.

• It is defined as the positive square root of the mean of the squares of deviations of the given observations from their arithmetic mean.

Coefficient of variation It is a relative measure of dispersion.

Coefficient of variation is the percentage variation in mean, standard deviation being considered as the total variation in the mean and symbolically denoted as C.V. or It is a ratio between standard deviation and mean of sample expressed in percentage.

Coefficient of variation s CV= Ă— 100 X S= Standard deviation of the sample X = mean of the sample

The following are the haemoglobin values 9g/100ml) of piglets receiving treatment for piglet anaemia: 9.1,10,11.4,12.4,9.8,8.3,9.9,9.1,7.5,6.7 .

Calculate the sample mean, variance, standard deviation and coefficient of variation.

X

X2

9.1

82.81

10

100

11.4

129.96

12.4

153.76

9.8

96.04

8.3

68.89

9.9

98.01

9.1

82.81

7.5

56.25

6.9

44.89

∑X=94.2

∑X2=913.89

Mean= ∑X n

94.2 = = 10

9.42 (∑X)2

2 ∑X 2 Variance= S =

913.42- 8873.6 10

n

n-1 Standard Deviation=

= 2.90

10-1

2.90

=1.70

Coefficient of variation (C.V)= S

X 100

X =

1.70 X 100 = 18.046% 9.42

Formulas for variability Note the similarity of these formulas Average deviation

Variance

σ2

Standard deviation

X ∑ AD =

− µ

N

(X ∑

=

σ=

) −µ

2

N

(X ∑

) −µ

N

2

(Σx ) Σx −

2

2

σ=

n n −1

Coeff .range =

C.V =

S .E =

σ × 100 X

sd n

L−S × 100 L+S