MEASURES OF DISPERSION

Dispersion means scatterness. The degree to which numerical data tend to spread about an average value is called the dispersion of the data. There are four measures of dispersion.

Range:

Range = L – S, where L = largest value; S = smallest value.

Coefficient of Range = $\frac{L-S}{L+S}$

Illustration 1: The range of the following set of observations 2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3, is

(A) 11 (B) 7

(C) 5.5 (D) 6

Solution: (B)

Quartile Deviation:

Quartile deviation = $\frac{Q_3-Q_1}{2}$

Coefficient of Quartile Deviation = $\frac{Q_3-Q_1}{Q_3+Q_1}$

Mean Deviation:

It is the average of the modulus of the deviations of the observations in a series taken form mean or median.

Methods for Calculation of Mean Deviation:

Case I: For Ungrouped Data.

In this case the mean deviation is given by the formula

Mean Deviation = M.D. = $\frac{\Sigma |x-A|}{n}=\frac{\Sigma |d|}{n}$,

where 'd' stands for the deviation from the mean or median and |d| is always positive whether d itself is positive or negative and n is the total number of items.

Case II: For Grouped data.

Let x₁, x₂, x₃, …, x_n occur with frequencies f₁, f₂, f₃, ,f_n respectively and let Sf = n and M can be either Mean or Median, then the mean deviation is given by the formula.

Mean Deviation = $\frac{\Sigma f|x-M|}{\Sigma f}=\frac{\Sigma f|d|}{n}$

Where d = |x – M| and Sf = n.

Coefficient of Mean Deviation = $\frac{MeanDeviation}{Median}$

or = $\frac{MeanDeviation}{Mean}$ (In case the deviations are taken from mean)

Standard Deviation:

The Positive square root of the average of squared deviations of all observations taken from their mean is called standard deviation. It is generally denoted by the Greek alphabet s or s.

Variance:

The square of the standard deviation is called variance and is denoted by s².

Coefficient of Standard Deviation:

It is the ratio of the standard deviation to it's A.M. i.e., Coefficient of standard deviation = $\frac{\sigma }{\bar{x}}$.

Standard Deviation for Ungrouped Data:

Direct method:

In case of individual series, the standard deviation can be obtained by the formula.

$\sigma =\sqrt{\frac{{(x-\bar{x})}^2}{n}}=\sqrt{\frac{\Sigma d^2}{n}}$ [First Form]

where d = x - $\bar{x}$ and x = value of the variable or observation, $\bar{x}$= arithmetic mean, n = total number of observations.

Short-cut Method:

This method is applied to calculate Standard deviation, when the mean of the data comes out to be a fraction. In that case it is very difficult and tedious to find the deviations of all observations from the mean by the earlier method. The formula used is

$\sigma =\sqrt{\frac{\Sigma d^2}{n}-{\left(\frac{\Sigma d}{n}\right)}^2}$ [Second Form] where d = x – A, A = assumed mean, n = total number of observations.

Standard deviation for grouped data:

It is calculated by the following formula.

$\sigma =\sqrt{\frac{\Sigma f{(x-\bar{x})}^2}{n}}$ [Third Form]

where $\bar{x}$ is A.M., x is the size of the item, and f is the corresponding frequency in the case of discrete series.

But when the mean has a fractional value, then the following formula is applied to calculate S.D.

$\sigma =\sqrt{\frac{\Sigma f{d}^2}{n}-{\left(\frac{\Sigma fd}{n}\right)}^2}$ [Fourth Form]

where d = x – A, A = assumed mean, $n=\sum f=$total frequency.

Standard deviation in continuous series:

Direct Method. The standard deviation in the case of continuous series is obtained by the following formula.

$\sigma =\sqrt{\frac{\Sigma f{(x-\bar{x})}^2}{n}}$ [Fifth Form]

where x = mid-value, $\bar{x}$ = A.M., f = frequency, n = total frequency.

Combined Standard Deviation:

Let $\sigma$₁ and $\sigma$₂ be the S.D. of the two groups containing n₁ and n₂ items respectively. Let ${\bar{x}}_{1}and{\bar{x}}_2$be their respective A.M. Let x and s be the A.M. and S.D. of the combined group respectively. Then

$\bar{x}=\frac{n_1{\bar{x}}_1+n_2\bar{x}}{n_1+n_2}$.

$\sigma =\sqrt{\frac{n_1{\sigma }_1^2+n_2{\sigma }_2^2+n_1{d}_1^2+n_2{d}_2^2}{n_1+n_2}}$, where $d_1={\bar{x}}_1-\bar{x}and{d}_2={\bar{x}}_2-\bar{x}$.

Variance = $\frac{\Sigma {(x-\bar{x})}^2}{n}$, or Variance = s²

$\therefore$ = $\sqrt{Variance}$

Variance = $\left[\frac{\Sigma f{d^{\prime }}^2}{n}-{\left(\frac{\Sigma f{d}^{\prime }}{n}\right)}^2\right]$ x i² (Continuous Series)

Standard Deviation of n Natural Number:

$\sigma$ = ${\left(\frac{1}{12}(n^2-1)\right)}^{1/2}$

Imperical Relation:

$\bullet$Mean deviation = $\frac{4\sigma }{5}$

$\bullet$Semi-intererquatile range = $\frac{2\sigma }{3}$

Illustration 2: The S.D of 7 scores 1, 2, 3, 4, 5, 6, 7 is

(A) 4 (B) 2

(C) $\sqrt{7}$ (D) none of these

Solution: S.D of first n natural number is $\sqrt{\frac{n^2-1}{12}}$

For n = 7 this value = $\sqrt{\frac{7^2-1}{12}}=\sqrt{4}=2$

Symmetric and Skew-symmetric

In a symmetrical distribution, Mean, Median, Mode coincide. Here, frequencies are symmetrically distributed on both sides of some central value.

A distribution which is not symmetrical, is called skew-symmetrical. In a moderately skew-symmetric distribution,

Mean - Mode = 3 (Mean - Median)

In a positively skew-symmetric distribution, the value of mean is maximum and that of mode is least, and the median lies between the two.

In a negatively skew-symmetric distribution, the value of mode is maximum and that of mean is least, and the median lies between the two.

In a negatively skew-symmetric distribution, the value of mode is maximum and that of mean is least, and the median lies between the two.

Absolute Measures of skewness are

$(i)\bar{x}-M_{e,}(ii)\bar{x}-M_0,(iii)Q_3+Q_1-2Q_2.$

Relative Measures of Skewness are

(i) Karl Pearson's coefficient of skewness.

$=\frac{\bar{x}-M_0}{\sigma }$. It is lies between –1 and 1.

(ii) Bowley's coefficient of skewness

$=\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1}$. It also lies between –1 and 1.

(iii) Kelley's cofficient of skewness $=\frac{P_{10}+P_{90}-2Q_2}{Q_{90}-P_{10}}.$

AREA RESULT

For a symmetrical distribution (normal curve),

(i) the interval $(\bar{x}-\sigma ,\bar{x}+\sigma )$ contains 68.27% items.

(ii) the interval $(\bar{x}-2\sigma ,\bar{x}+2\sigma )$ contains 95.45% items.

(iii) the interval $(\bar{x}-3\sigma ,\bar{x}+3\sigma )$ contains 99.74% items.