Dispersion is a statistical term that describes the size of the range of values expected for a particular variable.
It is mainly useful to find the relationship between the set of data. The tendency of a data scattered over a range is called as Dispersion. It is one of the most important frequency distribution. In other simple words, it can be called as a spread or variation.

Dispersion is defined as the measurement including the average deviation, variance and the standard deviation. The most widely used measure of dispersion are standard deviation and variance.

The mean value of a data gives the information about the set. Many sets have the same values. And to find out how sets are different, we need more information. And that can be done by looking at, how the data is spread out, or dispersed.

There are many ways to measure these dispersion and few of them are:
  1. Mean Absolute Deviation
  2. Range
  3. Standard Deviation
  4. Variance
  5. Inter-quartile range

Mean Absolute Deviation: The difference between the individual values in the data set and the mean got from the average absolute values is called as Mean Absolute Deviation.
 
This method measure the average distance between the values in the data and the mean.

$Mean\ Absolute\ Deviation = $$\frac{\sum(x_i - \overline{x})}{n}$

Here $\bar{x}$, is the arithmetic mean of data set.

Range: This is one of the most simplest method of measuring dispersion. Range can be found by subtracting lowest value from highest value and for this reason the range is often unreliable for the measure of dispersion as it depended on only two values in the set.

For example:
Range of following data: 22, 35, 37, 42, 30, 46, 48, 53, 54, 64
Highest value = 64
Lowest value = 22
=> Range = 64 - 22 = 42
The Range is 42. This gives us a very less knowledge on how the values are dispersed.

Standard Deviation: In simple words, standard deviation is the square root of the variance.
Formula of Standard Deviation
Variance: The variance is the measure of how far a set of number is spread out. It is the average squared difference of the data from the mean score of a distribution.
Formula of Variance
Inter-Quartile Range: Inter-quartile range is the difference between the first and the third quartile range. It measures the spread of the middle 50% of an ordered data set.
Formula Inter Quartile RangeHere,
Q3 is the third quartile
Q1 is the first quartile
A dispersion graph represents a variation from the mean. This graph gives an clear idea on how one variable can affect the other or we can say that this graph helps us to know the relationship between the two variables.

During the construction of the graph, as an initial step we should clearly explain the variables that are to be evaluate. This graph places individual data values along a line, hence presenting the position of each data value in relation to the other data values.
Graph of Dispersion
The coefficient of dispersion is used to know the whether the data distribution in a Poisson or Binomial. The coefficient of dispersion is pure number independent of the units of the measurement. It gives us the clear idea about the scattering of the values in the data about the average. 
Some measure are very important in the study of dispersion as they tell us whether the dispersion is small or large. Types of dispersion are given below:

(a)  Relative Measure of Dispersion 
(b) Absolute Measure of Dispersion

Relative Measure of Dispersion:
As mentioned earlier, the absolute measure of dispersion fails to compare the amount of dispersion, when there are two set of observation. In such situation, we use relative measure of dispersion. This measure of dispersion helps us to compare the dispersion in two or more than two sets of observations. Here, the units in which original data is measured is free from them. If the original data is in dollar or kilogram, then we do not use this unit in the answer found from this measure.

Absolute Measures of Dispersion:
This measure of dispersion gives us an clear idea on the amount of dispersion in a set of observation. The answer found here will be in the same units as the units of the original observation. Say, if the observation is in kilograms, then the absolute measure will be also in kilograms. The only drawback of this measure is that we cannot compare the amount of dispersion when we have 2 sets of observation. The absolute measures which are commonly used are:
  1.  Mean Deviation
  2. Range
  3. Standard deviation and Variance
  4. Quartile Deviation
The composite dispersion calculates the range of the returns from larger amount of dispersion to the smaller amount. The composite internal dispersion is a measure of the variability of portfolio-level returns for only those portfolios that are included in the composite for the full year around the composite return. The range is calculated as a lot, if not, performance related statistics from the reported composite average. Then all the expectations are based on the average weighted return in the composite.
Below are few problems based on the Dispersion:

Solved Examples

Question 1: Find the standard deviation and the variance for the following values:
2, 4, 5, 6, 6, 8, 10, 15.
Solution:
 
Let us first find the mean for the above given data:

Mean = $\frac{2+4+5+6+6+8+10+11}{8}$

= $\frac{56}{8}$

= 7

$\therefore$ Mean ($\bar{x}$)= 7

Now, square the difference of the mean value from given values.

(x - $\bar{x}$)  $({x-\bar{x}})^2$ 
 2  -5 25
 4 -3 9
 5 -2 4
 6-1 1
 6 -1
 8 1
 10 3
 11 4 16 
 Total   66

Now, calculate the variance and the standard deviation
Given N= 8

= $\frac{66}{8}$

Variance = 8

and standard deviation is the square root of the variance

S.D = $\sqrt{8}$

Standard deviation = 2.83

 

Question 2: For the following data, measure the square of standard deviation.
448, 452, 468, 472.

Solution:
 
Let us first calculate the mean

$\bar{x}$ = $\frac{\sum_{K=1}^{n}(x_{k})}{N}$

$\bar{x}$ = $\frac{448+452+468+472}{4}$

$\bar{x}$ = $\frac{1840}{4}$

$\bar{x}$ = 460

Now let us find out the standard deviation using the formula,

$\sigma$ = $\sqrt{\frac{\left ( \sum_{K=1}^{n}(x-\bar{x}) \right )^2}{n}}$

= $\sqrt{\frac{(448-460)^2+(452-460)^2+(468-460)^2+(472-460)^2}{4}}$

= $\sqrt{\frac{144+64+64+144}{4}}$

= $\sqrt{104}$

= 10.19.

Square of standard deviation:

$(\sigma)^2$ = 10.192

= 103.83