**Measures of Dispersion**

A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution. The measure of dispersion displays and gives us an idea about the variation and the central value of an individual item.

In other words, dispersion is the extent to which values in a distribution differ from the average of the distribution. It gives us an idea about the extent to which individual items vary from one another, and from the central value.

The variation can be measured in different numerical measures, namely:

**(i) Range:** It is the simplest method of measurement of dispersion and defines the difference between the largest and the smallest item in a given distribution. If Y max and Y min are the two ultimate items, then

Range = Y max – Y min

**(ii) Quartile deviation:** It is known as semi-interquartile range, i.e., half of the difference between the upper quartile and lower quartile. The first quartile is derived as Q, the middle digit Q1 connects the least number with the median of the data. The median of a data set is the (Q2)second quartile. Lastly, the number connecting the largest number and the median is the third quartile (Q3). Quartile deviation can be calculated by

Q = ½ × (Q3 – Q1)

**(iii) Mean deviation:** Mean deviation is the arithmetic mean (average) of deviations **⎜**D**⎜**of observations from a central value (mean or median).

Mean deviation can be evaluated by using the formula: A = 1⁄n [∑i|xi – A|]

**(iv) Standard deviation:** Standard deviation is the square root of the arithmetic average of the square of the deviations measured from the mean. The standard deviation is given as,

σ = [(Σi (yi – ȳ) ⁄ n] ½ = [(Σ i yi 2 ⁄ n) – ȳ 2] ½

Apart from a numerical value, graphics methods are also applied for estimating dispersion.

**Types of Measures of Dispersion**

(1) Absolute measures

- Absolute measures of dispersion are expressed in the unit of variable itself, like kilograms, rupees, centimeters, marks,

(2) Relative measures

- Relative measures of dispersion are obtained as ratios or percentages of the average.
- These are also known as coefficients of dispersion.
- These are pure numbers or percentages that are totally independent of the units of

**Characteristics of a Good Measure of Dispersion**

- It should be easy to calculate and simple to understand.
- It should be based on all the observations of the series.
- It should be rigidly defined.
- It should not be affected by extreme values.
- It should not be unduly affected by sampling fluctuations.
- It should be capable of further mathematical treatment and statistical analysis.

*Related Topics:*

- Measures of Central Tendency- Median
- Measures of Central Tendency- Mean
- Measures of Central Tendency- Mode

**What are the objectives of computing dispersion?**

(1) Comparative study

- Measures of dispersion give a single value indicating the degree of consistency or uniformity of distribution. This single value helps us in making comparisons of various distributions.
- The smaller the magnitude (value) of dispersion, higher is the consistency or uniformity and vice-versa.

(2) Reliability of an average

- A small value of dispersion means low variation between observations and average. It means that the average is a good representative of observation and very reliable.
- A higher value of dispersion means greater deviation among the observations. In this case, the average is not a good representative and it cannot be considered reliable.

(3) Control the variability

- Different measures of dispersion provide us data of variability from different angles, and this knowledge can prove helpful in controlling the variation.
- Especially in the financial analysis of business and medicine, these measures of dispersion can prove very useful.

(4) Basis for further statistical analysis

- Measures of dispersion provide the basis for further statistical analysis like computing correlation, regression, test of hypothesis, etc.

**What are the various ‘absolute measures’ of Dispersion?**

Following are the different ‘absolute measures’ of dispersion:

**(1) Range**

- It is the simplest method of measurement of dispersion.
- It is defined as the difference between the largest and the smallest item in a given distribution.
**Range = Largest item (L) – Smallest item (S)**

**(2) Interquartile Range**

- It is defined as the difference between the Upper Quartile and Lower Quartile of a given distribution.

**Interquartile Range = Upper Quartile (**Q_{3}**)–Lower Quartile(**Q_{1}**)**

**(3) Quartile Deviation**

- It is known as Semi-Inter-Quartile Range, i.e. half of the difference between the upper quartile and lower quartile.
**Quartile Deviation =**_{\(\begin{array}{l}\frac{Upper\, Quartile\left ( _{Q3} \right )\, -\, Lower\, Quartile\left ( _{Q1} \right )}{2}\end{array} \)}

**(4) Mean Deviation**

- Mean deviation is the arithmetic mean (average) of deviations
**⎜**D**⎜**of observations from a central value {Mean or Median}.

**(5) Standard Deviation**

- Standard deviation is the Square Root of the Arithmetic Average of the squared of the deviations measured from the mean.

**(6) Lorenz Curve**

- The Lorenz Curve is a graphic method of measuring estimated dispersion.
- This curve is often used to measure the inequalities of income or wealth in a society.

**What are the various ‘relative measures’ of Dispersion?**

Following Are the Relative Measures of Dispersion:

**(1) Coefficient of Range**

It refers to the ratio of the difference between two extreme items of the distribution to their sum.

Coefficient of Range =

**(2) Coefficient of Quartile Deviation**

It refers to the ratio of the difference between Upper Quartile and Lower Quartile of a distribution to their sum.

Coefficient of Quartile Deviation = _{}

**\(\begin{array}{l}\frac{_{Q3}\, -\, _{Q1}}{_{Q3}\, +\, {Q1}}\end{array} \)**

**(3) Coefficient of Mean Deviation**

- Mean deviation is an absolute measure of dispersion.
- In order to transform it into a relative measure, it is divided by the particular average, from which it has been calculated.
- It is then known as the Coefficient of Mean Deviation.
- Coefficient of Mean Deviation from Mean
_{\(\begin{array}{l}\left ( \bar{X} \right )\, =\,\frac{MD_{\bar{X}}}{\bar{X}}\end{array} \)} - Coefficient of Mean Deviation from Median
_{\(\begin{array}{l}\left ( ME \right )\, =\,\frac{MD_{Me}}{Me}\end{array} \)}

**(4) Coefficient of Standard Deviation**

- It is calculated by dividing the standard deviation
by the mean_{\(\begin{array}{l}\left ( \sigma \right )\end{array} \)}of the data.\(\begin{array}{l}\left ( \bar{X} \right )\end{array} \) - Coefficient of Standard Deviation
_{\(\begin{array}{l}=\, \frac{\sigma }{X}\end{array} \)}

**(5) Coefficient of Variation**

- It is used to compare two data with respect to stability (or uniformity or consistency or homogeneity).
- It indicates the relationship between the standard deviation and the arithmetic mean expressed in terms of percentage.
- Coefficient of Variation (C.V.)
_{\(\begin{array}{l}=\, \frac{\sigma }{X}\, X\, 100\end{array} \)} - Where,
**C.V.**= Coefficient of Variation;= Standard Deviation;_{\(\begin{array}{l}\sigma\end{array} \)}

** _{}**

**\(\begin{array}{l}\bar{X}\end{array} \)**

**What Are the merits and demerits of range?**

**Merits**

- It is very easy to calculate and simple to understand.
- No special knowledge is needed while calculating range.
- It takes the least time for computation.
- It provides a broad picture of the data at a glance.

**Demerits**

- It is a crude measure because it is only based on two extreme values (highest and lowest).
- It cannot be calculated in the case of open-ended series.
- Range is significantly affected by fluctuations of sampling, i.e. it varies widely from sample to sample.

**Merits and demerits of Quartile Deviation**

**Merits**

- It is also quite easy to calculate and simple to understand.
- It can be used even in case of open-end distribution.
- It is less affected by extreme values so, it a superior to ‘Range’.
- It is more useful when the dispersion of the middle 50% is to be computed.

**Demerits**

- It is not based on all the observations.
- It is not capable of further algebraic treatment or statistical analysis.
- It is affected considerably by fluctuations of sampling.
- It is not regarded as a very reliable measure of dispersion because it ignores 50% observations.

**What Are the merits and demerits of mean deviation?**

**Merits**

- It is based on all the observations of the series and not only on the limits like Range and QD.
- It is simple to calculate and easy to understand.
- It is not much affected by extreme values.
- For calculating mean deviation, deviations can be taken from any average.

**Demerits**

- Ignoring + and – signs is bad from the mathematical viewpoint.
- It is not capable of further mathematical treatment.
- It is difficult to compute when the mean or median is in fraction.
- It may not be possible to use this method in case of open ended series.

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