Measure of dispersion with types, formula

Measure of dispersion in statistics is a descriptive statistical measure used to measure the scatterness or deviation or variation in the data set. It is also called measure of variation (variability). The term ‘Dispersion’ refers to the scatterness or variability or deviation. Thus, the measures of dispersion gives an idea about how the variables are deviated from the central value. It is another important property of a frequency distribution.

The major types measures of dispersion are range, quartile deviation, coefficient of quartile deviation, mean deviation, coefficient of mean deviation, standard deviation, variance and coefficient of variation. Among these measures, standard deviation is considered as the best measure of dispersion because it satisfies the most of all requisites of a good measure of dispersion.

Requisites of a good measure of Dispersion

According to Prof. Yule, a measure of dispersion in statistics is said to be a good measure if it satisfies the following requisites(requirements);

• It should be rigidly defined.
• It should be easy to calculate and simple to understand.
• It should be based upon all the observations.
• It should be suitable for further mathematical calculations.
• It should be least fluctuation by sampling.
• It should be least affected by the extreme observations.

What is measure of Absolute and relative dispersion ?

Absolute measure of dispersions: It is a measure which is expressed in terms of original units of the data. These measures are used to compare the variability among the variables having same measurement units. The common absolute measure of dispersion in statistics are range, quartile deviation, mean deviation and standard deviation or variance.
Relative measure of dispersion: A relative measure of dispersion is a measure which is independent with the original units of data. These measures are used to compare the variability among the variables having different measurement units. The common relative measures of dispersion are coefficient of range, coefficient of quartile deviation, coefficient of mean deviation and coefficient of variation.

What is Range in statistics?

Range is a commonly used measure of dispersion. It is more appropriate to calculate the variability of the data set which are related to the temperature, rainfall and maximum & minimum value. Mathematically, range is defined as the difference between two extreme values. In other words, it is the difference between the maximum and minimum values.

Measurements formula of range in statistics

❑Absolute measure of range:

1. For individual and ungrouped frequency distribution:
   Range (R) = (X_max)-(X_min)
   Where, X_max = Maximum value and X_min = Minimum value
2. For grouped frequency distribution:
   Range (R) = Upper limit of the highest class – lower limit of the lowest class
   Range (R) = Mid value of the highest class – Mid value of the lowest class

❑ Relative measure:

Coefficient of range = (X_max – X_min) / (X_max+X_min)Where, the symbols have their usual meanings.

Introduction to Quartile deviation in statistics

Quartile deviation is another measure of dispersion in statistics. This measure is more appropriate for the frequency distribution having open ended class intervals. Quartile deviation is also called semi-inter quartile range. Measurements formula of quartile Deviation

 Absolute measure:

For all types of frequency distribution, Inter-quartile range =Q₃ -Q₁
Then, Semi-inter quartile range or Quartile deviation(Q.D)=1/2(Q₃ -Q₁) ,(Q.D.) = where, the symbols have their usual meanings

Relative measure :

Coefficient of quartile deviation =(Q₃ -Q₁)/(Q₃ +Q₁)

What is Mean Deviation (M.D.) in statistics

Mean deviation is another measure of dispersion in statistics and, it is able to measure the deviations from arithmetic mean or median or mode. It is also called ‘average deviation’ or ‘mean absolute deviation’. Mathematically, it is defined as the arithmetic mean of the absolute deviations of the items taken from their average value (mean or median or mode). The mean deviation taken from the median is the leas so if it is asked to calculate only mean deviation, generally, the mean deviation from the median is calculated.

Mean Deviation formula

A. Absolute measure:

For individual frequency distribution
Mean deviation taken from mean $M{D}_{\stackrel{-}{X}} =\frac{1}{n}\sum \left|\begin{array}{c}X–\stackrel{-}{X}\end{array}\right|$
Mean deviation taken from median $M{D}_{M_D} =\frac{1}{n}\sum \left|\begin{array}{c}X–M_d\end{array}\right|$
Mean deviation taken from mode $M{D}_{M_0} =\frac{1}{n}\sum \left|\begin{array}{c}X–M_0\end{array}\right|$
For ungrouped frequency distribution

Mean deviation taken from mean $M{D}_{\stackrel{-}{X}} =\frac{1}{N}f\sum \left|\begin{array}{c}X–\stackrel{-}{X}\end{array}\right|$
Mean deviation taken from median $M{D}_{M_D} =\frac{1}{N}f\sum \left|\begin{array}{c}X–M_d\end{array}\right|$
Mean deviation taken from mode $M{D}_{M_0} =\frac{1}{N}f\sum \left|\begin{array}{c}X–M_0\end{array}\right|$
For grouped frequency distribution

Mean deviation taken from mean $M{D}_{\stackrel{-}{X}} =\frac{1}{N}\sum \left|\begin{array}{c}\mathrm{x}–\stackrel{-}{X}\end{array}\right|$
Mean deviation taken from median $M{D}_{M_D} =\frac{1}{N}\sum \left|\begin{array}{c}\mathrm{x}–M_d\end{array}\right|$
Mean deviation taken from mode $M{D}_{M_0} =\frac{1}{N}\sum \left|\begin{array}{c}\mathrm{x}–M_0\end{array}\right|$

Relative measure formula mean deviation

Coefficient of mean deviation from mean
${\mathit{\text{Coeff.of MD}}}_{\overline{\mathit{\text{x}}}}\mathit{\text{ =}}\frac{\mathit{\text{M D from mean}}}{\mathit{\text{Mean}}}\mathit{\text{=}}\frac{{\mathit{\text{MD}}}_{\overline{\mathit{\text{x}}}}}{\overline{\mathit{\text{x}}}}$
Coefficient of mean deviation from median
$\mathit{\text{Coeff}}\mathit{\text{.}}\mathit{\text{of}}{\mathit{\text{MD}}}_{{\mathit{\text{m}}}_{\mathit{\text{d}}}}\mathit{\text{ =}}\frac{\mathit{\text{M}}\mathit{\text{D}}\mathit{\text{from}}\mathit{\text{median}}}{\mathit{\text{median}}}\mathit{\text{=}}\frac{{\mathit{\text{MD}}}_{{\mathit{\text{m}}}_{\mathit{\text{d}}}}}{{\mathit{\text{m}}}_{\mathit{\text{d}}}}$
Coefficient of mean deviation from mode
${\mathit{\text{Coeff.ofMD}}}_{{\mathit{\text{m}}}_o}\mathit{\text{ =}}\frac{\mathit{\text{MD}}\mathit{ }\mathit{\text{from}}\mathit{ }\mathit{\text{mode}}}{\mathit{\text{median}}}\mathit{\text{=}}\frac{{\mathit{\text{MD}}}_{{\mathit{\text{m}}}_o}}{{\mathit{\text{m}}}_o}$

what is Standard Deviation (S.D) in statistics

Standard deviation is one of the powerful and widely used measure of dispersion. It is also considered as the best measure of dispersion because it satisfies the most of all requisites of a good measure of dispersion. This measure provides the information about how the variable values are deviated or scattered from the central value of the given data set. Mathematically, standard deviation is defined as the positive square root of the average of the square of the deviations of the items from their mean.

standard deviation formula in statistics

$standard deviation\left(\sigma \right)=\sqrt{\frac{1}{n}\sum (x–\overline{x}{)}^2}$
$standard deviation\left(\sigma \right)= \sqrt{\frac{1}{n}\sum d^2–(\frac{\sum d}{n}{)}^2}for individual frequecy distribution where d=X–A$
$standard deviation\left(\sigma \right)=\sqrt{\frac{1}{N}\sum \mathrm{f}(x–\overline{x}{)}^2}$
standard deviation=$\sqrt{\frac{1}{n}\sum \mathrm{f}{d}^2–(\frac{\sum \mathrm{f}d}{n}{)}^2}$

What are Variance in statistics

Variance is another absolute measure of dispersion. It is the square of standard deviation. Mathematically, variance is defined as the average of the square of the deviations of the items taken from their mean.

Coefficient of variation (CV)

Coefficient of variation is a widely used relative measure of dispersion based on standard deviation. It is a number and always expressed in percentage. Mathematically, coefficient of variation is defined as the ratio of the standard deviation to the mean and multiplied by 100.

$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \left(\mathrm{CV}\right)=\frac{\mathrm{\sigma }}{\overline{\mathrm{x}}}\times 100\%$