Spread of dispersion.

Spread of dispersion

Averages give us the idea of the concentration of the items around the central part of the distribution . But the averages only do not give the clear pictures about the distribution because two distribution with same averages may differ in the scatteredness of the item from the central valueExample. Mean Median Mode

. Mean Median Mode

From this table, we see that mean , mode and median of two series . A and B are same. Only with these results , we cannot say the two series . A and B are similar . Because the difference of the items from the averages in B is more in comparison to A . so in series A , items are concentrating more on the central value but the scattered off the items from a central value in series B is more . Hence , though two series Aand B have same averages , they cannot be said similar because they may differently be constituted.

Actually, the meaning of dispersion is the scatteredness of the items from the central value, so dispersion is defined as the measure of variation in the item from the central value.

Spread or dispersion is necessary for data analysis because.

• It helps to determine the reliability.
• It devises a system of quality control.
• It compares two or more series with regard to their variability.
• It helps in using other statistical tools.

Same important measure of dispersion are;

• Range
• Mean deviation
• Standard deviation

Range

The range is the simplest possible measure of dispersion and is defined as the difference between the values of the extreme items of a series. Thus, Range=( Highest value of an item in a series)- lowest value of items in a series. The utility of range is that it gives an idea of the very quickly , but the drawback it that range is affected very greatly by the fluctuation of sampling. Its value is never stable , Being based on only two values of the variable . as such range is mostly used the rough measure of variability and is not considered as an appropriate measure in serious research studies.

Mean Deviation

Mean deviation is the average of difference to the value of items from some average of the series . Such a difference is technically described as the deviation. In calculating mean deviation we ignore the minus sign of deviation is thus, obtained as under:

$$Mean\,deviation\,from\,mean(δ_\overline{X}) =\frac{\sum X_i -\overline{X}}{n}, If\,deviation\,{\sum X_i -\overline{X}}\,are\,from\,arithmetic\,average$$

$$Mean\,deviation\,from\,median(δ_m) =\frac{\sum X_i -M}{n}, If\,deviation\,{\sum X_i -M}\,are\,from\,median$$

$$Mean\,deviation\,from\,mode(δ_z) =\frac{\sum X_i -Z}{n}, If\,deviation\,{\sum X_i -Z}\,are\,from\,mode$$

$$Where\,δ=Symbols\,for\,mean\,deviation\,$$

$$X=i^{th}\,values\,of\,the\,variable\,X$$

$$n=Number\,of\,items$$

$$\overline{X}=Arthimetic\,mean$$

$$M=Median$$

$$Z=mode$$

When to mean deviation is divided by the averages used in finding out the mean deviation itself. The Resulting quantity is described as the coefficient of mean deviation . The coefficient of mean deviation is the relative measure of dispersion and is comparable to a similar measure of other series. Mean Deviation and its coefficient are used in statistical studies for judging the variability and thereby under the study of central tendency of a series more precise by throwing light on the typicalness of in an average . It is a better measure of variability than range as it takes into consideration the values of all items of a series . Even then it is not a frequently used measure as it is not amenable to algebraic process.

Standard Deviation

Standard deviation is most widely used the measure of dispersion of series and is commonly denoted by the symbol (σ) (pounced as sigma).Standard deviation is defined as the square-root of the average of squares is deviations. When such deviations for the values of individual items in a series are obtained from the arithmetic average . It is worked out as under.

$$Standard\,Deviation (σ)=\sqrt{\frac{\sum(X_i-\overline{X^2)}}{n}}$$

$$Or\,Standard\,Deviation (σ)=\sqrt{\frac{\sum(X_i-\overline{X^2)}}{\sum{f_i}}}$$

In the case of frequency of distribution where f₁ means the frequency of the !the item.

When we divide the standard deviation the standard deviation by the arithmetic average of the series, the resulting quantity is known as the coefficient of standard deviation which happens to relative measure and is often used for comparing with the similar measure of other series. When this coefficient of standard deviation is multiplied by 100, the resulting figure is known a coefficient of variation.sometimes we work out the square of standard deviation, known as variance , which is frequently used in the context of analysis of variation.

The standard deviation (along with several related measures like variance, a coefficient of variance etc. is used mostly in research studied and is regarded as a very satisfactory measure of dispersion in a series. It is amenable to mathematical manipulation because the algebraic signs are not ignored in its calculations. As we ignore in case of mean deviation. It is affected by fluctuations of sampling. These advantages make of the scatteredness of a series. It is popularly used in the context of estimation of testing of hypothesis.

Reference

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.