Question

Which one of the given measures of dispersion is considered best?

Answer 1

Hint: We first explain the concept of standard deviation. We explain the pros. of using it for finding dispersion. We compare it with the rest of the measures of dispersion to establish its superiority.

Complete step-by-step solution:
The best measurement for dispersion is standard deviation.
Standard Deviation helps to make comparison between variability of two or more sets of data, testing the significance of random samples and in regression and correlation analysis.
It is based on the values of all the observations which makes use of every item in a particular distribution.
It is widely used measure of dispersion as all data distribution is nearer to the normal distribution.
It enables algebraic treatment. It has corrected mathematical processes in comparison to range, quartile deviation and mean deviation – the other parts of measures of dispersion.
Let us take the given set of $n$ observations where we have ${{x}_{i}},i=1\left( 1 \right)n$.
We can take the mean of the given observations as $\overline{x}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}$.
The standard deviation will be according to the formula of \[\sigma =\sqrt{\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}}\].
If there are frequencies then we take the weighted form. We get \[\sigma =\sqrt{\dfrac{1}{N}\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}{{f}_{i}}}-{{\left( \overline{x} \right)}^{2}}}\] where $\overline{x}=\dfrac{1}{N}\sum\limits_{i=1}^{n}{{{x}_{i}}{{f}_{i}}}$ and $N=\sum\limits_{i=1}^{n}{{{f}_{i}}}$.

Note: Standard Deviation has a precise value and is a well-defined and definite measure of dispersion. That is, it is rigidly defined as it is independent of the origin.