Variance and Standard Deviation

Variance:

The measure of distributional spread is called variance. The mean of the squared deviation is another name for it. It is used to examine a distribution's consistency. Variance of the distribution X, variation of X, and simply variation are all phrases that can be used to define the variance. It comes from:

Var(X)= frac{\sum(x-u)^2}{N}

It is also written as σ^2.

Example 1:

Calculate the variance from the following data:

Here,

u=\(\frac{\sum(x)}{n}\)

=\(\frac{108}{6}\)

=18

Var(X)=\frac{\sum(x-u)^2}{N}

= \(\frac{70}{6}\)

= 11.67

Therefore, the required variance is 11.67.

Standard Deviation:

Although the variance is a measure of dispersion, it is not always suitable. If a distribution is related to a company's profit, the variance is expressed in (Rs) and not rupees. The unit of variance for another distribution that relates to student grades in business statistics is (marks)2. To solve this issue, the standard deviation—a more accurate indicator of dispersion—is produced by taking the variance's square root. We take the square root of the variance using our earlier example of single observations. It is indicated by σ.

i.e,

For the above example,

S.D=\sqrt{σ^2}

=\sqrt{11.67}

=3.42

Variance and Standard deviation for grouped data with frequency

Var(X)=\frac{\sum{f}(x-\overline{x})^2}{n}

Which is equivalent to:

σ^2=\frac{\sum{f}{x^2}}{n}-\overline{x}^2

Example 2:

Find an estimate of the variance and standard deviation of the following data for the marks obtained in a test by 88 students.

Solution:

\overline{x}=\frac{\sum{fx}}{n}

=\(\frac{2510}{88}\)

Now,

σ^2=\frac{\sum{f}{x^2}}{n}-\overline{x}^2

=\frac{83800}{88}-\frac{2510}{8}^2

=138.727

Again,

S.D=\sqrt{138.727}

=11.78

Coefficient of Variation:

As it represents variation in the same units for a single distribution, the standard deviation is an absolute measure of dispersion. Therefore, it is not a good metric to employ when comparing two or more distributions. A relative measure of dispersion is what we employ. The coefficient of variation (also known as CV), which depicts the relationship between the standard deviation and the mean such that the standard deviation is given as a percentage of mean, is one such measure of relative dispersion. As a result, the measurement unit for the standard deviation is no longer used, and the new unit is changed to %.

CV=\(\frac{σ}{u}\times100\)

Example 3:

In a small business firm, there are two typists typist A and typist B. Typist A types out, on an average of 30 pages per day with a standard deviation of 6 while Typist B can type out on an average of 45 pages with a standard deviation of 10. Which typist shows greater consistency in his output?

Solution:

Coefficient of variation for A=\(\frac{σ}{u}\times100\)

=\(\frac{6}{30}\times100\)

=20%

Coefficient of variation for B=\(\frac{σ}{u}\times100\)

=\(\frac{10}{45}\times100\)

=22.22%

So, from the above example, we can conclude the following fact. Though the daily output of A is much less, he is more consistent than typist B. Thus we can conclude that the typist A shows greater consistency in his output. So we can finally conclude that the usefulness of the coefficient of variation becomes clear in comparing two groups of data having different means.

Reference:

• Kunda, Surinder. An Introduciton to business statistics. n.d.
• lboro.ac.uk/media/wwwlboroacuk/content/mlsc/downloads/var_stand_deviat_group.pdf