Question

What is the difference between the population standard deviation and the sample standard deviation?

Answer 1

Hint: To answer our question, we will first understand the meaning of standard deviation. Standard deviation is the measure of the dispersion of data from its mean. It measures the absolute variability of a distribution. The higher the dispersion of the data from the mean value, greater the standard deviation and vice versa. Now, we will classify the standard deviation and see the difference between them.

Complete step by step solution:
The standard deviation of a distribution is classified into two categories. The first one is called population standard deviation and the second one is called sample standard deviation.
Let us first see the mathematical difference between these two deviations, that is, the difference in formulas and using that as a base we will try to understand the need for having two standard deviations.
If ‘$\mu $’ is the mean of the population, the formula for the population standard deviation of a population with data entities as, ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},..............,{{x}_{N}}$ , is given by:
$\Rightarrow \sigma =\sqrt{\dfrac{\sum\limits_{i=1}^{N}{{{\left( {{x}_{i}}-\mu \right)}^{2}}}}{N}}$
And, in our second case, if the mean of the sample is denoted by ‘$\overline{x}$’, the formula for the sample standard deviation of the sample data, ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}},..............,{{x}_{N}}$ , is given by:
$\Rightarrow s=\sqrt{\dfrac{\sum\limits_{i=1}^{N}{{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}}}{N-1}}$
Now, we can see the difference in the formulas lies in the denominator only. The reason for this is that, on doing this, the sample variance “${{s}^{2}}$” is converted to a so called ‘unbiased estimator’ for the population variance “${{\sigma }^{2}}$”.
Hence, we can observe the mathematical and analytical difference between the population standard deviation and the sample standard deviation.

Note: If the size of population is very large and we have to take many random samples of the same size ‘n’ from that large population, the mean of many values of ${{s}^{2}}$will have an average very close to the value of ${{\sigma }^{2}}$. Now, the reason for why this is true involves a lot of concepts of higher mathematics which are irrelevant at this beginner stage.