Question

What’s the mathematical formula for the pooled variance of two populations?

Answer 1

Hint: Let us first understand what is meant by pooled variance of populations. The term pooled variance of two or more populations is used when we know the means of all the individual populations and their individual variances can be considered to be equal. The mathematical result emerging with the help of this method is called pooled variance.

Complete step-by-step solution:
So, from our above definition of pooled variance, we can say that, the pooled variance is just an approximation of estimating value of variance $\left( {{\sigma }^{2}} \right)$. The only significant of this pooled variance is that it underlies different populations that have different values of mean.
At first we will look at the generalized mathematical formula for the pooled variance calculation and then use it to derive an expression for the pooled variance of two populations.
The generalized formula of pooled variance for multiple populations is given by:
$\begin{align}
  & \Rightarrow {{V}_{p}}=\dfrac{\sum\limits_{i=1}^{k}{{{\left( {{n}_{i}}-1 \right)}^{2}}s_{i}^{2}}}{\sum\limits_{i=1}^{k}{\left( {{n}_{i}}-1 \right)}} \\
 & \therefore {{V}_{p}}=\dfrac{\left( {{n}_{1}}-1 \right)s_{1}^{2}+\left( {{n}_{2}}-1 \right)s_{2}^{2}+\left( {{n}_{3}}-1 \right)s_{3}^{2}+..........+\left( {{n}_{k}}-1 \right)s_{k}^{2}}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+........+{{n}_{k}}-k} \\
\end{align}$
Where, the different populations are having different indices as 1, 2, 3, ......, k.
Now, we shall use this above formula to calculate a mathematical formula for the pooled variance of two populations. Putting the value of ‘k’ as 2 in the above equation, we get our required result as:
$\Rightarrow {{V}_{p}}=\dfrac{\left( {{n}_{1}}-1 \right)s_{1}^{2}+\left( {{n}_{2}}-1 \right)s_{2}^{2}}{{{n}_{1}}+{{n}_{2}}-2}$
Hence, the mathematical formula for the pooled variance of two populations is given by, ${{V}_{p}}=\dfrac{\left( {{n}_{1}}-1 \right)s_{1}^{2}+\left( {{n}_{2}}-1 \right)s_{2}^{2}}{{{n}_{1}}+{{n}_{2}}-2}$

Note: While calculating the standard deviation and variance of a certain population or a binomial distribution, we should always take care of both the formulas and not confuse the one with the other. This is because there is only a mere difference of a square in between them. A simple way to remember them is that the square of standard deviation is variance.