Question

What is population variance?

Answer 1

Population variance shows us how data points in a specific population are spread out. It is the average of the distances from each data point in the population to the mean, squared. Population variance has different formulas for grouped and ungrouped data.

Complete step-by-step answer:
Population variance shows us how data points in a specific population are spread out. We can say that it is the average of the distances from each data point in the population to the mean, squared. We will denote population variance as ${{\sigma }^{2}}$ . We can write this mathematically as
${{\sigma }^{2}}=\dfrac{1}{n}{{\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\overline{x} \right)}}^{2}}$
Where, x is the item given in the data, $\overline{x}$ is the mean of the data, n is the total number of items. We will use the above formula for ungrouped data.
We can express the population variance for grouped data as shown below.
${{\sigma }^{2}}=\dfrac{1}{n}{{\sum\limits_{i=1}^{n}{{{f}_{i}}\left( {{m}_{i}}-\overline{x} \right)}}^{2}}$
Where f is the frequency of the class and m is the midpoint of the class.
There are certain properties associated with population variance. If we add a constant to every data point, the population variance remains unchanged. The second property is that if we take the square root of the population variance, then we get the population standard deviation, which represents the average distance from the mean. Lastly, Since the population variance measures spread, it will be 0 for a set of identical points.

Note: Students must not get confused with population variance and sample variance. The sample variance is an estimate of population variance, and we use this in situations where calculating the population variance would be too cumbersome. The only difference in the way we calculate the sample variance is that the sample mean is used, then the deviations are summed up over the sample, and the sum is divided by $n-1$ , where n is the number of sample points. We can write the sample variance for ungrouped data as
${{s}^{2}}=\dfrac{1}{n-1}{{\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\overline{x} \right)}}^{2}}$
Similarly, the sample variance for grouped data is given as
${{s}^{2}}=\dfrac{1}{n-1}{{\sum\limits_{i=1}^{n}{{{f}_{i}}\left( {{m}_{i}}-\overline{x} \right)}}^{2}}$