how to find population variance

Population Variance Formula

Population Variance Formula (Table of Contents)

Population Variance Formula
Examples of Population Variance Formula (With Excel Template)

Population Variance Formula

In statistics, a variance is basically a measure to find the dispersion of the data set values from the mean value of the data set. It measures the distance of that data point and the mean. So higher the variance, higher will be the dispersion and data points will tend to far from the mean. Similarly, lower variance indicates that data points will be closer to the mean. It is very useful in comparing data sets which may have the same mean value but a different range. Population variance, in the same sense, indicates how the population data points are spread out. It is the average of the distances from each data point in the population to the mean, squared. Usually, calculate the variance of population data but sometimes population data is so huge that it does not make economic sense to find the variance for that. In that case, sample variance is calculated and that will become the representative of the population variance.

Suppose you have a population data set X with data points {X1, X2……..Xn}. The formula for Population Variance is given by:

Population Variance = Σ (X_i – X_m)² / N

Where:

X_i – i^th value of data set
X_m – Mean value of data set
N – Total number of data points

The formula may look confusing at first, but it is really to work on. Following are the steps which can be followed to calculate Population Variance:

Find whether the data set you are working is sample or population.
Find the number of points in the data set i.e. n for the population.
The next step is to find the mean value. It is basically the average of all the values.
After that, for each data point, find the difference of that from the mean and then square it.
Take sum all values in the above step and divided that by a number of points calculated in point 2.

There is another way to calculate variance by using VAR.P() function for population variance and VAR.S () function for sample variance in excel.

Examples of Population Variance Formula (With Excel Template)

Let's take an example to understand the calculation of the Population Variance Formula in a better manner.

You can download this Population Variance Formula Excel Template here – Population Variance Formula Excel Template

Population Variance Formula – Example #1

Let's say we have two sample data sets A & B and each contains 20 random data points. Calculate population variance for both the data sets.

Data Set:

Population Variance Formula Example 1-1

Mean is calculated as:

Population Variance Formula Example 1-2

Mean of Data Set A = 51.2
Mean of Data Set B = 46.95

Now, we need to calculate the difference between the data points and the mean value.

Population Variance Formula Example 1-3

Similarly, calculate for all the data set of A.

Example 1-4

Similarly, calculate it for data set B also.

Formula Example 1-5

Calculate the square of the difference for both the data sets A and B.

Example 1-6

Population Variance is calculated using the formula given below

Population Variance = Σ (X_i – X_m)² / N

Population Variance Formula Example 1-7

So if you see here, B has more variance that A, which means that data points of B are more dispersed than A.

Population Variance Formula – Example #2

Let say you are a very risk-averse investor and you looking to invest money in the stock market. Since your risk appetite is low, you want to invest in safe stocks which have lower variance.

You want to analyze the stocks based on their past results, so we have decided to take a sample of 15 years and work on that data. Your financial advisor has suggested you 4 stocks from which you can choose from. You want to select 2 stocks among those 4 and you will decide that on the basis of lower variance.

You have got information on their historical returns for the last 15 years.

Population Variance Formula Example 2-1

Population Variance is calculated using excel Formula

Population Variance Formula Example 2-2

Based on the information, you will choose stock X and Z to invest since they have the lowest variance.

Explanation

We discuss the meaning of variance from a statistical standpoint but it also helps us in understanding various financial ratios also. Variance is foundation stone for standard deviation which is calculated by taking the square root of variance. Standard deviation is a measure of risk an investment carries and how risky that investment is. Based on the risk an investment has, investors can then calculate the minimum return they require to compensate that risk. Variance value, since it is square of a number will always be positive. This can be zero for data set which has all the identical items.

Relevance and Uses of Population Variance Formula

Variance helps the investors and analyst to determine standard deviation which further helps in finding risk and reward ratio or Sharpe ratio for an investment. Basically, anyone can earn a risk-free rate of return by investing in Treasury and risk-free securities. But return over and above this is the excess return and to achieve that.

So as to higher the Sharpe ratio, better is the investment.

As we said that variance helps in finding standard deviation which measures risk, but lower standard deviation value is not always preferred. If an investor has a higher risk appetite and wants to invest more aggressively, he will be willing to take more risk and prefer a relatively higher standard deviation than a risk-averse investor. So it all depends on what level of risk an investor is willing to take.