Population Variance | How to Calculate?

Population Variance | How to Calculate?

Population variance is a measure of the spread of population data. Hence, population variance can be defined as the average of the distances from each data point in a particular population to the mean, squared and it indicates how data points are spread out in the population. Population variance is an important measure of dispersion used in statistics. Statisticians calculate variance to determine how individual numbers in a data set relate to each other.

While calculating population variance, the dispersion is calculated with reference to the population mean. Hence, we have to find out the population mean in order to calculate population variance. One of the most popular notifications of the population variance is σ2. This is pronounced as sigma squared.

Population variance can be calculated by using the following formula:

Population Variance How to Calculate
Population Variance How to Calculate

where

  • σis population variance,
  • x1, x2, x3,…..xare the observations,
  • N is the number of observations,
  • µ is the mean of the data set.

Population Variance Calculator

The formula for population variance can be calculated by using the following five simple steps:

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Step 1: Calculate the mean (µ) of the given data. In order to calculate the mean, add all the observations and then divide that by the number of observations (N).

Step 2: Make a table. Please note that constructing a table is not compulsory, but presenting it in a tabular format would make the calculations easier. In the first column, write each observation (x1, x2, x3,…..xn).

Step 3: In the second column, write the deviation of each observation from the mean (x– µ).

Step 4: In the third column, write the square of each observation from the mean (x– µ)2. In other words, square each of the numbers obtained in column 2.

Step 5: Subsequently we need to add the numbers obtained in the third column. Find the sum of the squared deviationsSquared Deviationsand divide the sum so obtained by the number of observations (N). This will help us to obtainPopulation Variancewhich is the population variance.

Population Variance formula
Population Variance formula

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 (Xi – Xm)2 / N

Where:

  • Xi – ith value of data set
  • Xm – 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.

Population Variance Symbol

Since the population variance measures spread, σ2 for a set of identical points is 0.

If you add a constant to every data point the σ2 remains unchanged. For instance, suppose you study the birth years of senior citizens in New York and decide to switch calendars from the standard Gregorian one to a calendar where 1900 was year 1, the σ2 would stay the same.

The square root of the population variance is the population standard deviation, which represents the average distance from the mean.

The population variance is a parameter of the population and is not dependent on research methods or sampling practices.

How To Find Population Variance

We discuss the meaning of variance from a statistical standpoint but it also helps us in understanding various financial ratios also. Variance is the 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.

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.

How To Calculate Population Variance

Ruby is a third-grade teacher. She has been working with her students on reading. The students can read out loud and to themselves. Ruby has been giving the students timed reading comprehension tests. Each student is timed to read six pages in the classroom reading books. She can average the reading speeds among her students, but she doesn’t feel like this information paints a clear picture with her students. Some of her students can read very fast, and some of her students read about average speed. The numbers are the number of minutes it takes for the students to read six pages in the classroom reading books.

Ruby will need to know how to find the population and sample variance of her data. Variance is how far a set of numbers is spread out. This is very different from finding the average, or the mean, of a set of numbers.

For example, take a look at the following set of numbers: 12, 8, 10, 10, 8, 12. If you add these numbers together and divide by the total numbers in the data set, which in this case is 6, you will get an average of 10. Notice that these numbers are all pretty close to the number 10.

Now take a look at this set of data: 28, 4, 6, 4, 2, 16. You’ll notice that there is a greater difference between the numbers in the second set of data versus the first set of data. However, both sets of data have an average of ten. We show these differences in data by using variance.

There are two main types of variance: population and sample. The population is all members of a specified group. If we were to collect data on just the members of your household, then everyone living in your house would be considered the population. The sample is a part of a population used to describe the whole group.

If we were collecting data on the members of your household and only collected data about two members out of five members, then this would be considered a sample. Other examples of population and samples would be the total members of a school (the population) versus only the members of a class in the school (the sample), or a random selection of 50 members of a school, which would also be a sample.

Sample Variance Vs Population Variance

The sample variance is an estimate of σ2 and is very useful in situations where calculating the population variance would be too cumbersome. The only difference in the way the sample variance is calculated is that the sample mean is used, the deviations are summed up over the sample, and the sum is divided by n-1 (Why use n-1?). When calculating sample variance,  n is the number of sample points (vs N for population size in the formula above).

Unlike the population variance, the sample variance is simply a statistic of the sample. It depends on the research methodology and on the sample chosen. A new sample or a new experiment will likely give you a different sample variance, although if your samples are both representative your sample variances should be good estimates of the population variance and so close to each other.

How Do You Calculate The Variance Of A Population?

To calculate the variance follow these steps: Work out the Mean (the simple average of the numbers) Then for each number: subtract the Mean and square the result (the squared difference). Then work out the average of those squared differences.

What Is The Difference Between Sample Variance And Population Variance?

Summary: Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. As a result, both variance and standard deviation derived from sample data are more than those found out from population data.

Is Population Variance The Same As Standard Deviation?

The standard deviation is the square root of the variance. The standard deviation is expressed in the same units as the mean is, whereas the variance is expressed in squared units, but for looking at distribution, you can use either just so long as you are clear about what you are using.

How Do You Calculate Population Variance And Standard Deviation?

How to Calculate Population Standard Deviation
  1. Calculate the mean (simple average of the numbers).
  2. For each number: Subtract the mean. Square the result.
  3. Calculate the mean of those squared differences. This is the variance.
  4. Take the square root of that to obtain the population standard deviation.