how to find sample variance

Formula to Calculate Population Variance

Population variance formula is a measure by the average distances of population data and it is calculated by finding out the mean of population formula and variance is calculated by Sum of the square of variables minus mean which is divided by a number of observations in population.

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 Statistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance. read more . 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 The population mean is the mean or average of all values in the given population and is calculated by the sum of all values in population denoted by the summation of X divided by the number of values in population which is denoted by N. read more 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:

σ² = ∑ⁿ _i=1 (x_i – μ)² / N

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For eg:
Source: Population Variance Formula (wallstreetmojo.com)

where

σ²is population variance,
x_1,x₂, x_3,…..x_nare the observations
N is the number of observations,
µ is the mean of the data set

Step by Step Calculation of Population Variance

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

Step 1:Calculate the mean (µ) of the given data. In order tocalculate the mean Mean refers to the mathematical average calculated for two or more values. There are primarily two ways: arithmetic mean, where all the numbers are added and divided by their weight, and in geometric mean, we multiply the numbers together, take the Nth root and subtract it with one. read more , add all the observations and then divide that by the number of observations (N).
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 (x_1,x₂, x_3,…..x_n).
In the second column, write the deviation of each observation from the mean (x_i– µ).
In the third column, write the square of each observation from the mean (x_i– µ)². In other words, square each of the numbers obtained in column 2.
Subsequently, we need to add the numbers obtained in the third column. Find the sum of the squared deviations and divide the sum so obtained by the number of observations (N). This will help us to obtain which is the population variance.

Examples

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

Example #1

Calculate the population variance from the following 5 observations: 50, 55, 45, 60, 40.

Solution:

Use the following data for the calculation of population variance.

There are a total of 5 observations. Hence, N=5.

µ=(50+55+45+60+40)/5 =250/5 =50

So, the Calculation of population variance σ²can be done as follows-

σ²= 250/5

Population Variance σ² will be-

Population Variance (σ² ) = 50

The population variance is 50.

Example #2

XYZ Ltd. is a small firm and consists of only 6 employees. The CEO believes that there should not be high dispersion in the salaries of these employees. For this purpose, he wants you to calculate the variance of these salaries. The salaries of these employees are as under. Calculate the population variance of the salaries for the CEO.

Solution:

Use the following data for the calculation of population variance.

There are a total of 6 observations. Hence, N=6.

=(30+27+20+40+32+31)/6 =180/6 =$ 30

So, the Calculation of population variance σ²can be done as follows-

σ²= 214/6

Population Variance σ² will be-

Population Variance (σ² ) = 35.67

The population variance of the salaries is 35.67.

Example #3

Sweet Juice Ltd manufactures different flavors of juice. The Management Department purchases 7 big containers for storing this juice in the factory. The Quality Control Department has decided that it will reject the containers if the variance of the containers is above 10. Given are the weights of 7 containers in kg: 105, 100, 102, 95, 100, 98, and 107. Please advise the Quality Control Department on whether it should reject the containers.

Solution:

Use the following data for the calculation of population variance.

There are a total of 7 observations. Hence, N=7

=(105+100+102+95+100+98+107)/7 =707/7 =101

So, the Calculation of population variance σ²can be done as follows-

σ²= 100/7

Population Variance σ² will be-

Population Variance (σ² )= 14.29

Since the variance (14.29) is more than the limit of 10 decided by the Quality Control Department, the containers should be rejected.

Example #4

The management team of a hospital named Sagar Healthcare recorded that 8 babies had been born in the first week of March 2019. The doctor wanted to evaluate the health of the babies as well as the variance of the heights. The heights of these babies are as follows: 48 cm, 47 cm, 50 cm, 53 cm, 50 cm, 52 cm, 51 cm, 60 cm. Calculate the variance of the heights of these 8 babies.

Solution:

Use the following data for the calculation of population variance.

So, the Calculation of population variance σ²can be done as follows-

In Excel, there is an inbuilt formula for population variance that can be used to calculate the population variance of a group of numbers. Select a blank cell and type this formula =VAR.P(B2: B9). Here, B2: B9 is the range of cells you want to calculate the population variance from.

Population Variance σ² will be-

Population Variance (σ² ) = 13.98

Relevance and Use

Population variance is used as a measure of dispersion. Let us consider two population sets with the same mean and number of observations. Data set 1 consists of 5 numbers – 55, 50, 45, 50, and 50. Data set 2 consists of 10, 50, 85, 90, and 15. Both the data sets have the same mean, which is 50. But, in data set 1, the values are close to each other while data set 2 has dispersed values. The variance gives a scientific measure of this closeness/dispersion. Data set 1 has a variance of only 10, while data set 2 has a huge variance of 1130. Thus, a large variance indicates that the numbers are far from the mean and from each other. A small variance indicates that the numbers are close to each other.

Variance is used in the field of portfolio management while carrying out asset allocation Asset Allocation is the process of investing your money in various asset classes such as debt, equity, mutual funds, and real estate, depending on your return expectations and risk tolerance. This makes it easier to achieve your long-term financial goals. read more . Investors calculate the variance of asset returns to determine optimal portfolios Portfolio optimization refers to the strategic process of making improvements or positive changes to the investor's portfolio containing various assets. The Sharpe ratio helps to analyze the returns from an optimal portfolio. read more by optimizing the two major parameters – return and volatility. Volatility measured by variance is a measure of the risk of a particular financial security.