Understanding Variance in Statistics: Definition, Calculating Steps, & Examples

Monday - January 22, 2024

The variance of the data set measures how much each point differs from the mean or average. It can be defined as the average of the squared deviations from the mean value. It is the positive square of the standard deviation. The abbreviation “Var” is frequently used to represent variance in a statistics context. 

Variance is a valuable concept to estimate the variability of data points about the mean. Investors and finance professionals use this tool to measure the uncertainty of investments. In this article, we will talk about the definition and formulas of variance. We will provide different examples to show how to find variance.

What is Variance?

In statistics, variance is the measure of how much the individual data points deviate from the average value. It is defined as the average squared distance from the mean. It shows how far apart the data are from the average. 

Higher variance indicates greater dispersion from the mean while lower variance suggests less deviation.  Essentially, variance helps in understanding the variability within a dataset.

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Types of Variance 

The variance of the given dataset can be classified into two types.

  1. Population Variance
  2. Sample Variance

Let’s learn more about them.

Population Variance 

Variance of population measures the dispersion of individual data points within a population. Population refers to a collection of entire objects or events of interest. It is used when determining the amount of deviation of each data point from the population mean. 

It is expressed by the square of the Greek letter sigma (σ²). Population variance offers insight into the variability present within the entire population.

The formula for population variance is as follows:

σ² = Σ (xi – μ)² / N

Where,

  • σ² = The population variance
  • Σ = Symbol for the sum of the terms that follow
  • xi = each observation in the population
  • μ = population mean
  • (xi – μ)² = The squared difference between the observation xi and the population mean μ.
  • N = total number of observations in population

Sample Variance

If the number of observations in a population set is extremely large then the calculation of the population variance can become challenging. In such cases, we opt to select a subset of the population data that can describe the attributes of the entire group. 

The variance of this subset data is referred to as sample variance. Sample variance is calculated by squaring the deviations of each data point in the sample from the sample mean and then averaging those squared deviations. It is denoted by the square of the Latin letter s (s²).

Population variance formula is as follows:

s² = (Σ (xi – x̄) ²) / (n – 1)

Where, 

  • s² = sample variance
  • xi =  each observation in the sample dataset
  • x̄ = sample mean (average of all of the observations in the sample)
  • (xi – x̄)² =  the squared difference between the observation xi and the sample mean x̄ 
  • n = Sample size (or number of observations in the sample)

Formulas of Variance for both Population and Sample

The following table contains formulas that can be used to determine the variance of a population or sample. 

Properties of Variance

Here are some important properties of variance that will help us in solving its numerical problems.  

  • The variance is always non-negative.
  • The variance will be equal to zero if and only if all the data points are the same.
  • A larger variance shows that the data is far off from the average. Similarly, a smaller variance indicates that all the data points are near to mean.
  • Var(X + a) = Var(X), where X is a random variable and “a” is any constant.
  • Var(aX) = a
  • Var(a) = 0
  • Var(X1 + X2 +…+ Xn) = Var(X1) + Var(X2) +…+ Var(Xn),  Where X1, X2, …, Xn are independent random variables.
  • The square root of the variance is a measure of the standard deviation of the data set.

How to Calculate Variance?

Calculating variance is not complex at all. It comprises merely three steps. By following these steps, you can easily determine the variance for populations and samples. 

Step 1: Find the mean or average of the given dataset by the following formula:

Mean = Sum of the observation/ Number of the observation

Step 2: Subtract the mean from each data point to get the deviations, and square those deviations.

Step 3: Find the average of the squared deviations to get the population variance. To calculate sample variance, the square deviation is divided by n – 1.

Solved Example of Population and Sample variance 

Let’s solve some examples to understand the steps explained above.

Example 1: (For population Variance) 

Calculate the variance for the given population dataset.

10, 19, 21, 13, 16

Solution:

Step 1: First, we need to find the mean of the population data.

µ = 10 + 19 + 21 + 13 + 16 / 5

µ = 79/5

µ = 15.8

Step 2: Subtract the population mean from each population data point to get the deviations, and square those deviations.

Step 3: Calculate the average of the squared deviations to get the population variance.

σ² = Σ (xi – μ)² / N = (33.64 + 10.24+ 27.04 + 7.84 + 0.04) / 5

σ² = 78.8 / 5

Thus, Population Variance = σ² = 15.76

Example 2: (For Sample Variance) 

Calculate the sample variance for a random sample of 6 data points. 5, 6, 12, 9, 14, and 11

Solution:

Step 1: First, calculate the sample mean. 

Sample Mean (x̄) = (5 + 6 + 12 + 9 + 14 + 11) / 6

x̄ = 9.5

Step 2: Subtract the sample mean from each sample data point to get the deviations, and square those deviations.

Step 3: Divide the sum of square deviation by n – 1. 

s² = (Σ (xi – x̄) ²) / (n – 1) = (20.25 + 12.25 + 6. 25 + 0.25 + 20. 25 + 2.25) / 5

s² = 61.5/5

Thus, Sample variance = s² = 12.3

Conclusion 

In this article, we have talked about the concept of variance by examining its definition and various types. We explored both sample and population variance formulas and discussed their properties that help in problem solving. We have also learned methods for calculating variance in samples and populations. 

Multiple examples were provided to understand how to find variance numerically. This article will help you in finding variance confidently.

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