Variance Formula: Population Variance & Sample Variance

Variance Formula measures how far a set of numbers are spread out in Statistics and probability theory. Variance is a statistical measurement that is used to indicate how widely individuals vary within a group. If individual observations differ significantly from the group mean, the variance is large and vice versa. The variance formula for the ungrouped data and the grouped data is different.

A variance of zero indicates that all the values are equal (identical). It must be noted that variance is always non-negative. A small variance indicates that the data points are very close to the mean and, therefore, close to each other, while a high variance indicates that the data points are widely spread around the mean and from each other. 

Read More: NCERT Solutions For Class 11 Maths Statistics

Key Terms: Variance, Variance Formula, Population Variance, Sample Variance, Standard Deviation, Dispersion, Grouped Data, Ungrouped Data

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What is Variance?

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Variance is a measure of dispersion that is used to measure how far a set of numbers is spread. It is a quantity that is used to check the variability of data about an average value. The data can be of two types namely grouped and ungrouped. The data expressed in the form of class intervals is grouped data while the data consisting of individual data points is called ungrouped data. 

There are two types of variance:

1. Population Variance
2. Sample Variance

Population and Sample

Population Variance

• A population means all the members of a particular group. 
• When we want to find out how each data point in a given population varies or is spread out then we use the population variance. 
• Population variance is used to give the squared distance of each data point from the population mean.

Sample Variance 

• A sample is a portion of the population used to describe the entire group.
• It is used in case the size of the population is too large then it is difficult to take each data point into consideration. 
• The sample variance is defined as the average of the squared distances from the mean. The variance is calculated with respect to the sample mean.

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Variance Formula

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The variance is calculated for either grouped or ungrouped data. Variance can be of two types which are:

• Variance of a Population
• Variance of a Sample

The variance of a population is denoted by σ², while the variance of a sample is denoted by s².

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Variance Formulas for Ungrouped Data

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The variance formula for ungrouped data is as follows: 

Population Variance

σ² = \(\frac{1}{N} \sum _{i=1}^N (x_i - \mu)^2\)

Where

• σ² = Variance
• x_i = ith observation of given data
• μ = Population mean
• N = Size of Population or Total number of observations

Sample Variance

S² = \(\frac{1}{n-1} \sum _{i=1}^N (x_i - \bar{x})^2\)

Where

• s² = Sample variance
• x_i = ith observation of given data
• x̄ = Sample mean
• n = Size of Sample or Total number of data values taken in sample

Variance Formulas for Ungrouped Data

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Variance Formulas for Grouped Data

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The variance formula for grouped data is: 

Formula for Population Variance

The formula of population variance for grouped data is: 

σ² = ∑ f (m − x̄)² / n

Formula for Sample Variance

The formula for sample variance for grouped data is: s² = ∑ f (m − x̄)² / n − 1

Where

• f = frequency of the class
• m = midpoint of the class

The formulas of population variance and sample variance can also be written as:

Population Variance

σ²= \(\frac{1}{N}[ \sum _{i=1}^N f_ix_i^2-(\frac{ \sum _{i=1}^N f_ix_i^2}{N})^2]\)

Where

• σ² = Variance
• x_i = Midvalue of ith class
• f_i = Frequency of ith class
• N = Size of Population or Total number of observations

Sample Variance

S² = \(\frac{1}{n-1}[ \sum _{i=1}^N f_ix_i^2-(\frac{ \sum _{i=1}^N f_ix_i^2}{n})^2]\)

Where

• s² = Sample variance
• x_i = Midvalue of ith class
• f_i = Frequency of ith class
• n = Size of Sample or Total number of data values taken in sample

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Variance Formula Solved Examples

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Example 1: Calculate the variance for the given data:

Solution: The variance will be calculated as:

s² = \(\frac{1}{n-1}[ \sum _{i=1}^N f_ix_i^2-(\frac{ \sum _{i=1}^N f_ix_i^2}{n})^2]\)

= [1/ (70 – 1)] [2855149.5 – (1/70) (14137)2]

= 1.179

Example 2: Calculate the variance for the following set of data representing the heights of some trees in feet: 3, 21, 98, 203, 17, 9

Solution: To calculate the variance, we will follow the given steps:

Step 1: Find the sum of all the values in the given data set.

3 + 21 + 98 + 203 + 17 + 9 = 351

Step 2: Now, square the value received.

351 × 351 = 123201

Divide the obtained value by the number of items, such that

123201/6 = 20533.5

Step 3: Take up the original numbers from Step 1 and square them individually.

3 × 3 + 21 × 21 + 98 × 98 + 203 × 203 + 17 × 17 + 9 × 9

Now, add up the squares together 

9 + 441 + 9604 + 41209 + 289 + 81 = 51,633

Step 4: Subtract the value in Step 2 from the value in Step 3.

51633 – 20533.5 = 31,099.5

Step 5: Now, subtract 1 from the number of items in your data set. 

6 – 1 = 5

Step 6: Divide the value obtained in Step 4 by the value in Step 5. 

31099.5/5 = 6219.9

Thus, the variance of the given data set is 6219.9.

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Properties of Variance 

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Following are some properties that can help in solving both simple and complex sum problems:

• If the variance is zero, it indicates that all the data points in the data set have equal value.
• If the variance is large, it means that the data is more vastly spread out from the mean. Also, a small variance is a sign that the data point values are close to one another as well as clustered around the mean.
• Var (X + C) = Var(X), where X is a random variable and C is a constant.
• Var (aX + b) = a².Var(X), where a and b are constants.
• Var (CX) = C².Var(X), where C is a constant.
• Var (X₁ + X₂ +……+ X_n) = Var(X₁) + Var(X₂) +……. +Var (X_n) (where X₁, X₂… X_n are random and independent variables.)

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Relationship between Variance and Standard Deviation 

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The relationship between variance and standard deviation for a sample data set is shown below:

• Variance estimates the average squared deviations from the mean value of data, while standard deviation estimates the square root of that value.
• Both the variance and the standard deviation estimate variability in a distribution.

Variance and Standard Deviation

• The standard deviation has the same units as the original values, such as minutes or meters, while the variance has much larger units, like meters squared.
• The variance will have an equal value to the square of the standard deviation, or the standard deviation can be referred to as the square root of the variance.

Variance = Square of Standard Deviation 

V = σ²

Here

• V stands for variance.
• σ stands for standard deviation.

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Things to Remember

• Variance is a measure of dispersion that is used to check the variability of data about an average value. 
• Variance can be of grouped data or ungrouped data. 
• The two types of variance are population variance and sample variance. 
• The variance of a population is denoted by σ², while the variance of a sample is denoted by s².
• Population Variance for Ungrouped Data: σ² =\(\frac{1}{N} \sum _{i=1}^N (x_i - \mu)^2\)
• Sample Variance for Ungrouped Data:s² = \(\frac{1}{n-1} \sum _{i=1}^N (x_i - \bar{x})^2\)
• Population Variance for Grouped Data:σ² = \(\frac{1}{N}[ \sum _{i=1}^N f_ix_i^2-(\frac{ \sum _{i=1}^N f_ix_i^2}{N})^2]\)
• Sample Variance for Grouped Data:s² = \(\frac{1}{n-1}[ \sum _{i=1}^N f_ix_i^2-(\frac{ \sum _{i=1}^N f_ix_i^2}{n})^2]\)

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Previous Years’ Questions (PYQs)

• The variance of the first 50 even natural numbers is… (JEE Main 2014)
• The variance of the first 20 natural numbers is…
• The mean of 5 observations is 5 and their variance is 124. If three… (JEE Main 2016)
• If the mean and the variance of a binomial variate… (JEE Main 2015)
• The variance of first n even natural numbers is… (JEE Main 2009)
• A measure of dispersion is indicative of the reliability…
• The mean and variance of 7 observations are 8 and…
• The mean and variance of 20 observations are found to… (JEE Main 2020)
• The variance of the numbers 2, 3, 11 and x is…
• The mean and standard deviation of marks obtained…