How variance is calculated for sample and population data? Explained with solution

Variance functions as a fundamental statistical measurement that quantifies the spread of data points around a central mean. This calculation is essential for understanding data dispersion within both complete populations and representative samples.

While the core logic of squaring deviations remains constant, the mathematical treatment of the denominator changes based on the data source. For a population, the total number of observations (N) is used, whereas for a sample, researchers use n-1 to account for potential bias.

This adjustment, known as Bessel’s correction, ensures that sample variance provides an accurate and unbiased estimate of the wider population variability. This article provides a comprehensive overview of these formulas, the conceptual logic behind the squared differences and practical solutions for manual calculation.

Key Takeaways

• Variance measures the average squared distance of each data point from the arithmetic mean of a set.
• Population variance uses the total count N as the divisor to represent the entire group variability.
• Sample variance employs n-1 to correct for underestimation and ensure an unbiased estimate of population spread.
• Squaring deviations prevents positive and negative values from cancelling out during the summation process.
• Standard deviation is derived by taking the square root of the calculated variance to return to original units.

In descriptive statistics, there are various types that are used to calculate the nature and spread of the data values. The variance is one of them. It is frequently used to measure the spread of sample and population data values. 

The standard deviation is another type of statistics that is used to measure the distribution of data values from the mean. It is more accurate than the variance as it is measured in single units while the variance is measured in squared units.

In this post, we will learn the basics of variance along with its types, formulas, and examples with solutions.

What is the variance?

The variance is the subtype of descriptive statistics that is used to measure the variation of data values from the expected value. It depends on the nature of the data sets such as population or sample data values used in this subtype of descriptive statistics.

The population set is the set that is taken from the whole set of observations taken from the whole observations under consideration.

While the sample is taken from the whole set under consideration. It is taken to ease up the calculations and avoid the larger set of observations to find the estimated result of the whole population.

Types of variance

There are further two types of variance one is the sample variance while the other is the population variance.

Population variance

The data set taken from the population and to measure the spread of data values from the average value of the whole set of data is known as population variance. It is calculated by taking the difference of the data values from the population mean.

And after that take the square of the deviation to make them positive and add all the squared deviations to evaluate the sum of squares and in the end, find the quotient of the sum of the squares and the total number of population data

Sample variance

The sample data set taken from the population and to measure the spread of data values from the mean value of the sample set of data is known as sample variance. It is calculated by taking the difference of the data values from the sample mean.

And after that take the square of the deviation to make them positive and add all the squared deviations to evaluate the sum of squares. In the end, find the quotient of the sum of the squares and the total number of sample data.

Formulas of variance

Here are the general formulas to calculate the variance of sample and population data.

Population variance formula

The formula for finding the population variance is:

σ² = [∑ (y_i – μ)²/N]

Sample variance formula

The formula for finding the sample variance is:

s² = [∑ (y_i – ȳ)²/N]

A variance calculator can be used to find the variance of sample and population data with just a single click according to the above formulas.

How to calculate the variance of sample and population data?

The variance can be calculated by using the formulas of sample and population. Let’s take a few examples of calculating the variance of sample and population data manually.

Example 1: for calculating the variance from sample data

Evaluate the given sample data values to calculate the sample variance.

5, 7, 10, 11, 14, 16, 17, 23, 25, 27, 28, 33

Solution

Step 1: First of all, add all the sample observations and divide them by the total number of observations.

Sum = 5 + 7 + 10 + 11 + 14 + 16 + 17 + 23 + 25 + 27 + 28 + 33

Sum = 216

Total number of observation = N = 12

Sample Mean = ȳ = 216/12 = 108/6 = 54/3

Sample Mean = ȳ = 18

Step 2: Now subtract the given observations from the sample mean (ȳ) and take the square of the outputs of each subtraction to make them positive.

Data values	yi – ȳ	(yi – ȳ)2
5	5 – 18 = -13	(-13)2 = 169
7	7 – 18 = -11	(-11)2 = 121
10	10 – 18 = -8	(-8)2 = 64
11	11 – 18 = -7	(-7)2 = 49
14	14 – 18 = -4	(-4)2 = 16
16	16 – 18 = -2	(-2)2 = 4
17	17 – 18 = -1	(-1)2 = 1
23	23 – 18 = 5	(5)2 = 25
25	25 – 18 = 7	(7)2 = 49
27	27 – 18 = 9	(9)2 = 81
28	28 – 18 = 10	(10)2 = 100
33	33 – 18 = 15	(15)2 = 225

Step 3: Now calculate the sum of squares by adding the above-squared deviations.  

∑ (y_i – ȳ)² = 169 + 121 + 64 + 49 + 16 + 4 + 1 + 25 + 49 + 81 + 100 + 225

∑ (y_i – ȳ)² = 904

Step 4: Now take the quotient of the sum of squared differences and the total number of observations decreased by one. This will give you the result of the variance.

∑ (y_i – ȳ)² / N – 1 = 904 / 12 – 1

∑ (y_i – ȳ)² / N – 1 = 904 / 11

∑ (y_i – ȳ)² / N – 1 = 82.18

Example 2: for calculating the variance from population data

Evaluate the given sample data values to calculate the population variance. 

5, 7, 9, 12, 15, 17, 19, 22, 25, 27, 29

Solution

Step 1: First of all, add all the population observations and divide them by the total number of observations.

Sum = 5 + 7 + 9 + 12 + 15 + 17 + 19 + 22 + 25 + 27 + 29

Sum = 187

Total number of observation = N = 11

Population Mean = µ = 187/11

Population Mean = µ = 17

Step 2: Now subtract the given observations from the population mean (µ) and take the square of the outputs of each subtraction to make them positive.

Data values	yi – µ	(yi – µ)2
5	5 – 17 = -12	(-12)2 = 144
7	7 – 17 = -10	(-10)2 = 100
9	9 – 17 = -8	(-8)2 = 64
12	12 – 17 = -5	(-5)2 = 25
15	15 – 17 = -2	(-2)2 = 4
17	17 – 17 = 0	(0)2 = 0
19	19 – 17 = 2	(2)2 = 4
22	22 – 17 = 5	(5)2 = 25
25	25 – 17 = 8	(8)2 = 64
27	27 – 17 = 10	(10)2 = 100
29	29 – 17 = 12	(12)2 = 144

Step 3: Now calculate the sum of squares by adding the above-squared deviations. 

∑ (y_i – µ)² = 144 + 100 + 64 + 25 + 4 + 0 + 4 + 25 + 64 + 100 + 144

∑ (y_i – µ)² = 674

Step 4: Now take the quotient of the sum of squared differences and the total number of observations. This will give you the result of the variance.

∑ (y_i – µ)² / N = 674 / 11

∑ (y_i – µ)² / N = 61.27

Wrap up

Now you can get all the basics for calculating the variance by using the sample and population data values. In this lesson, we have discussed all the basic concepts that are necessary to solve the sample and population variance problems.

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Understanding variance in statistical analysis

Variance is a critical component of descriptive statistics and serves as the foundation for complex inferential tests. It provides a numerical value that describes how far individuals in a group deviate from the average.

In practical applications, a low variance indicates that data points are clustered closely around the mean, suggesting high consistency. Conversely, a high variance signals that data points are widely scattered, indicating significant diversity or instability within the observed phenomenon.

The role of squared deviations

The calculation of variance necessitates squaring the difference between each data point and the mean. If the differences were simply summed, the positive and negative deviations would result in a total of zero, providing no information regarding spread.

By squaring these values, all deviations become positive, allowing for an aggregate measure of total variability. This mathematical approach also places higher weight on outliers, which are values significantly distant from the mean.

Population vs sample variance distinction

A population includes every member of a defined group, such as every student in a school or every part produced in a factory. When data for the entire population is available, the variance is calculated by dividing the sum of squares by the total number of observations (N).

In contrast, a sample is a subset taken from a larger population. Because samples are unlikely to capture the full range of extremes found in a population, using the standard average of squared deviations tends to underestimate the true variability. By dividing by n-1 instead of n, the variance is slightly increased, correcting this bias and providing a more reliable estimate.

Step-by-step calculation guide

To calculate variance manually, follow a structured sequence of mathematical operations.

1. Calculate the Mean: Sum all data points and divide by the total count.
2. Determine Deviations: Subtract the mean from each individual data point.
3. Square the Deviations: Multiply each difference by itself to ensure positive values.
4. Sum the Squares: Add all the squared values together to get the Sum of Squares (SS).
5. Apply the Divisor: Divide by N for population data or n-1 for sample data.

Numerical example solution

Consider a sample of five test scores: 70, 80, 85, 90, and 75.

• Step 1: Mean = (70 + 80 + 85 + 90 + 75) / 5 = 80.
• Step 2 & 3: * (70 – 80)² = 100
  • (80 – 80)² = 0
  • (85 – 80)² = 25
  • (90 – 80)² = 100
  • (75 – 80)² = 25
• Step 4: Sum of Squares = 100 + 0 + 25 + 100 + 25 = 250.
• Step 5: Sample Variance = 250 / (5 – 1) = 62.5.

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