how to calculate population variance

2 min read 16-03-2025

Understanding population variance is crucial in statistics. It measures how spread out a population's data points are from the mean (average). A large variance indicates high variability, while a small variance suggests data points cluster closely around the mean. This article will guide you through calculating population variance step-by-step.

Understanding the Concepts

Before diving into calculations, let's clarify some key terms:

Population: The entire group you're studying. This could be anything from the height of all students in a school to the GDP of all countries in the world.
Population Mean (μ): The average of all data points in the population.
Variance (σ²): The average of the squared differences from the mean. Squaring ensures that both positive and negative deviations contribute positively to the overall variance.

Step-by-Step Calculation of Population Variance

Here's how to calculate population variance:

1. Calculate the Population Mean (μ):

First, find the average of your data set. Add all the values together and divide by the number of values (N).

Formula: μ = Σx / N

Where:

Σx = Sum of all data points
N = Total number of data points

2. Find the Deviation of Each Data Point from the Mean:

For each data point (x), subtract the mean (μ) to find its deviation.

Formula: x - μ

3. Square Each Deviation:

Square each deviation calculated in step 2. This removes negative signs, ensuring that all deviations contribute positively to the variance.

Formula: (x - μ)²

4. Sum the Squared Deviations:

Add up all the squared deviations calculated in step 3.

Formula: Σ(x - μ)²

5. Divide by the Population Size (N):

Finally, divide the sum of squared deviations by the total number of data points (N) to get the population variance.

Formula: σ² = Σ(x - μ)² / N

Example Calculation

Let's say we have the following population data representing the number of hours students study per week: {5, 6, 7, 8, 9}

Calculate the Mean (μ): (5 + 6 + 7 + 8 + 9) / 5 = 7
Find Deviations:
- 5 - 7 = -2
- 6 - 7 = -1
- 7 - 7 = 0
- 8 - 7 = 1
- 9 - 7 = 2
Square Deviations:
- (-2)² = 4
- (-1)² = 1
- (0)² = 0
- (1)² = 1
- (2)² = 4
Sum Squared Deviations: 4 + 1 + 0 + 1 + 4 = 10
Calculate Variance (σ²): 10 / 5 = 2

Therefore, the population variance (σ²) for this data set is 2.

Sample Variance vs. Population Variance

It's crucial to differentiate between population variance and sample variance. When working with a sample (a subset of the population), you use a slightly different formula:

Sample Variance (s²): s² = Σ(x - x̄)² / (n - 1)

Where:

x̄ = sample mean
n = sample size

The denominator (n-1) is used in sample variance due to Bessel's correction, which provides a less biased estimate of the population variance.

Using Software for Calculation

Calculating variance by hand can be tedious, especially with large datasets. Statistical software packages like R, SPSS, Excel, and Python (with libraries like NumPy and Pandas) offer functions to easily calculate both population and sample variance.

Conclusion

Calculating population variance helps understand the spread of data within a population. This knowledge is vital in various fields, from finance and economics to biology and social sciences. Remember the key steps: calculate the mean, find deviations, square them, sum them, and divide by the population size (N). Understanding the difference between population and sample variance is also essential for accurate statistical analysis.