how to find the sample variance

3 min read 13-03-2025

Understanding variance is crucial in statistics. It measures how spread out a dataset is. This guide will walk you through calculating the sample variance, which is an estimate of the population variance based on a sample of data. We'll cover the formula, the steps, and provide examples to make it clear.

Understanding Sample Variance

Before diving into calculations, let's clarify what sample variance represents. The population variance describes the spread of an entire population's data. However, it's often impossible to collect data from every member of a population. Instead, we use a sample – a subset of the population – to estimate the population variance. This estimate is the sample variance.

A larger sample variance indicates greater data dispersion; a smaller variance suggests the data points cluster more closely around the mean.

Calculating Sample Variance: A Step-by-Step Approach

The formula for sample variance (denoted as s²) is:

s² = Σ(xi - x̄)² / (n - 1)

Let's break down each component:

Σ: This symbol means "summation." We'll add up the results of the calculations within the parentheses.
xi: This represents each individual data point in your sample.
x̄: This is the sample mean (average) of your data. Calculate this by summing all data points and dividing by the number of data points.
n: This is the number of data points in your sample.
(n - 1): This is known as Bessel's correction. It's used to provide an unbiased estimate of the population variance. Using 'n' instead would underestimate the population variance.

Step-by-Step Calculation

Here's a detailed, step-by-step process to calculate sample variance:

1. Calculate the Sample Mean (x̄):

Sum all the data points in your sample.
Divide the sum by the number of data points (n).

2. Find the Deviation from the Mean (xi - x̄) for Each Data Point:

Subtract the sample mean (x̄) from each individual data point (xi). This gives you the deviation of each point from the average.

3. Square Each Deviation [(xi - x̄)²]:

Square each of the deviations calculated in the previous step. Squaring ensures that negative deviations don't cancel out positive ones.

4. Sum the Squared Deviations [Σ(xi - x̄)²]:

Add up all the squared deviations.

5. Divide by (n - 1):

Divide the sum of squared deviations by (n - 1), which is one less than the sample size. This is Bessel's correction for an unbiased estimate.

Example Calculation

Let's say we have the following sample data representing the heights (in inches) of five students: 62, 65, 68, 70, 72.

1. Calculate the Mean: (62 + 65 + 68 + 70 + 72) / 5 = 67.4 inches

2. Calculate Deviations from the Mean:

62 - 67.4 = -5.4
65 - 67.4 = -2.4
68 - 67.4 = 0.6
70 - 67.4 = 2.6
72 - 67.4 = 4.6

3. Square the Deviations:

(-5.4)² = 29.16
(-2.4)² = 5.76
(0.6)² = 0.36
(2.6)² = 6.76
(4.6)² = 21.16

4. Sum the Squared Deviations: 29.16 + 5.76 + 0.36 + 6.76 + 21.16 = 63.10

5. Divide by (n - 1): 63.20 / (5 - 1) = 15.775

Therefore, the sample variance (s²) is approximately 15.775 square inches.

Using Software and Calculators

Many statistical software packages (like R, SPSS, Excel) and even some calculators can directly compute sample variance. This can save time and reduce the chance of errors, especially with larger datasets. Check your software's documentation for the specific functions.

Conclusion

Calculating sample variance provides a valuable measure of data dispersion. Understanding the formula and the steps involved is crucial for interpreting statistical results and making informed decisions based on data analysis. Remember Bessel's correction for an unbiased estimate of the population variance, and consider using software for ease and accuracy, particularly with larger datasets.