How to Calculate Sample Variance with Ss and N

Sample variance is a fundamental measure in statistics that quantifies the amount of variation or dispersion in a set of data points. When you have the sum of squares (SS) and the sample size (n), you can calculate the sample variance directly. This guide explains how to perform this calculation, including the formula, step-by-step instructions, and practical examples.

What is Sample Variance?

Sample variance measures how far each number in a dataset is from the mean (average) of the dataset. A higher variance indicates that the data points are more spread out, while a lower variance indicates that the data points are closer to the mean.

Variance is particularly useful in statistical analysis because it provides insight into the consistency of data. For example, in quality control, low variance might indicate consistent product performance, while high variance could signal issues that need investigation.

The Formula

The sample variance (s²) can be calculated using the sum of squares (SS) and the sample size (n) with this formula:

Sample Variance Formula:

s² = SS / n

Where:

s² = sample variance
SS = sum of squares (sum of each data point's squared deviation from the mean)
n = sample size (number of data points)

This formula assumes you already have the sum of squares calculated. If you don't have SS, you would first calculate it by finding the mean of your data points, then summing the squared differences between each data point and the mean.

How to Calculate Sample Variance

Step 1: Gather Your Data

First, collect your dataset and determine the sample size (n). For example, you might have test scores from a class of 20 students.

Step 2: Calculate the Sum of Squares (SS)

If you don't already have SS, calculate it by:

Finding the mean (average) of your data points
For each data point, subtract the mean and square the result
Sum all these squared values to get SS

Step 3: Apply the Formula

Once you have SS and n, plug these values into the sample variance formula:

s² = SS / n

Step 4: Interpret the Result

The resulting value is your sample variance. A higher value indicates greater dispersion in your data, while a lower value indicates more consistent data points around the mean.

Worked Example

Let's calculate the sample variance for the following dataset of exam scores: 85, 90, 78, 92, 88.

Step 1: Calculate the Mean

Mean = (85 + 90 + 78 + 92 + 88) / 5 = 433 / 5 = 86.6

Step 2: Calculate the Sum of Squares (SS)

Score	Deviation from Mean	Squared Deviation
85	85 - 86.6 = -1.6	(-1.6)² = 2.56
90	90 - 86.6 = 3.4	(3.4)² = 11.56
78	78 - 86.6 = -8.6	(-8.6)² = 73.96
92	92 - 86.6 = 5.4	(5.4)² = 29.16
88	88 - 86.6 = 1.4	(1.4)² = 1.96
Total SS		2.56 + 11.56 + 73.96 + 29.16 + 1.96 = 120.2

Step 3: Calculate Sample Variance

Using the formula:

s² = SS / n = 120.2 / 5 = 24.04

Step 4: Interpretation

The sample variance of 24.04 indicates that, on average, the exam scores deviate from the mean by approximately 4.9 points (the square root of 24.04). This suggests moderate variability in the exam performance.

FAQ

What is the difference between sample variance and population variance?: Sample variance uses n (sample size) in the denominator, while population variance uses N (total population size). For large samples, the difference is small.
When would I use sample variance instead of standard deviation?: Sample variance is used when you need to measure the spread of data in squared units, while standard deviation (the square root of variance) is used when you want the spread in the same units as the original data.
Can sample variance be negative?: No, sample variance is always a non-negative value because it's based on squared deviations. However, individual deviations can be negative.
What if my sample size is small?: For small samples, it's often recommended to use the sample variance formula with n-1 in the denominator (Bessel's correction) to get an unbiased estimate of population variance.
How does sample variance relate to confidence intervals?: Sample variance is a key component in calculating confidence intervals for the population mean, as it provides an estimate of the population variance.