Sample Variance Equation, Derivation & Rearrangements

Core idea

Overview

Sample variance is a measure of the dispersion or spread of data points around the mean within a specific subset of a population. In psychological research, it serves as an unbiased estimator of population variance by incorporating Bessel's correction, which uses degrees of freedom instead of the total count.

When to use: Use sample variance when you are analyzing a subset of a larger population and need to estimate the degree of individual differences. It is a fundamental requirement for inferential statistics such as t-tests and ANOVA, assuming the data is measured on an interval or ratio scale.

Why it matters: It allows psychologists to quantify how much scores vary from the average, which is crucial for determining if experimental effects are significant or due to chance. By using n - 1, the formula corrects for the systematic underestimation of variability that occurs when only a small group is studied.

Symbols

Variables

s^2 = Sample Variance, SS = Sum of Squares, n = Sample Size

s^{2}

Sample Variance

V a r iab l e

S S

Sum of Squares

V a r iab l e

n

Sample Size

V a r iab l e

Walkthrough

Derivation

Formula: Sample Variance

The average squared deviation from the sample mean, corrected for bias using n − 1.

Data are measured on an interval or ratio scale.
The sample is drawn from a normally distributed population (for inference).

1

Calculate the mean:

Find the arithmetic mean of the sample.

\overset{x}{ˉ} = \frac{\sum x _{i}}{n}

2

Sum the squared deviations:

Square each deviation from the mean and add them up.

S S = i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

3

Divide by n − 1 (Bessel's correction):

Dividing by n − 1 rather than n gives an unbiased estimate of the population variance.

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Note: n − 1 is the degrees of freedom for a single-sample variance.

Result

s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

Source: University Psychology — Statistics

Free formulas

Rearrangements

Solve for $s^{2}$

Sample Variance

s^{2} = \frac{S S}{n - 1}

Simplify the formula for sample variance by introducing the Sum of Squares (SS) notation.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine data points scattered along a number line; the sample mean is the central balancing point, and the sample variance quantifies the average 'spread' or 'width' of these points around that center, where larger

s^{2}

A measure of the average squared deviation of individual data points from the sample mean.

Quantifies how spread out the data points are around their average value within a sample; a larger value indicates greater variability.

x

An individual observation or score within the sample dataset.

Each 'x' is a single piece of information whose position relative to the mean contributes to the overall variability.

\overset{x}{ˉ}

The arithmetic mean of all individual data points in the sample.

Represents the central tendency or typical value of the sample, serving as the reference point for measuring deviations.

(x - \overset{x}{ˉ})^{2}

The squared difference between an individual data point and the sample mean.

Measures the 'distance' of each point from the mean, with squaring ensuring positive values and giving disproportionately more weight to larger deviations.

\sum (x - \overset{x}{ˉ})^{2}

The sum of all squared deviations from the mean for every data point in the sample.

Represents the total amount of variability or dispersion present across all data points in the sample.

n - 1

The degrees of freedom, representing the number of independent pieces of information available to estimate the population variance.

This correction factor makes the sample variance an unbiased estimator of the population variance by accounting for the fact that the sample mean itself is estimated from the data.

Signs and relationships

^2: Squaring the deviation (x - $\overset{x}{ˉ}$ ) ensures that all differences contribute positively to the variance, regardless of whether a data point is above or below the mean.
n - 1: This is Bessel's correction. Dividing by n - 1 instead of n provides an unbiased estimate of the population variance from a sample. This is because the sample mean $\overset{x}{ˉ}$ is used in the calculation, which inherently

Free study cues

Insight

Canonical usage

The unit of sample variance is the square of the unit of the original data points.

Common confusion

A common mistake is forgetting that the unit of variance is the square of the original data's unit, while the standard deviation (the square root of variance) has the same unit as the original data.

Dimension note

Sample variance is dimensionless if the original data points (e.g., Likert scale ratings, test scores, counts) are dimensionless.

Unit systems

$x$ U · Represents an individual data point. Its unit (U) and dimension (D) depend on the specific quantity being measured (e.g., seconds for reaction time, dimensionless for a Likert score).

$\overset{x}{ˉ}$ U · Represents the sample mean. It shares the same unit and dimension as the individual data points (x).

$n$ dimensionless · Represents the sample size, which is a count and therefore dimensionless.

$s^{2}$ U^2 · The unit of sample variance is the square of the unit of the original data points. If the original data points (x) are dimensionless, then the sample variance (s^2) is also dimensionless.

One free problem

Practice Problem

A clinical psychologist measures the anxiety scores of 10 patients. The Sum of Squares (ss) for these scores is calculated to be 180. What is the sample variance for this group?

Sum of Squares180

Sample Size10

Solve for: $v a r$

Hint: Divide the Sum of Squares by the degrees of freedom, which is the sample size minus one.

The full worked solution stays in the interactive walkthrough.

Study smarter

Tips

Always use n - 1 for samples to ensure the estimate is unbiased.
The units of variance are the square of the original measurement units.
A variance of zero indicates that all scores in the dataset are identical.
Variance is the square of the standard deviation.

Avoid these traps

Common Mistakes

Using N instead of n-1 for samples.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The average squared deviation from the sample mean, corrected for bias using n − 1.

Use sample variance when you are analyzing a subset of a larger population and need to estimate the degree of individual differences. It is a fundamental requirement for inferential statistics such as t-tests and ANOVA, assuming the data is measured on an interval or ratio scale.

It allows psychologists to quantify how much scores vary from the average, which is crucial for determining if experimental effects are significant or due to chance. By using n - 1, the formula corrects for the systematic underestimation of variability that occurs when only a small group is studied.

Using N instead of n-1 for samples.

Always use n - 1 for samples to ensure the estimate is unbiased. The units of variance are the square of the original measurement units. A variance of zero indicates that all scores in the dataset are identical. Variance is the square of the standard deviation.

References

Sources

Wikipedia: Sample variance
Discovering Statistics Using IBM SPSS Statistics by Andy Field
Statistics for Psychology by Arthur Aron, Elaine N. Aron, and Elliot J. Coups
Wikipedia: Variance
Statistical Methods for Psychology (David C. Howell)
Discovering Statistics Using R (Andy Field)
Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences (10th ed.). Cengage Learning.
Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.