PsychologyDescriptive StatisticsA-Level

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Variance (Sample)

The square of the standard deviation.

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s^{2} = \frac{Σ ( x - x ˉ ) ^{2}}{n - 1}

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Core idea

Overview

Sample variance measures the average squared deviation of data points from their mean within a specific subset of a population. It utilizes Bessel's correction, dividing by n - 1, to provide an unbiased estimate of the true population variability.

When to use: This formula is used when you are working with a sample of data rather than an entire population. It is applicable for interval or ratio data in psychological research where quantifying the spread of scores is necessary for inferential testing.

Why it matters: Variance serves as the mathematical foundation for more complex statistical analyses like ANOVA and regression. In psychology, understanding variance helps researchers distinguish between meaningful experimental effects and random individual differences.

Symbols

Variables

s^2 = Sample Variance, \Sigma(x-\bar{x})^2 = Sum of Squares, n = Sample Size

s^{2}

Sample Variance

V a r iab l e

Σ (x - \overset{x}{ˉ})^{2}

Sum of Squares

V a r iab l e

n

Sample Size

V a r iab l e

Walkthrough

Derivation

Derivation/Understanding of Variance (Sample)

This derivation explains how sample variance is calculated to provide an unbiased estimate of the population variance, highlighting the reason for dividing by n-1.

We have a random sample drawn from a larger population.
We want to estimate the variance of the population using this sample.

The Concept of Variance:

It quantifies the spread or dispersion of a dataset, with larger values indicating greater spread. For a population, this is typically denoted by $σ^{2}$ .

Variance measures the average squared deviation of data points from their mean.

Initial Calculation Using Sample Mean:

If we simply used the sample mean $\overset{x}{ˉ}$ and divided by the sample size n, we would get a measure of spread within that specific sample.

A natural first thought for sample variance might be \frac{Σ ( x - x ˉ ) ^{2}}{n}

Why Dividing by n is Biased:

Using $\overset{x}{ˉ}$ as the reference point makes the sum of squares artificially small. This means simply dividing by n would consistently underestimate the true population variance.

The sample mean \overset{x}{ˉ} is the value that minimises the sum of squared deviations for that specific sample. Therefore, Σ (x - \overset{x}{ˉ})^{2} will always be smaller than Σ (x - μ)^{2} (where μ is the true population mean).

Applying Bessel's Correction:

To correct for this systematic underestimation and provide an unbiased estimate of the population variance, we divide by n - 1 instead of n. This adjustment is known as Bessel's correction.

s^{2} = \frac{Σ ( x - x ˉ ) ^{2}}{n - 1}

Result

s^{2} = \frac{Σ ( x - x ˉ ) ^{2}}{n - 1}

Source: AQA A-level Psychology Specification

Free formulas

Rearrangements

Solve for $s^{2}$

Make $s^{2}$ the subject

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

This rearrangement demonstrates how to express the formula for sample variance using the shorthand notation for the Sum of Squares (SS). The initial formula explicitly shows the sum of squared differences from the mean, which is then replac...

Difficulty: 2/5

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

Visual intuition

Graph

Graph unavailable for this formula.

The graph of variance plotted against the independent variable x forms a parabola that opens upwards. This parabolic shape occurs because the formula squares the difference between each independent variable value and the mean, ensuring that all deviations contribute positively to the total variance.

Graph type: parabolic

Why it behaves this way

Intuition

Imagine data points scattered along a number line. The sample mean is the central balancing point, and the variance quantifies the average squared distance of these points from that center, giving a measure of how spread

s^{2}

A measure of the average squared deviation of individual data points from the sample mean. It quantifies the dispersion or spread of scores within a sample.

A larger

s^{2}

indicates that data points are, on average, more spread out from the mean, signifying greater variability in the sample.

The value of a single observation or data point within the sample.

Represents an individual score or measurement taken from the dataset.

\overset{x}{ˉ}

The arithmetic average of all data points in the sample. It serves as the measure of central tendency for the sample.

The "balancing point" or typical value around which the data points in the sample are centered.

(x - \overset{x}{ˉ})^{2}

The squared difference between an individual data point and the sample mean. It quantifies an individual data point's deviation from the mean, with the squaring operation ensuring

Measures how far an individual score deviates from the average, with larger values indicating greater individual spread. Squaring prevents positive and negative deviations from cancelling out.

Σ (x - \overset{x}{ˉ})^{2}

The sum of all squared differences between each data point and the sample mean. This is also known as the Sum of Squares (SS).

Represents the total amount of variability within the sample before it is averaged. A larger sum indicates more overall spread among the data points.

n - 1

The degrees of freedom for the sample variance. This term (Bessel's correction) is used in the denominator to provide an unbiased estimate of the population variance from the

Dividing by n - 1 instead of n slightly increases the variance for small samples, correcting for the fact that the sample mean

\overset{x}{ˉ}

is used, which makes the sample deviations appear slightly smaller than if the true

Signs and relationships

^2: Squaring the deviation (x - $\overset{x}{ˉ}$ ) ensures that all differences contribute positively to the total sum of squares, regardless of whether an individual data point is above or below the mean.
n - 1: The subtraction of 1 from the sample size n (known as Bessel's correction) is applied to make the sample variance an unbiased estimator of the true population variance.

Free study cues

Insight

Canonical usage

The unit of sample variance is the square of the unit of the original data points, reflecting the average squared deviation from the mean.

Common confusion

A common mistake is forgetting to square the unit when reporting variance, or confusing the units of variance with those of standard deviation.

Unit systems

$x$ Varies depending on measured quantity (e.g., seconds, items, scores) · Represents individual data points in the sample.

$s^{2}$ Unit of x squared (e.g., s^2, items^2, scores^2) · The calculated sample variance.

$n$ dimensionless · The number of observations in the sample.

One free problem

Practice Problem

A clinical psychologist measures the stress levels of 10 patients. The calculated Sum of Squares (SS) for their scores is 180. What is the sample variance?

Sum of Squares180

Sample Size10

Solve for: $V A R$

Hint: Divide the Sum of Squares by the degrees of freedom, which is n - 1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

If the Sum of Squares for 10 reaction times is 180, Variance = 180 / 9 = 20 ms^2.

Study smarter

Tips

Always use n - 1 for samples to avoid underestimating population variability.
The Sum of Squares (SS) represents the numerator and is always a non-negative value.
Variance is expressed in squared units; take the square root if you need the standard deviation.

Avoid these traps

Common Mistakes

Reporting variance as the final measure of spread instead of standard deviation.
Dividing by n instead of n-1 for sample data.

Common questions

Frequently Asked Questions

This derivation explains how sample variance is calculated to provide an unbiased estimate of the population variance, highlighting the reason for dividing by n-1.

This formula is used when you are working with a sample of data rather than an entire population. It is applicable for interval or ratio data in psychological research where quantifying the spread of scores is necessary for inferential testing.

Variance serves as the mathematical foundation for more complex statistical analyses like ANOVA and regression. In psychology, understanding variance helps researchers distinguish between meaningful experimental effects and random individual differences.

Reporting variance as the final measure of spread instead of standard deviation. Dividing by n instead of n-1 for sample data.

If the Sum of Squares for 10 reaction times is 180, Variance = 180 / 9 = 20 ms^2.

Always use n - 1 for samples to avoid underestimating population variability. The Sum of Squares (SS) represents the numerator and is always a non-negative value. Variance is expressed in squared units; take the square root if you need the standard deviation.

References

Sources

Wikipedia: Variance
Wikipedia: Bessel's correction
Britannica: Variance
Gravetter, F. J., Wallnau, L. B., Forzano, L. B., & Witnauer, J. E. (2021). Statistics for the Behavioral Sciences (11th ed.). Cengage.
AQA A-level Psychology Specification

Variance (Sample)

Overview

Variables

Derivation

The Concept of Variance:

Initial Calculation Using Sample Mean:

Why Dividing by n is Biased:

Applying Bessel's Correction:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Standard Deviation (Sample)

Frequently Asked Questions

Sources