Variance

TL;DR

• Quantifies how spread out values are around the mean.
• Computed as the average of squared differences from the mean.
• The square root of variance is the standard deviation; higher variance means greater dispersion or volatility.

Definition

Variance is a measure of how much a set of numbers or observations differ from the mean or average of that set. It describes the dispersion or spread of the data.

The formula for variance (population form, as given) is:

$$\text{Variance} = \frac{1}{n}\sum (x - \mu)^2$$

where x is each data point, μ is the mean, and n is the total number of data points.

Explanation

To calculate variance:

• Find the mean (average) of the data.
• Subtract the mean from each data point to obtain deviations.
• Square each deviation to make all values nonnegative and to weight larger deviations more heavily.
• Take the average of those squared deviations.

Because variance uses squared units, its square root is commonly used to return to the original units; this square root is the standard deviation. A high variance (and correspondingly high standard deviation) indicates a greater spread of values around the mean; a low variance indicates values clustered more closely around the mean.

Examples

Example: Class grades

• If the class average is a B and individual grades are close to that average, the variance will be low.
• Grades: A, A, B, B, C, C — the variance will be low because the grades are close to the average.
• Grades: A, A, A, B, C, F — the variance will be higher because the grades show a wider range.

Example: Stock prices

• If a company’s stock price fluctuates within a small range, variance will be low.
• If the stock price is $50 for a few months with only minor fluctuations of a few dollars, the variance will be low.
• If the stock price is $50 one month,$30 the next month, and $70 the following month, the variance will be higher because fluctuations cover a wider range.

Use cases

• Identifying trends, patterns, and anomalies in data.
• Supporting predictions or decisions that depend on the spread or volatility of values.

Notes or pitfalls

• Variance is expressed in squared units of the original data; interpretability is commonly improved by taking the square root (standard deviation).
• High variance corresponds to high volatility or dispersion; low variance corresponds to values clustered near the mean.

Related terms

• Standard deviation