Variance Calculator

Variance Calculator – Measure How Spread Out Your Data Really Is

Variance is one of the core measures of dispersion in statistics. While the mean describes the center of a dataset, variance describes how far values spread out around that center. This Variance Calculator on MyTimeCalculator brings together population variance, sample variance and standard deviation, with modes for raw data, grouped data, summary statistics and two-dataset comparison.

Throughout this article x₁, x₂, …, x_n denote data values, x̄ is the sample mean, μ is the population mean, n is the sample size and N is the population size. Variance is always expressed in squared units, while standard deviation uses the original units of the data.

Population Variance and Sample Variance

For a population of N values x₁, x₂, …, x_N with mean μ, the population variance is defined as

σ² = (1 / N) Σ (x_i − μ)²

This formula describes the exact spread of the population. There is no sampling uncertainty because every member of the population is observed.

For a sample of n values x₁, x₂, …, x_n with sample mean x̄, the sample variance is defined as

s² = (1 / (n − 1)) Σ (x_i − x̄)²

The denominator n − 1 instead of n ensures that s² is an unbiased estimator of the population variance when the sample is drawn at random. The same formulas apply to grouped data if each value is weighted by its frequency.

Standard Deviation as the Square Root of Variance

Because variance is in squared units, many people prefer to use standard deviation, which brings the units back to the original scale. For a population the standard deviation is

σ = √σ²

and for a sample it is

s = √s²

Standard deviation sits in the same units as the data and is often easier to interpret, but variance is algebraically more convenient when combining or transforming variables.

Ungrouped Data: Variance From Raw Values

When you have raw data values the variance formulas take the form

• Population variance: σ² = (1 / N) Σ (x_i − μ)²
• Sample variance: s² = (1 / (n − 1)) Σ (x_i − x̄)²

The calculator’s single dataset tab uses these definitions directly. It parses your values, computes the mean, subtracts the mean from each value, squares the deviations, takes the appropriate average and then square-roots to obtain standard deviation. It also reports minimum, maximum and range for quick context.

Grouped Data: Variance With Frequencies

In grouped data each representative value x_j occurs with frequency f_j. Let N = Σ f_j be the total frequency and

μ = (1 / N) Σ f_j x_j

Then the grouped population variance is

σ² = (1 / N) Σ f_j (x_j − μ)²

and the grouped sample variance is

s² = (1 / (N − 1)) Σ f_j (x_j − x̄)²

The grouped data tab implements exactly these formulas. You enter up to eight rows of midpoints and frequencies, and the calculator returns the mean, population variance, sample variance and their square roots.

Variance From Summary Statistics (Σx and Σx²)

When working with large datasets or intermediate steps from statistical software, you might know only the count n, the sum of values Σx and the sum of squared values Σx². In that case it is convenient to use computational formulas for variance.

First compute the mean

μ = Σx / n

Then the population variance can be written as

σ² = Σx² / n − μ²

and the sample variance as

s² = [Σx² − n μ²] / (n − 1)

The summary statistics tab uses these formulas to produce mean, variance and standard deviation directly from n, Σx and Σx².

Comparing the Spread of Two Datasets

Variance and standard deviation are especially useful for comparing how spread out two datasets are. A higher variance means the data are more dispersed around the mean, while a lower variance means they are tightly clustered.

In the comparison tab the calculator parses two datasets A and B and computes for each:

• n, the number of data points
• Mean
• Population and sample variance
• Population and sample standard deviation
• Minimum, maximum and range

It then reports which dataset has the larger variance and the ratio between the larger and smaller variance. This gives an intuitive sense of how much more spread out one dataset is compared to the other.

Population vs Sample: Which Variance Should You Use?

The choice between population and sample variance depends on how you interpret your data.

• If the data includes every member of the group of interest, use population variance with denominator N.
• If the data is a sample from a larger population and you want to estimate its variance, use sample variance with denominator n − 1.

The calculator always reports both, but in practice you should decide which interpretation is appropriate for your situation. For many real-world problems where data is collected from a subset of cases, sample variance is the default choice.

Linear Transformations and Variance

Variance responds predictably to linear transformations. If you define a new variable Y = a + bX, where a and b are constants, the variance satisfies

Var(Y) = b² Var(X)

Adding a constant does not change variance because it shifts all values equally. Multiplying by a constant scales variance by the square of that constant. This property underlies many techniques in standardizing, rescaling and comparing variables on different units.

How to Use the Variance Calculator Effectively

• Use the single dataset tab when you have raw values in a list and want quick variance and standard deviation.
• Use the grouped data tab when your values are summarized in a frequency table or class midpoints.
• Use the summary statistics tab when working with large datasets where only n, Σx and Σx² are available.
• Use the comparison tab when you want to see which of two datasets is more variable.
• Use the population vs sample and formulas tabs as a reference when interpreting results or writing up your findings.

This calculator is intended for learning, homework checking, teaching and quick analysis. It does not replace full statistical software, but it makes the variance formulas concrete and easy to explore.