Inferences About the Population Variance

Statistics • Chi Square Tests

Written by STEM Calculators Team Published December 25, 2025 Updated February 24, 2026

Task

These procedures use the chi-square distribution and are appropriate when the population is (approximately) normally distributed.

Data entry

Raw data can be separated by spaces, commas, or new lines.

Sample data

What is computed from raw data?

The calculator computes the sample size n, the sample mean x̄, and the unbiased sample variance s² (dividing by n − 1).

Confidence level

Variance interval form: \[ \left(\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}},\; \frac{(n-1)s^2}{\chi^2_{\alpha/2}}\right) \]

Ready

χ² model visualization (updates with df)

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The shaded area updates after calculation.

When this method applies

Use these procedures when:

You have a simple random sample from the population.
The population distribution is (approximately) normal.
You want inference about σ² (variance) or σ (standard deviation).

Key formulas

\[ \begin{aligned} \chi^2 &= \frac{(n-1)s^2}{\sigma^2}\quad \text{(sampling distribution)} \\ \text{CI for }\sigma^2 &: \left(\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}},\; \frac{(n-1)s^2}{\chi^2_{\alpha/2}}\right) \end{aligned} \]

A CI for σ is obtained by taking square roots of the CI endpoints for σ².

Enter values and click Calculate.

Batch mode: paste CSV data (compute many rows at once)

Paste rows as CSV (comma-separated). Header is optional. Supported columns: task (ci_var/ci_sd/ht_var/ht_sd), conf, alpha, alt (two/gt/lt), n, s2, s, null (σ²₀ or σ₀ depending on task), and optional data (raw sample values separated by spaces/semicolons; if present, it overrides n/s2/s).

CSV input

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Inferences About the Population Variance (σ²)

This calculator helps build (1) a confidence interval for the population variance σ² (and optionally the population standard deviation σ), and (2) a hypothesis test about σ², using the chi-square (χ²) distribution.

When it applies

You have a simple random sample from one population.
The population is approximately normal (important for χ² methods).
You know (or can compute) the sample size n and sample variance s².

What to enter

Sample size (n): number of observations in the sample.
Sample variance (s²) (or paste raw data if the calculator supports it): variance computed from the sample.
Task: choose Confidence Interval or Hypothesis Test.
For tests: enter the hypothesized variance σ₀², choose the alternative, and pick α.

Key formulas used

If the population is normal, the statistic below follows a chi-square distribution with df = n − 1.

\[ \chi^2 = \frac{(n-1)s^2}{\sigma^2}, \qquad df=n-1 \]

Confidence interval for σ²

Choose a confidence level (for example 95%). The calculator finds two χ² critical values and then computes the interval endpoints for σ².

\[ \left(\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}},\; \frac{(n-1)s^2}{\chi^2_{\alpha/2}}\right) \]

If the calculator also reports the interval for σ, it takes the positive square roots of both endpoints.

Hypothesis tests about σ²

Select the alternative form and enter σ₀² (the value claimed under H₀). The test uses:

\[ \chi^2_{\text{obs}}=\frac{(n-1)s^2}{\sigma_0^2} \]

Right-tailed (variance too large): H₁: σ² > σ₀²
Left-tailed (variance too small): H₁: σ² < σ₀²
Two-tailed (variance different): H₁: σ² ≠ σ₀²

The calculator can show the decision using the p-value, critical values, or both. The visualization shades the relevant tail area(s) of the χ² curve so you can see where the test statistic falls.

Batch mode (CSV)

Use the batch section to compute many rows at once. Paste CSV data directly into the textarea (header optional). Typical columns are: n, s2 (or s^2), and optionally task (ci/ht), conf, alpha, sigma0sq, alt. After running batch, you can copy the results table.

Tip

Variance tests and intervals are sensitive to non-normal data. If the population is strongly non-normal, χ²-based inference for σ² may be unreliable.

Frequently Asked Questions

What is meant by inferences about the population variance?

It means using sample data to estimate or test the variability of a population, measured by the population variance sigma^2 or the population standard deviation sigma. These procedures quantify uncertainty using a confidence interval or a hypothesis test.

Why does inference for sigma^2 use the chi-square distribution?

If the population is normal, the statistic (n - 1)s^2 / sigma^2 follows a chi-square distribution with df = n - 1. This relationship allows computing critical values, p-values, and confidence intervals for sigma^2.

How is a confidence interval for sigma^2 computed?

A (1 - alpha) interval is formed using chi-square cutoffs: ((n - 1)s^2 / chi2_right, (n - 1)s^2 / chi2_left), where chi2_left and chi2_right are chi-square critical values based on df = n - 1. The calculator applies the correct tail areas for the chosen confidence level.

Can this calculator give an interval or test for sigma instead of sigma^2?

Yes, sigma is the square root of sigma^2, so results for variance can be converted to results for standard deviation by taking square roots of the interval endpoints. Hypothesis tests can also be framed using sigma0 or sigma0^2.

What assumptions are important for chi-square variance inference?

The key assumption is that the underlying population is approximately normal (or that the sample comes from a normal process). With strong non-normality, chi-square methods for variance can be unreliable.

Inferences About the Population Variance

Hypotheses

Decision settings

When this method applies

Calculation steps

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Frequently Asked Questions

Hypotheses

Decision settings

When this method applies

Calculation steps

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Frequently Asked Questions

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