How to Use the One-Sample Variance Test Calculator
This calculator allows you to perform a one-sample chi-square test for a population variance or standard deviation using summary data or raw observations.
1. Summary Data
If you already have the sample size and sample variance or standard deviation, follow these steps:
- Enter the variable name (optional) and significance level (α).
- Select the alternative hypothesis: two-tailed, right-tailed, or left-tailed.
- Choose whether the hypothesized value is a population variance or standard deviation, then enter its value.
- Under Summary Data, enter the sample size (n).
- Select Sample Variance or Sample Standard Deviation and enter the corresponding value.
- Click Calculate.
The calculator will return the chi-square statistic, degrees of freedom, p-value, critical value, and hypothesis test decision. It also provides descriptive statistics, confidence intervals, and an APA-formatted result.
2. Raw Data
If you have the individual observations rather than summary statistics, select Raw Data. You can then choose between two options.
Paste Data: Enter or paste the numerical observations into the data box. Separate values using commas, spaces, semicolons, or line breaks. The calculator will automatically find the sample size, sample variance, and sample standard deviation before performing the hypothesis test.
Upload Dataset: Upload a CSV, TXT, XLS, or XLSX file. Once the file has been loaded, select the numeric variable you want to analyze. You can also select multiple variables or perform separate tests within groups.
When uploading a dataset, the calculator provides a data preview so that you can check the observations before interpreting the results.
What Is a One-Sample Variance Test?
A one-sample variance test is a statistical hypothesis test used to determine whether the variance of a population differs from a specified value.
For example, suppose a manufacturer claims that the variance in the weights of its products is 16 g². A researcher collects a random sample and obtains a sample variance of 25 g².
Although the sample variance is greater than the claimed population variance, this difference does not necessarily mean that the manufacturer’s claim is incorrect. It could have occurred because of random sampling variation.
A one-sample variance test helps us determine whether the observed difference is statistically significant. The classical test uses the chi-square (χ²) distribution and is therefore also called the one-sample chi-square test for variance.
It can be used to test whether a population variance is:
- Different from a hypothesized variance (two-tailed test).
- Greater than a hypothesized variance (right-tailed test).
- Less than a hypothesized variance (left-tailed test).
The same procedure can also test a claim about a population standard deviation because the standard deviation is the square root of the variance.
One-Sample Variance Test Formula
The test statistic formula for a one-sample variance is:
χ² = (n − 1)s² / σ₀²
Where:
- χ² is the chi-square test statistic.
- n is the sample size.
- s² is the sample variance.
- σ₀² is the hypothesized population variance.
The test statistic follows a chi-square distribution with df = n − 1. These degrees of freedom determine which chi-square distribution we use to find the p-value and critical value.
Sometimes a question provides the sample standard deviation (s) and the hypothesized population standard deviation (σ₀) instead of their variances.
In this case, we can write the formula as:
χ² = (n − 1)(s / σ₀)²
This is equivalent to the first formula. The only difference is that we square the standard deviations to obtain their respective variances.
For example, if the sample standard deviation is 6 and the hypothesized population standard deviation is 4, then:
- Sample variance, s² = 6² = 36
- Hypothesized population variance, σ₀² = 4² = 16
We would therefore use 36 and 16 in the chi-square test statistic formula.
If you have individual observations and need to calculate their variance first, you can use the Variance Calculator.
Hypotheses for a One-Sample Variance Test
The null hypothesis assumes that the population variance is equal to the hypothesized value.
H₀: σ² = σ₀²
The alternative hypothesis depends on the claim being tested and can be either of these:
- Two-tailed: H₁: σ² ≠ σ₀²
- Right-tailed: H₁: σ² > σ₀²
- Left-tailed: H₁: σ² < σ₀²
For example, if we want to determine whether the population variance differs from 25, we use a two-tailed test with H₁: σ² ≠ 25. However, if the purpose is to determine whether the population variance exceeds 25, we use a right-tailed test with H₁: σ² > 25.
Remember. You should select the correct alternative hypothesis based on the research question and not after examining the sample results.
How to Perform a One-Sample Variance Test by Hand
To perform a one-sample variance test manually, we follow the six steps of hypothesis testing:
- State the hypotheses.
- State the significance level.
- Calculate the test statistic.
- Find the p-value and critical value.
- Make the decision.
- Write the conclusion.
Example 1
A packaging company claims that the population variance in the weights of its products is 16 g². A random sample of 20 packages has a sample variance of 25 g².
Assuming the package weights are normally distributed, test whether the population variance differs from 16 g² at the 5% significance level.
Solution
Since we want to compare one sample variance with a hypothesized population variance, the appropriate test is a one-sample chi-square test for variance.
To perform the test by hand, follow these steps:
Step 1: State the Hypotheses
The question asks whether the population variance is different from 16 g². Therefore, we use a two-tailed test.
The hypotheses are:
- H₀: σ² = 16
- H₁: σ² ≠ 16
Step 2: State the Significance Level
From the question, the significance level is 5%. Therefore, α = 0.05.
Step 3: Calculate the Test Statistic
The one-sample variance test statistic formula is:
χ² = (n − 1)s² / σ₀²
From the question, we know that:
- Sample size, n = 20
- Sample variance, s² = 25
- Hypothesized population variance, σ₀² = 16
Substituting these values into the formula gives:
χ² = (20 − 1) × 25 / 16
= 19 × 25 / 16
= 29.6875
The degrees of freedom are: df = n − 1
= 20 − 1
= 19
Therefore, the calculated chi-square statistic is 29.6875 with 19 degrees of freedom.
Step 4: Find the P-Value and Critical Values
For a two-tailed chi-square variance test, we use the two tails of the chi-square distribution.
With χ² = 29.6875 and df = 19, and using the chi-square p-value calculator, the two-tailed p-value for the test is 0.111825.
We can also use the critical-value method. In this case, you can either use the Chi-Square Critical Value Calculator or a chi-square distribution table.
At α = 0.05 and df = 19, the two critical chi-square values are:
- Lower critical value = 8.9065
- Upper critical value = 32.8523
Step 5: Make the Decision
We can make the decision using either the p-value or the critical values as follows:
- P-value method: Since the p-value (0.111825) is greater than the significance level, α = 0.05, we fail to reject H₀.
- Critical-value method: Since the test statistic value (χ² = 29.6875) lies between 8.9065 and 32.8523, we fail to reject H₀.
As you can see, both methods lead to the same decision.
Step 6: Write the Conclusion
At the 5% significance level, there is insufficient statistical evidence to conclude that the population variance in package weights differs from 16 g². Therefore, although the observed sample variance was 25 g², the difference was not statistically significant.
How to Interpret the One-Sample Variance Test Results
After running the test, the calculator provides the chi-square statistic, degrees of freedom, p-value, critical values, and confidence interval.
The chi-square statistic compares the sample variance with the hypothesized population variance. Values much smaller or greater than the degrees of freedom may provide evidence against the null hypothesis, depending on the alternative hypothesis.
The p-value determines whether the difference is statistically significant. Compare it with the selected significance level:
- If p < α, reject H₀.
- If p ≥ α, fail to reject H₀.
The critical values give the boundaries of the rejection region. Unlike z and t distributions, the chi-square distribution is not symmetric. Thus, the lower and upper critical values for a two-tailed test are not simply negatives of each other.
The calculator also provides confidence intervals for the population variance and standard deviation. For a two-tailed test, it reports a two-sided interval. However, for a one-tailed test, it reports the corresponding lower or upper confidence bound.
For example, in the first worked example, the 95% confidence interval for the population variance is approximately: 95% CI for σ² = [14.459, 53.332]
Since the hypothesized variance of 16 lies within this interval, the result agrees with the decision to fail to reject H₀.
Assumptions of the One-Sample Variance Test
Before performing a one-sample chi-square test for variance, you should check the following assumptions.
- Random sampling: The sample should be representative of the population about which you want to draw conclusions. Random sampling is a common way to support this requirement.
- Independence: Each observation should be independent of the others. Repeated measurements of the same subject should not be treated as independent observations without considering the study design.
- Numerical data: The observations should measure a quantitative variable, such as height, weight, time, or test score.
- Normality: The observations should come from a normally distributed population.
The normality assumption is particularly important. Unlike some tests for population means, the classical chi-square test for variance is highly sensitive to non-normal data. Even with a large sample, departures from normality can affect the accuracy of its p-value.
If you have raw observations, consider assessing their distribution using the Shapiro-Wilk Test Calculator and inspecting the data for skewness and outliers.
One-Sample Variance Test vs. F-Test and Levene’s Test
Although these tests are all concerned with variability, they answer different research questions. The table below provides a quick summary of the common variance tests and their uses.
| Statistical Test | Purpose |
|---|---|
| One-sample variance test | Tests whether one population variance equals a specified value |
| F-test for two variances | Compares the variances of two independent populations |
| Levene’s test | Tests whether two or more independent groups have equal variances |
For example, if you want to determine whether the population variance in package weights differs from 16 g², use the one-sample variance test. However, if you have two independent samples and want to compare their population variances, use the F-Test Calculator.
If your study involves two or more independent groups and you want to assess the homogeneity of variances assumption, the Levene’s Test Calculator may be more appropriate, especially when departures from normality are a concern.
Frequently Asked Questions
A one-sample variance test calculator performs a chi-square hypothesis test to determine whether a population variance differs from a hypothesized value. This calculator returns the chi-square statistic, degrees of freedom, p-value, critical values, confidence interval, hypothesis decision, and APA-formatted result.
Yes. A hypothesis test for a population standard deviation uses the same chi-square procedure as a test for population variance. Simply square the sample and hypothesized population standard deviations before substituting them into the variance test formula. This calculator also allows you to enter standard deviations directly.
A one-sample chi-square variance test compares a sample variance with a hypothesized population variance. In contrast, a chi-square test of independence determines whether two categorical variables are associated. Although both tests use the chi-square distribution, their test statistics and research questions are different.
The degrees of freedom are calculated using df = n − 1, where n is the sample size. For example, a sample of 20 observations has 19 degrees of freedom.
The classical chi-square variance test assumes that the population is normally distributed. When this assumption is violated, the p-value and confidence interval may be unreliable. In such cases, a statistical method that is less sensitive to non-normality may be needed.
