CI for Variance and Standard Deviation
Computes the confidence interval for the variance and standard deviation of a normal population, from a sample. While most tools estimate the mean, this one estimates the spread — useful in quality control, where consistency matters as much as the central value. The calculation uses the chi-square distribution, which is asymmetric, so the interval is not centered on the sample variance. Enter the sample and the confidence level.
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CI for Variance and Standard Deviation
Computes the confidence interval for the variance and standard deviation of a normal population, from a sample. While most tools estimate the mean, this one estimates the spread — useful in quality control, where consistency matters as much as the central value. The calculation uses the chi-square distribution, which is asymmetric, so the interval is not centered on the sample variance. Enter the sample and the confidence level.
Estimating the spread, not just the mean
Most analyses focus on the mean, but in many contexts the spread is what really matters. In quality control, a process with a perfect mean but high variability delivers inconsistent parts. Estimating the population variance with a margin of uncertainty is, then, as valuable as estimating the center — and that's what this confidence interval does.
The calculation rests on the chi-square distribution, which describes how the sample variance is distributed around the true one. There's a detail that often confuses: because the chi-square is asymmetric, the interval is not centered on the sample variance, and the bounds are obtained by dividing the sum of squares by two different quantiles of the distribution, with the tails crossed.
Enter the sample and the confidence level. The tool returns the interval for the variance and also for the standard deviation, which is simply the square root of the bounds. The usual caveat applies: the method assumes the data come from an approximately normal population, and it is sensitive to departures from that assumption, more so than the intervals for the mean.
Related Tools
One-Sample Variance Test (Chi-Square)
Computes the chi-square test for the variance of a single sample, which checks whether the population variance equals a reference value. It's the dispersion counterpart of the one-sample t-test for the mean. It shows up often in quality control: is a process's variability within the specified limit, or has it increased? The statistic compares the sample variance with the hypothesized one and follows a chi-square distribution. Enter the sample and the hypothesized variance (σ₀²).
CI for the Ratio of Two Variances
Computes the confidence interval for the ratio between the variances of two normal populations, σ₁²/σ₂². Instead of just testing whether the variances differ, it gives a plausible range for how many times one is larger than the other. If the interval contains the number 1, there's no evidence the spreads are different. The calculation uses the F distribution, and because it's asymmetric, the bounds come from F quantiles with the degrees of freedom swapped. Enter the two samples and the confidence level.
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Compute the Z-score (standard score) of a value given population mean and standard deviation. Includes corresponding percentile. Everything in your browser.
Mood's Median Test
Computes Mood's median test, a non-parametric test that checks whether several groups share the same median. The idea is simple and robust: you compute the grand median of all the data together and count, in each group, how many values fall above it. If the groups had the same median, those counts would be proportional to the sizes; an imbalance produces a large chi-square statistic. It's quite resistant to outliers, working only with counts. Enter the groups separated by semicolons.
Chi-Square Test of Independence
Computes the chi-square test of independence for a contingency table, checking whether two categorical variables are associated or independent. For example: does preference for a product depend on age group? The test compares the observed frequencies with those that would be expected if there were no relationship at all, and the chi-square statistic measures the size of that mismatch. The tool returns the statistic, the degrees of freedom and the p-value. Enter the table: columns by comma, rows by semicolon.
CI for Correlation (Fisher z Transform)
Computes the confidence interval for a correlation coefficient using the Fisher z transformation. The sampling distribution of r is skewed, especially near −1 or +1, which prevents applying the usual symmetric margin. Fisher solved this with a transformation that makes the distribution approximately normal; you build the interval on that scale and then map back to r. The result is an asymmetric interval, narrower on the side near the extremes. Enter the coefficient r and the sample size.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.