Permutation Standard Deviation Calculator

Enter Sample Data

Sample A Values

Use commas, spaces, or line breaks.

Sample B Values

Both groups should contain numeric values.

Statistic Type

Alternative Hypothesis

Resampling Mode

Simulation Count

Confidence Level (%)

Random Seed

What This Calculator Does

It builds a permutation distribution by relabeling combined values across two groups. Then it measures the distribution spread using the standard deviation of the resampled statistic.

It also reports the observed statistic, permutation mean, z score, p value, and a central interval for quick interpretation.

Best Use Cases

Permutation testing for two independent samples
Robust comparisons with fewer distribution assumptions
Teaching resampling variability and inference
Auditing simulation-based statistical workflows

Plotly Graph

The histogram below shows the permutation distribution. The red line marks the observed statistic. The green line marks the permutation mean.

Example Data Table

Observation	Sample A	Sample B
1	14	11
2	18	12
3	19	15
4	21	17
5	24	20

Formula Used

Let the resampled statistic from permutation i be T_i, and let there be M evaluated permutations. The permutation mean is:

μ_perm = ( Σ T_i ) / M

The permutation standard deviation is estimated from the resampled statistics:

SD_perm = √[ Σ (T_i − μ_perm)² / (M − 1) ]

The reported z score is:

z = (T_obs − μ_perm) / SD_perm

The p value depends on the selected alternative. For two-sided testing, the calculator counts permutation statistics with absolute distance from the permutation mean at least as large as the observed distance.

Available Statistics

Mean Difference: mean(Sample A) − mean(Sample B)
Median Difference: median(Sample A) − median(Sample B)
Sum Difference: sum(Sample A) − sum(Sample B)
Variance Ratio: variance(Sample A) / variance(Sample B)

How to Use This Calculator

Enter numeric values for Sample A and Sample B.
Select the comparison statistic you want to study.
Choose a two-sided, greater, or less alternative.
Set Auto, Exact, or Monte Carlo resampling mode.
Enter a simulation count for Monte Carlo runs.
Choose the confidence level for the central interval.
Submit the form to generate the results section.
Review the graph, summary table, and export files.

Frequently Asked Questions

1) What is permutation standard deviation?

It is the standard deviation of the statistic values generated from repeated label rearrangements of the pooled sample. It measures how much the test statistic naturally varies under the permutation model.

2) When should I use this method?

Use it when you want fewer parametric assumptions, especially for small samples, skewed data, or unusual distributions. It is valuable for teaching, validation, and nontraditional inference workflows.

3) What does the p value represent here?

The p value estimates how often the relabeled data produce a statistic at least as extreme as the observed one, based on the chosen alternative hypothesis and permutation distribution.

4) Why can exact mode be unavailable?

Exact enumeration grows rapidly with sample size because the number of possible relabelings can become very large. Auto mode switches to Monte Carlo simulation when exact evaluation becomes impractical.

5) What does the z score tell me?

It shows how far the observed statistic lies from the permutation mean in permutation standard deviation units. Larger absolute values indicate greater separation from the relabeling reference distribution.

6) Should I use many permutations?

Yes. More permutations usually improve stability of the estimated distribution, standard deviation, and p value. For routine work, several thousand simulations often give a useful balance between speed and precision.

7) Can I compare medians or variances instead of means?

Yes. This calculator supports mean difference, median difference, sum difference, and variance ratio. Different statistics answer different questions, so choose the one matching your analytic goal.

8) Why use a random seed?

A seed helps reproduce the same Monte Carlo sequence later. That is useful for audits, teaching demonstrations, documentation, and debugging when you need consistent numerical outputs.