Runs Test Calculator

Example Data Table

This example uses a numeric sequence and the median as the classification threshold.

Position	Value	Median comparison	Code
1	12	Above	A
2	9	Below	B
3	15	Above	A
4	13	Above	A
5	8	Below	B
6	16	Above	A

Formula Used

The runs test checks whether the order of two categories appears random. A run is a consecutive block of the same category.

Expected runs: μ = 1 + (2 n₁ n₂) / (n₁ + n₂) Variance: σ² = [2 n₁ n₂ (2 n₁ n₂ - n₁ - n₂)] / [(n₁ + n₂)² (n₁ + n₂ - 1)] Standardized statistic without continuity correction: z = (R - μ) / σ With continuity correction: z = (R - μ ± 0.5) / σ

Where:

R = observed number of runs
n₁ = count in category A
n₂ = count in category B
μ = expected runs under randomness
σ² = variance of runs
z = standardized test statistic

How to Use This Calculator

Choose numeric mode for measured values or binary mode for two-category sequences.
For numeric mode, select median, mean, or a custom threshold.
Choose how ties should be handled when a value equals the threshold.
Select the alternative hypothesis and significance level.
Paste your sequence into the appropriate input box.
Press Run Test to see the result above the form.
Review the table, graph, p values, and decision summary.
Use the CSV or PDF buttons to export the report.

FAQs

1. What does the runs test measure?

It checks whether the order of two categories looks random. It does not measure average size, only how values alternate or cluster through the sequence.

2. When should I use numeric mode?

Use numeric mode when you have measured values and want to convert them into above-threshold and below-threshold groups using the mean, median, or a custom cut point.

3. Why are ties important?

Ties occur when a value equals the threshold. Different tie rules change category assignments, which can slightly change run counts, z scores, and p values.

4. What does a low number of runs mean?

Too few runs usually suggests clustering, persistence, or trend behavior. Similar categories stay together longer than randomness would predict.

5. What does a high number of runs mean?

Too many runs usually suggests rapid alternation between categories. The sequence switches more often than expected under random ordering.

6. Should I rely on the exact or normal p value?

For small samples, exact p values are preferable because they use the discrete distribution directly. Larger samples commonly rely on the normal approximation.

7. Does the test need equal group sizes?

No. The formulas already account for unequal counts in the two categories. Very unbalanced groups can still reduce sensitivity.

8. Can this prove a process is perfectly random?

No. It only checks one aspect of randomness: ordering. A sequence may pass the runs test yet still fail other randomness checks.