Runs Test Calculator

The Runs Test (also called the Wald–Wolfowitz runs test) is a non-parametric statistical test used to determine whether a sequence of values is random. A 'run' is a consecutive series of identical values or values on the same side of the median. Too few or too many runs compared to what is expected by chance suggests the sequence is not random.

A run is an unbroken sequence of identical elements or elements sharing the same characteristic (e.g., all 1s or all 0s, or all values above the median). For example, in the sequence 1 1 0 0 1 0, there are 4 runs: [1 1], [0 0], [1], [0].

The null hypothesis (H₀) states that the sequence is random. The alternative hypothesis (H₁) is that the sequence is not random. If the absolute value of the Z-statistic exceeds the critical value at your chosen significance level, you reject H₀ and conclude the sequence is not random.

The Z-statistic measures how many standard deviations the observed number of runs is from the expected number. If |Z| > critical value (e.g., 1.96 at α = 0.05), the sequence is statistically non-random. A large positive Z suggests too many runs (alternating pattern); a large negative Z suggests too few runs (clustering pattern).

The most commonly used significance level is α = 0.05, which corresponds to a 95% confidence level and a critical Z-value of ±1.96. Use α = 0.01 (critical Z = ±2.576) for stricter tests, or α = 0.10 (critical Z = ±1.645) for a more lenient test. The choice depends on how much risk of a Type I error you are willing to accept.

Choose 'Binary' if your sequence already consists of only two values (e.g., 0s and 1s, H/T, success/failure). Choose 'Numeric (split at median)' if you have continuous or discrete numeric data — the calculator will automatically classify each value as above or below the median before performing the test.

The Runs Test assumes that the data can be classified into exactly two categories (dichotomous), observations are independent, and the sequence order matters. The normal approximation used for the Z-statistic is reliable when both n₁ and n₂ are greater than 10. For very small samples, consult exact runs test tables instead.

Yes. The Runs Test is commonly applied to stock price movements (up/down days) to test whether price changes are random, which is a component of the Efficient Market Hypothesis. It is also used in quality control, simulation validation, and any field where the randomness of a sequence is important.