Runs Test Calculator

StatisticsLast updated: August 22, 2026

The Wald-Wolfowitz runs test asks whether the ORDER of the observations in a sequence is random. The values are coded in binary form as "above" or "below" a cut point (the median by default); each block of consecutive identical signs is called a run, and the number of runs R is compared with the value expected under randomness.

What sets it apart from other tests is that it examines the pattern rather than the central tendency: the same set of observations gives a different result when supplied in a different order. FEWER runs than expected mean clustering or a trend (autocorrelation); MORE runs mean excessive alternation (oscillation) — both are departures from randomness. This calculator reports R, n1, n2, the expected R, the standard deviation, z and p with and without the continuity correction, a table coding the sequence as above/below, and a set of run charts.

Enter the values in the order they were observed (time, production order, row order). The order must be meaningful; sorted data carries no information for this test.
Each observation is coded according to whether it lies above or below this value. Observations exactly equal to it are left out of the analysis.
When n1 or n2 is below 20, the continuity-corrected version gives a more accurate result.
Sets the false-positive risk. In the social sciences 5% is the usual choice.

Wald-Wolfowitz Runs Test Formulas

Coding:            xᵢ > cut → "A" ,  xᵢ < cut → "B" ,  xᵢ = cut → dropped
n₁, n₂:            counts of observations coded above and below
Number of runs:    R = the number of consecutive same-sign blocks
Expected value:    E(R) = 2·n₁·n₂ / (n₁ + n₂) + 1
Variance:          σ²_R = 2n₁n₂(2n₁n₂ − n₁ − n₂) / [ (n₁+n₂)² (n₁+n₂−1) ]
z statistic:       z = (R − E(R)) / σ_R
Continuity corr.:  z_c = (|R − E(R)| − 0.5) / σ_R
Acceptance range:  E(R) ± z_{α/2} · σ_R

The null hypothesis is that the order of the observations is random, and the test is two-tailed: both too few and too many runs lead to rejecting H₀. The normal approximation is regarded as reliable when n₁ ≥ 10 and n₂ ≥ 10; for smaller counts, use the continuity-corrected version or an exact runs distribution table.

How to Calculate

  1. Enter the values in the order they were observed: the time, production number, or row order must be preserved — sorted data carries no information for this test.
  2. Choose the cut-point criterion: the median is the most common choice, and a manual threshold of 0 is the natural one for regression residuals.
  3. Decide which p-value to report; if n1 or n2 is below 20, prefer the continuity-corrected version.
  4. Compare the number of runs R with the expected E(R) and look at the acceptance interval: if R falls outside it, randomness is rejected.
  5. Interpret the direction: too few runs mean clustering and trend, too many mean excessive alternation.
  6. Study the long blocks in the sign table and the sequence chart to see which part of the sequence the pattern comes from.

Worked Examples

A shift in quality control measurements

The median is 12.50; 10 observations lie above it and 10 below. Only 5 runs are counted while the expected value is 11.00: with z = -2.757 and p = 0.006 the sequence is not random. The process ran high in the first half and low in the second, so the measurements are clustered; this is a typical shift pattern.

Number of runs (R): 5 · Expected number of runs E(R): 11.000 · Standard deviation of the number of runs: 2.1764

An excessively alternating sequence — too many runs

The values switch regularly between low and high: 14 runs are counted in 14 observations, whereas the expected value is 8.00. With a continuity-corrected z = 3.060 and p = 0.002, randomness is rejected. This pattern usually points to an alternating measurement error or to negative autocorrelation.

Number of runs (R): 14 · Expected number of runs E(R): 8.000 · Standard deviation of the number of runs: 1.7974

A measurement sequence that looks random

The median is taken as 22.50; 10 observations lie above that value and 10 below. 12 runs are observed while the expected value is 11.00, and σ_R = 2.1764; z = 0.459 gives p = 0.646, so nothing contradicts randomness. The observed R stays inside the 95% acceptance interval (6.73 – 15.27).

Number of runs (R): 12 · Expected number of runs E(R): 11.000 · Standard deviation of the number of runs: 2.1764

Frequently Asked Questions

What exactly does the runs test examine?
It examines the ORDER of the observations, not their magnitude: the null hypothesis is that the sequence is in random order. Supplying the same numbers in a different order changes the test statistic completely. Sorting the data from largest to smallest therefore makes the test meaningless; the order of observation must be preserved.
What is the difference between too few and too many runs?
Fewer runs than expected show that similar values are clustered together: the sequence contains a trend, a shift, or positive autocorrelation. More runs than expected show that the values keep switching above-below-above, that is, excessive oscillation or negative autocorrelation. Because the test is two-tailed, both extremes lead to rejecting randomness.
What should I use as the cut point?
The median is the most common and the safest choice; it makes the two groups roughly equal in size and maximizes the power of the test. When you are testing the randomness of regression residuals, the natural threshold is zero. The mean gives a result close to the median for symmetric distributions, but it unbalances the groups in skewed data.
What happens to observations equal to the cut point?
They are left out of the analysis, and the calculator reports how many were dropped. With an odd number of observations and the median selected, the median value itself is dropped; if many values repeat, the effective sample can shrink considerably. In such cases, try the mean or a manually chosen threshold.
When is this test used?
Its main uses are checking the randomness of measurements in statistical process control, looking for patterns in regression residuals, testing independence in time series, and testing random number generators. The Durbin-Watson test complements it for regression residuals, as does the autocorrelation function for time series.
How do I get the same result in SPSS and R?
In SPSS, choose Analyze → Nonparametric Tests → One Sample → Runs; the cut point can be set to the median, the mean, the mode, or a custom value, and the asymptotic p is reported. In R, runs.test(x) from the randtests package or runs.test(factor(x > median(x))) from tseries produces the same z and p. Excel has no built-in function; the signs and block counting must be set up by hand.
Does a non-significant result prove the sequence is random?
No. The runs test is sensitive only to the number of runs, so it may miss regular patterns that leave that number unchanged — cycles repeating on a fixed period, for instance. A non-significant result means "no departure from randomness was found by this criterion". A thorough check needs the autocorrelation function and graphical inspection as well.