Sign Test
Test a median or paired differences using only the direction of each difference. The sign test is the simplest non-parametric test, making almost no distributional assumptions, ideal when data is ordinal or badly non-normal.
Run Sign Test →What is the Sign Test?
The sign test is a simple non-parametric test that uses only the sign, positive or negative, of differences, ignoring their magnitude. It can test whether a population median equals a hypothesized value, or whether paired observations tend to increase or decrease.
For paired data, it counts how many differences are positive and how many are negative, discards any that are zero, and tests whether positives and negatives are equally likely under the binomial distribution. Because it relies only on direction, it makes almost no assumptions about the shape of the distribution.
This minimal assumption is the sign test's main advantage and its main limitation. It is extremely robust and works even for ordinal data where only order is meaningful, but by discarding the magnitude of differences it has less statistical power than tests that use that information, such as the Wilcoxon signed-rank test.
In plain terms: The sign test only asks: did it go up or down? It counts pluses and minuses and checks whether they're roughly balanced. Because it ignores how much things changed, it's very robust but not very sensitive, so if your data allows it, the Wilcoxon test usually does better.
Key Points
Direction Only
Uses only the sign of each difference, not its size. Extremely robust and assumption-light.
Binomial Basis
Under the null, positive and negative signs are equally likely, so the count follows a binomial distribution.
Low Power Trade-off
Ignoring magnitude costs power. The Wilcoxon signed-rank test is stronger when magnitudes are meaningful.
Key Formulas
Interpreting the Result
The null hypothesis is that positive and negative differences are equally likely (the median difference is zero). A small p-value means one direction dominates more than chance allows, indicating a real tendency to increase or decrease.
The sign test's robustness makes it a safe fallback for ordinal data or heavily skewed differences. But if the magnitudes of the differences are meaningful and roughly symmetric, prefer the Wilcoxon signed-rank test, which uses that extra information for greater power.
Assumptions & Validation
Paired or Single Sample
Data are paired differences or a single sample tested against a median.
If violated: Use independent-group tests for unpaired data.
Independent Observations
Pairs are independent of one another.
If violated: Address dependence between pairs.
At Least Ordinal
Differences can be classified as positive or negative.
If violated: The test needs a meaningful direction for each difference.
⚠️ Check assumptions first
The sign test's great virtue, using only direction, is also its weakness: by discarding the size of each difference it throws away information and has lower power than alternatives. Use it when magnitudes are unavailable or unreliable, as with ordinal data, or when the differences are so skewed that even the Wilcoxon test's symmetry assumption fails. Otherwise the Wilcoxon signed-rank test is usually the better paired choice.
When NOT to Use Sign Test
Meaningful Magnitudes
When difference sizes are meaningful and roughly symmetric, the Wilcoxon signed-rank test has more power.
Independent Groups
For two independent groups, use the Mann-Whitney U test.
Normal Differences
When differences are approximately normal, the paired t-test is most powerful.
Industry Applications
Ordinal Ratings
Test whether paired ordinal ratings tend to improve or worsen after a change.
Robust Median Test
Test a median against a target when the distribution is badly non-normal.
Quick Directional Check
Assess whether a change consistently moves values in one direction.
Preference Comparisons
Test whether one option is preferred over another across paired judgments.
Frequently Asked Questions
What is the sign test used for?
The sign test is a simple non-parametric test that examines whether paired differences tend to be positive or negative, or whether a population median equals a hypothesized value. It uses only the direction of each difference, not its magnitude, which makes it applicable even to ordinal data and highly robust to unusual distributions. It is often used as a quick, assumption-light check of a directional tendency.
How does the sign test work?
For paired data, it counts how many differences are positive and how many are negative, discarding any that are exactly zero. Under the null hypothesis, positive and negative differences are equally likely, so the count of positives follows a binomial distribution with probability one-half. The test then computes the probability of observing a count as extreme as the one in your data.
What is the difference between the sign test and the Wilcoxon signed-rank test?
Both are non-parametric tests for paired data, but the sign test uses only the direction of differences, while the Wilcoxon signed-rank test uses both the direction and the relative magnitude by ranking the absolute differences. Because it uses more information, the Wilcoxon test is generally more powerful, but it assumes the differences are symmetric. The sign test makes fewer assumptions and works for purely ordinal data.
When should I prefer the sign test?
Prefer the sign test when the magnitudes of differences are unavailable or unreliable, as with ordinal ratings, or when the differences are so skewed that the symmetry assumption of the Wilcoxon signed-rank test does not hold. Its minimal assumptions make it a safe fallback, at the cost of lower power than tests that use the size of the differences.
Why are zero differences discarded?
A difference of exactly zero provides no directional information, since it is neither positive nor negative, so the standard sign test removes these pairs and reduces the effective sample size accordingly. This keeps the binomial model valid for the remaining pairs. When many differences are zero, the reduced sample size lowers the test's power, which is a consideration in its interpretation.
Does the sign test have low power?
Relative to tests that use the magnitude of differences, yes. By reducing each difference to just a sign, the sign test discards information, so it requires a larger effect or sample to reach significance than the Wilcoxon signed-rank test or the paired t-test would. This trade-off buys robustness and minimal assumptions, which is worthwhile when those stronger tests' assumptions cannot be met.
Test Direction With Minimal Assumptions
Run the sign test on paired differences or a median. Free during Beta.
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