Binomial Sign Test Calculator

Example Data Table

This example gives 5 positive signs, 2 negative signs, and 1 tie. The sign test uses the 7 non-tied pairs.

Formula Used

The binomial sign test compares the number of positive paired differences with the number expected under the null hypothesis.

Step 1: Compute each paired difference as After - Before.

Step 2: Label each difference as positive, negative, or tie.

Step 3: Exclude ties and set n = positives + negatives.

Step 4: Let X = number of positive signs. Under the null hypothesis, X ~ Binomial(n, p0).

Binomial probability: P(X = k) = C(n,k) × p0^k × (1 - p0)^(n-k)

One-sided p-value: use the lower tail for “less” and the upper tail for “greater”.

Two-sided exact p-value: sum all binomial outcomes having probability less than or equal to the observed outcome probability.

Approximate z-score: z = (X - n×p0) / √(n×p0×(1-p0)), with optional continuity correction.

Confidence interval: the calculator reports a Wilson interval for the observed positive-sign probability.

How to Use This Calculator

1. Select Paired values when you have two matched measurements for each case.
2. Paste the first sample into the first box and the second sample into the second box.
3. Select Summary counts when you already know positive, negative, and tied counts.
4. Choose the alternative hypothesis: two-sided, greater, or less.
5. Keep p0 = 0.5 for the standard sign test unless your study requires a different null probability.
6. Set the significance level and confidence level.
7. Click Calculate to view the decision, p-values, interval, and graph.
8. Use the CSV and PDF buttons to save the output.

About This Binomial Sign Test Calculator

This calculator is built for quick nonparametric analysis of paired data. It works well when you only trust the direction of change and do not want to assume normality for differences. Instead of using the actual magnitudes, the sign test focuses on whether the second value is larger or smaller than the first.

That makes it useful for before-and-after studies, repeated measurements, classroom demonstrations, quality checks, and small experimental comparisons. If a pair increases, it contributes one positive sign. If it decreases, it contributes one negative sign. If the two values match exactly, the pair is treated as a tie and removed from the effective sample size.

The page supports two workflows. You can paste full paired values and let the tool classify every difference, or you can enter ready-made summary counts if the sign tally already exists. Both routes lead to the same exact binomial framework, which gives a clean p-value under the null positive probability.

Along with the exact result, the calculator also shows a normal approximation, an effect size, and a confidence interval for the observed positive-sign proportion. The bar chart helps you compare positive, negative, and tied outcomes at a glance. Export buttons let you save the results for reports, review notes, or class material.

FAQs

1. What does the binomial sign test examine?

It tests whether positive and negative paired differences occur with the probability expected under the null hypothesis, usually 0.5 for each direction.

2. When should I use this calculator?

Use it for matched pairs, before-and-after observations, or repeated measurements when you trust only the direction of change and prefer a nonparametric method.

3. Are ties included in the test?

No. Tied pairs do not support either direction, so the calculator removes them from the effective sample size before computing probabilities.

4. Does the sign test require normal data?

No. That is one of its strengths. It does not require normally distributed differences because it uses signs instead of magnitudes.

5. What does a two-sided alternative mean?

A two-sided test checks for any directional imbalance. It flags both unusually many positive signs and unusually many negative signs.

6. Why are exact and approximate p-values both shown?

The exact result is based on the binomial distribution. The approximate result is a quick normal-based check that becomes more reliable with larger samples.

7. Can I enter counts instead of raw paired values?

Yes. Switch to summary mode and provide positive, negative, and tied counts directly. This is useful when your tally is already prepared.

8. What does the confidence interval describe?

It estimates the plausible range for the observed probability of a positive sign among non-tied pairs, based on the sample data.