Calculator
Formula Used
One proportion estimate: p̂ = x / n
One proportion z score: z = (p̂ - p₀) / sqrt[p₀(1 - p₀) / n]
Two proportion estimates: p̂₁ = x₁ / n₁ and p̂₂ = x₂ / n₂
Pooled proportion: p̂ = (x₁ + x₂) / (n₁ + n₂)
Two proportion z score: z = [(p̂₁ - p̂₂) - D₀] / SE
Pooled SE: sqrt[p̂(1 - p̂)(1 / n₁ + 1 / n₂)]
Unpooled SE: sqrt[p̂₁(1 - p̂₁) / n₁ + p̂₂(1 - p̂₂) / n₂]
The p value depends on the selected alternative hypothesis.
How to Use This Calculator
- Select one proportion or two proportions.
- Enter successes and sample sizes.
- Enter the null proportion or null difference.
- Choose the alternative hypothesis.
- Set alpha and confidence level.
- Choose pooled error and correction options.
- Press Calculate to view the result.
- Download the result as CSV or PDF.
Example Data Table
| Scenario | Sample 1 Successes | Sample 1 Size | Sample 2 Successes | Sample 2 Size | Alternative | Alpha |
|---|---|---|---|---|---|---|
| One sample survey | 56 | 100 | N/A | N/A | Two sided | 0.05 |
| Two campaign comparison | 130 | 400 | 105 | 390 | Greater than | 0.05 |
| Defect rate check | 18 | 300 | 25 | 320 | Less than | 0.01 |
About the Z Test for Proportions
A z test for proportions checks whether an observed rate differs from a claimed rate. It can also compare two independent rates. The method is common in surveys, experiments, product testing, and quality checks. This calculator keeps the process structured. It separates raw counts, assumptions, test direction, and confidence settings.
When This Calculator Helps
Use it when outcomes are counted as success or failure. Examples include clicks versus no clicks, voters versus non voters, defects versus acceptable units, or passes versus failures. The sample should be random. Observations should be independent. Each sample should be large enough for the normal approximation to behave well.
Why the Options Matter
A one proportion test compares one sample rate with a null proportion. A two proportion test compares two sample rates. The pooled option is usually used for the hypothesis test because the null claim says both population rates are equal. The unpooled standard error is more useful for confidence intervals, where the two rates are estimated separately.
Understanding the Output
The z score measures how many standard errors separate the estimate from the null value. A large absolute z score gives stronger evidence against the null claim. The p value converts that distance into probability language. A small p value suggests the observed difference is unusual if the null claim is true.
Planning Better Tests
Good planning improves every conclusion. Choose the null proportion before collecting data. Pick the alternative direction from the research question. Avoid changing it after seeing results. Record sample size targets early. Bigger samples narrow intervals. Balanced samples often improve two group comparisons. Keep measurement rules consistent across groups.
Checking Assumptions
The normal approximation works best when expected successes and failures are not too small. Many classes use five or ten as a quick rule. Very small samples may need exact binomial methods instead.
Using Results Carefully
The calculator reports the sample proportions, difference, standard error, z score, p value, confidence interval, and a decision. Still, statistical output needs context. A significant result may be small in practical size. A non significant result may come from low sample size. Always review design, sampling quality, and real world impact before making careful decisions.
FAQs
What is a z test for proportions?
It is a hypothesis test for a population proportion. It compares an observed sample rate with a claimed rate, or compares two independent sample rates.
When should I use a one proportion test?
Use it when one sample gives success and failure counts. The test checks whether that sample proportion differs from a fixed null proportion.
When should I use a two proportion test?
Use it when two independent groups have success and failure counts. The test checks whether their population proportions differ.
What does the p value mean?
The p value shows how unusual the observed result is under the null hypothesis. Smaller values give stronger evidence against the null claim.
What is the pooled proportion?
The pooled proportion combines successes and sample sizes from both groups. It is often used when the null hypothesis says both proportions are equal.
What is continuity correction?
Continuity correction adjusts a discrete count problem before using a continuous normal curve. It can make the z test more conservative.
What sample size is needed?
The normal method works better when expected successes and failures are large enough. Many courses use at least five or ten in each category.
Can I export my results?
Yes. After calculation, use the CSV or PDF buttons. The exported file includes key statistics, the p value, and the decision.