Updated Hypothesis Testing Tool

Z-Test Calculator

Run one-sample and two-sample z-tests for both means and proportions using summary statistics. Enter sample sizes, means, standard deviations or success counts to compute the z-statistic, p-value, effect size and a clear decision for your chosen significance level.

One-Sample Mean Z-Test Two-Sample Mean Z-Test One-Proportion Z-Test Two-Proportion Z-Test

Test Means & Proportions with Z-Statistics

This Z-Test Calculator provides four classical large-sample z-tests in one place. You can test a sample mean against a hypothesized population mean, compare two independent means, test a single population proportion, or compare two proportions. All inputs are based on summary statistics for fast manual entry from reports or homework problems.

p-values are computed from the standard normal distribution. For very small samples, a t-test may be more appropriate than a z-test.

Use the one-sample mean z-test when you compare a sample mean to a hypothesized population mean μ₀ and you can treat the population standard deviation σ as known or well-estimated.

Use the two-sample mean z-test to compare means from two independent groups when the population standard deviations σ₁ and σ₂ can be treated as known or well-estimated from large prior samples.

Use the one-proportion z-test when you test whether the proportion of “successes” in a sample differs from a hypothesized population proportion p₀.

Use the two-proportion z-test to compare proportions from two independent samples, such as conversion rates in an A/B test. The classical pooled test assumes H₀: p₁ = p₂.

Z-Test Calculator – Complete Guide to Mean and Proportion Z-Tests

The Z-Test Calculator on MyTimeCalculator brings together the four most common large-sample z-tests used in statistics, analytics and research. You can test a sample mean against a hypothesized mean, compare two independent means, test a population proportion, or compare two proportions from separate samples. The interface is fully based on summary statistics, which makes it convenient when raw data are not available.

Z-tests rely on the standard normal distribution and are typically appropriate when sample sizes are reasonably large or when population standard deviations are known from previous studies. The calculator returns z-statistics, p-values, effect sizes and a simple decision phrase for your chosen significance level α.

1. Z-Tests for Means

When you work with quantitative measurements such as heights, test scores or processing times, you often want to test claims about population means. The calculator includes:

  • One-sample mean z-test: tests whether the mean of a single sample differs from a hypothesized population mean μ₀ when the population standard deviation σ is treated as known or well-estimated.
  • Two-sample mean z-test: compares the means of two independent groups (for example, treatment vs. control) when σ₁ and σ₂ are treated as known or estimated from large historical data sets.

One-sample mean z-test formula

Let \(\bar{x}\) be the sample mean, σ the population standard deviation, n the sample size and μ₀ the hypothesized mean under H₀:

\( z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \)

Two-sample mean z-test formula

Let \(\bar{x}_1, \bar{x}_2\) be the sample means, σ₁ and σ₂ the population standard deviations, and n₁, n₂ the sample sizes. With a null difference Δ₀ = μ₁ − μ₂ under H₀:

z = \(\dfrac{\bar{x}_1 − \bar{x}_2 − \Delta_0}{\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}}\)

2. Z-Tests for Proportions

Many practical problems involve proportions or probabilities: pass rates, click-through rates, success percentages and more. The calculator implements:

  • One-proportion z-test: tests whether the proportion of successes in a sample differs from a hypothesized population proportion p₀.
  • Two-proportion z-test: compares proportions from two independent samples, such as conversion rates in an A/B test or pass rates in two groups.

One-proportion z-test formula

If x is the number of successes in a sample of size n, the sample proportion is \(\hat{p} = x/n\). With hypothesized proportion p₀ under H₀:

\( z = \frac{\hat{p} - p_0}{\sqrt{p_0 (1 - p_0) / n}} \)

Two-proportion z-test formula (pooled)

For two samples with x₁ successes out of n₁ and x₂ successes out of n₂, the pooled proportion under H₀: p₁ = p₂ is:

\( \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \)

The test statistic for H₀: p₁ = p₂ is:

z = \(\dfrac{\hat{p}_1 − \hat{p}_2}{\sqrt{\hat{p} (1 − \hat{p}) (1/n_1 + 1/n_2)}}\),   where \(\hat{p}_1 = x_1/n_1\) and \(\hat{p}_2 = x_2/n_2\).

3. Effect Sizes: Cohen’s d and Cohen’s h

Statistical significance alone does not tell you how large or important an effect is. This is why the calculator reports effect sizes:

  • Cohen’s d (means): a standardized mean difference, roughly equal to the difference between means divided by a standard deviation. For the one-sample mean test, d is (x̄ − μ₀)/σ; for the two-sample mean test, it uses the difference of sample means divided by a pooled standard deviation.
  • Cohen’s h (proportions): a standardized difference between two proportions defined by h = 2 arcsin(√p₁) − 2 arcsin(√p₂). For a one-proportion test, p₁ is the sample proportion and p₂ the hypothesized proportion p₀.

As a rough guide, |d| or |h| around 0.2 is often called “small”, 0.5 “medium” and 0.8 or higher “large”, but context and domain standards are always crucial when interpreting effect sizes.

4. How to Use the Z-Test Calculator

  1. Choose the test type: use the tabs to pick a one-sample mean, two-sample mean, one-proportion or two-proportion z-test.
  2. Set α and tail type: choose your significance level α (for example 0.05) and whether you want a two-tailed test (≠) or a directional one-tailed test (< or >).
  3. Enter summary statistics: for mean tests, enter sample sizes, means and population standard deviations. For proportion tests, enter success counts and sample sizes; the calculator computes the proportions for you.
  4. Specify the null hypothesis value: for one-sample mean tests, enter μ₀; for two-sample mean tests, enter the null difference Δ₀ (often 0); for one-proportion tests, enter p₀.
  5. Run the z-test: click the appropriate “Compute Z-Test” button. The results section displays the z-statistic, p-value, effect size and a simple hypothesis decision.
  6. Interpret the results: if the p-value is smaller than α, the result is considered statistically significant at that level. Effect sizes help you judge whether the magnitude of the difference is practically important.

5. When to Prefer a t-Test Instead of a Z-Test

In many real-world applications, the population standard deviation is not truly known. When σ is unknown and the sample size is not extremely large, a t-test is more appropriate than a z-test because it accounts for the extra uncertainty by using the t-distribution.

As a rough rule:

  • Use z-tests when sample sizes are large or when population standard deviations can be treated as known from previous, reliable studies.
  • Use t-tests when σ is unknown and sample sizes are modest, which is common in experiments and homework problems with raw data.

MyTimeCalculator provides both t-Test and Z-Test tools so you can choose the method that best matches your assumptions and data structure.

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Z-Test Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about when to use z-tests, how to enter summary statistics and how to read the z-statistics, p-values and effect sizes from this calculator.

Z-tests are typically used when sample sizes are large and/or when the population standard deviation is treated as known from prior information. t-tests are preferred when σ is unknown and must be estimated from the sample, especially when sample sizes are small or moderate. For proportions, large-sample z-approximations are standard when n is big enough that both np and n(1 − p) are reasonably large.

Many real-world examples and textbook exercises present only sample sizes, means and standard deviations, not the full raw data. This calculator is optimized for those situations. You simply type the summary values from your table or report, and the calculator computes the z-statistic, p-value and effect size without needing the original data set.

A two-tailed z-test checks for any difference from the null value (for example H₁: μ ≠ μ₀ or p ≠ p₀) and splits the rejection region into both tails of the normal distribution. A one-tailed z-test checks for a difference in only one direction (H₁: μ > μ₀, μ < μ₀, p > p₀, p < p₀), placing the entire rejection region in one tail. The calculator lets you choose the tail type and automatically adjusts the p-value and decision sentences accordingly.

p-values are computed from the standard normal distribution, which is the exact reference distribution for z-statistics under the usual large-sample assumptions. In practice, the accuracy depends on how well your situation matches those assumptions, such as sample size, independence of observations and approximate normality for mean tests or large effective counts for proportion tests. For very small samples, a t-distribution or exact test may be more appropriate.

Yes. Enter the summary statistics from the problem (for example n, mean and σ, or success counts and sample sizes) and compare the z-statistics and p-values to the ones you computed by hand. This is a fast way to verify multiple-choice answers or detect algebra mistakes. For written work, make sure you still show the formulas and steps, because the calculator focuses on clean numerical output rather than step-by-step derivations.

Cohen’s d and Cohen’s h are standardized effect sizes that focus on the magnitude of the difference rather than just statistical significance. Cohen’s d is used for mean differences and Cohen’s h for proportions. Small p-values can occur even for tiny effects in very large samples; effect sizes help you judge whether the observed difference is practically meaningful in your context, not just statistically detectable.

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