# Z-Test Calculator

URL: https://speedycalculator.com/calculators/statistics/z-test/
Last updated: 2026-10-30

> The z-test calculator tests a mean against a hypothesised value when the population standard deviation is known, from summary statistics or raw data. It also handles a known z-score, two means with known σ and one proportion, giving z, the p-value, critical values and the decision.

## Inputs

- Test
- Sample mean (x̄)
- Sample data
- Hypothesised mean μ₀ (or difference Δ₀ for two means)
- Population standard deviation (σ)
- Sample size (n)
- z-score
- Mean of sample 1
- σ of population 1
- Size of sample 1
- Mean of sample 2
- σ of population 2
- Size of sample 2
- Number of successes (x)
- Hypothesised proportion p₀ (%)
- Alternative hypothesis H₁
- Significance level (α)

## Results

- p-value
- z-score
- Decision
- Critical z
- Rejection region
- Estimate (x̄, x̄₁ − x̄₂ or p̂)
- Standard error
- Confidence interval (two-sided, 1 − α)
- Left-tail area Φ(z)
- Right-tail area 1 − Φ(z)
- Standard normal distribution with rejection region

## Formula

```
One-sample mean: z = (x̄ − μ₀) ÷ (σ/√n) · Two means: z = (x̄₁ − x̄₂ − Δ₀) ÷ √(σ₁²/n₁ + σ₂²/n₂) · One proportion: z = (p̂ − p₀) ÷ √(p₀(1 − p₀)/n) · two-tailed p = 2(1 − Φ(|z|))
```

## Explanation

A z-test compares an estimate with a hypothesised value using the standard normal distribution. It applies when the standard deviation of the population is known – for example, from a long-run manufacturing process or a standardised test – or when the sample is large enough that the estimate of the standard deviation is very precise. The z-score says how many standard errors the sample mean lies from the claimed value.

Because the standard normal distribution does not depend on the sample size, z critical values are easy to remember: ±1.96 for a two-tailed test at α = 0.05, 1.645 for a one-tailed test, ±2.576 at α = 0.01. The p-value is the area of the normal curve beyond the observed z, in one tail or both.

The z-test assumes independent observations and either normal data or a sample large enough for the central limit theorem (often n ≥ 30). If σ is unknown and the sample is small, use a t-test instead – it accounts for the extra uncertainty from estimating σ. For proportions, the one-sample z-test is accurate when n·p₀ and n·(1 − p₀) are both at least about 10.

### How to use the Z-Test Calculator

- **Test**
- **Sample mean (x̄)**
- **Sample data**
- **Hypothesised mean μ₀ (or difference Δ₀ for two means)**
- **Population standard deviation (σ)**
- **Sample size (n)**
- **z-score**
- **Mean of sample 1**
- **σ of population 1**
- **Size of sample 1**
- **Mean of sample 2**
- **σ of population 2**
- **Size of sample 2**
- **Number of successes (x)**
- **Hypothesised proportion p₀**
- **Alternative hypothesis H₁**
- **Significance level (α)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose what you are testing and enter the sample mean, the hypothesised mean, the known σ and n (or the raw data, a z-score, two group summaries or a count of successes). The calculator computes the standard error and z, then reads the p-value and critical values from the standard normal distribution.

### Worked example

A battery maker states a mean life of 20 hours with a known σ of 2.5 hours. A sample of 50 batteries averages 19.3 hours. SE = 2.5 ÷ √50 = 0.3536, so z = (19.3 − 20) ÷ 0.3536 = −1.980.

The two-tailed p-value is 0.0477, just below 0.05, and z lies beyond the critical value −1.960, so we reject H₀ at α = 0.05 – but not at α = 0.01 (critical ±2.576). The 95% confidence interval for the mean life is 18.61 to 19.99 hours.

### z critical values

Confidence / αTwo-tailedOne-tailed90% / 0.10±1.6451.28295% / 0.05±1.9601.64598% / 0.02±2.3262.05499% / 0.01±2.5762.32699.9% / 0.001±3.2913.090

## Worked example

Inputs — Test: One-sample mean – summary data; Sample mean (x̄): 19.3; Hypothesised mean μ₀ (or difference Δ₀ for two means): 20; Population standard deviation (σ): 2.5; Sample size (n): 50; Alternative hypothesis H₁: Two-tailed (H₁: not equal); Significance level (α): 0.05.

Results — **p-value = 0.05**; z-score = -1.9799; Decision = Reject H₀ at α = 0.05 – p = 0.0477 ≤ 0.05 and z = −1.9799 lies in the rejection region.; Critical z = ±1.96; Rejection region = z ≤ −1.96 or z ≥ 1.96; Estimate (x̄, x̄₁ − x̄₂ or p̂) = 19.3.

## Key terms

- **z-score**: The number of standard errors between the estimate and the hypothesised value.
- **Standard normal distribution**: The normal distribution with mean 0 and standard deviation 1, written N(0, 1).
- **Known σ**: A population standard deviation taken from long experience or a specification, not estimated from the sample.
- **Central limit theorem**: The result that sample means are approximately normal for large samples, whatever the shape of the data.

## FAQ

**When should I use a z-test instead of a t-test?**
When the population standard deviation is known, or for proportions. If you estimate σ from a small sample, use the t-test.

**What is the z critical value for α = 0.05?**
±1.960 for a two-tailed test, 1.645 for a right-tailed test and −1.645 for a left-tailed test.

**What is the p-value for z = −2?**
0.0228 for a left-tailed test and 0.0455 for a two-tailed test.

**Can I use a z-test with a sample of 15?**
Only if σ is known and the data are roughly normal. Otherwise the t-test is more honest about the uncertainty.

**What does a negative z mean?**
The sample estimate lies below the hypothesised value. In a two-tailed test only the size of z matters.

**How is the two-sample z-test different?**
It compares two means using both known population standard deviations: SE = √(σ₁²/n₁ + σ₂²/n₂).

**Is a z-test the same as a z-score?**
A z-score standardises one value; a z-test standardises a sample statistic (such as a mean) using its standard error and turns it into a p-value.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)