Z-Statistic Calculator: Analyze Means and Proportions with Confidence
Enter your sample data, pick α, and the calculator returns a z-statistic. When |z| > 1.96 you reject the null at the 5 % level because only 5 % of a standard normal distribution lies outside ±1.96 (NIST e-Handbook, 2012).
Results
How to use the tool
1. Pick your test type
Means z-test – compare an average. Example: x̄ = 130, μ = 125, σ = 12, n = 36.
Proportions z-test – compare a percentage. Example: p̂ = 0.58, p = 0.50, n = 180.
2. Choose the direction
Two-tailed – detects any difference.
Left-tailed – tests if the sample is smaller.
Right-tailed – tests if the sample is larger.
3. Set your significance level (α)
Common choices: 0.05, 0.01, 0.10. Enter any value between 0 and 1.
4. Enter the data
For means
Sample Mean (x̄)
Population Mean (μ)
Population Standard Deviation (σ)
Sample Size (n)
For proportions
Sample Proportion (p̂)
Population Proportion (p)
Sample Size (n)
5. Press “Calculate” and read the outputs
Z-Statistic
P-value
Critical value(s)
Decision – reject or fail to reject H₀.
Key formulas
Means test: $$z= rac{\bar{x}-\mu}{\sigma/\sqrt{n}}$$
The standard normal distribution has μ = 0 and σ = 1 (NIST e-Handbook, 2012).
A z-test for means is typically valid when n ≥ 30 or σ is known (OpenStax “Intro Stats”, 2022).
Common α levels: 0.05 in social sciences, 0.01 in genomics (ASA Statement on p-Values, 2016).
Proportion tests need np and n(1-p) ≥ 10 (UCLA Statistical Consulting, 2023).
FAQ
What is a z-statistic?
A z-statistic tells you how many standard deviations your sample result lies from the null-hypothesis value (NIST, 2012).
When should you use a one-sample z-test for means?
Use it when the population standard deviation is known and the sample is random and ≥ 30 observations (OpenStax, 2022).
How large must the sample be for a proportion z-test?
You need at least 10 expected successes and 10 failures so the normal approximation is reasonable (UCLA Statistical Consulting, 2023).
Why pick a one-tailed test?
Choose one-tailed when your research question predicts a specific direction; it gives more power in that direction (Gerard & Smith, Journal Stats Ed, 2021).
How do you interpret the p-value?
The p-value gives the probability of observing a statistic as extreme as yours if the null hypothesis is true (ASA, 2016).
What if σ is unknown?
Replace the z-test with a t-test that uses the sample standard deviation and t-distribution (NIST, 2012).
Does non-normal data matter?
For large samples (n ≥ 30), the Central Limit Theorem makes the z-test robust to non-normality (Rice, “Mathematical Statistics”, 2020).
Can I change α after seeing results?
No—α must be set before analysis to avoid inflating Type I error (ASA, 2016).
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