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The 2 sample z test (also written two-sample z-test or two sample z test) checks whether the means of two independent populations differ, using a standard normal test statistic. It applies when both population standard deviations are known, which is rare in practice; when they are not, the z-test vs t-test comparison explains the usual switch to a t-test. This page gives the formula, a worked example with every step of the arithmetic, one-tailed and two-tailed p-values, a confidence interval, an APA report and working code.
What is a 2 sample z test? (definition)
A two-sample z-test for means compares the average of one group with the average of another. The null hypothesis is that the two population means are equal (H0: μ1 − μ2 = 0). The test statistic is the observed difference between sample means divided by its standard error, and it follows a standard normal distribution when the assumptions below hold. A large absolute z means the observed difference would be unlikely if the population means were really equal. For a test of two proportions rather than means, see the two-proportion z-test.
Two-sample z-test formula and symbols
With known population standard deviations, the statistic is:
z = ((x̄1 − x̄2) − (μ1 − μ2)) / √(σ1²/n1 + σ2²/n2)
Under H0, μ1 − μ2 = 0, so the numerator reduces to x̄1 − x̄2.
- x̄1, x̄2: the two sample means.
- μ1 − μ2: the hypothesised difference between population means (0 under the usual null).
- σ1, σ2: the known population standard deviations (not sample standard deviations).
- n1, n2: the two sample sizes.
- √(σ1²/n1 + σ2²/n2): the standard error of the difference between the means.
Worked example of a two sample z test (right-tailed and two-tailed)
Take two independent groups. Group 1: x̄1 = 15, σ1 = 2.3, n1 = 35. Group 2: x̄2 = 13.3, σ2 = 2.7, n2 = 48. The question is whether the Group 1 mean is greater than the Group 2 mean, at α = 0.05. This setup is adapted from the diet example in Berberyan, Nguyen and Swan (see Sources).
- Hypotheses. H0: μ1 − μ2 = 0. H1: μ1 − μ2 > 0 (right-tailed).
- Difference of means. 15 − 13.3 = 1.70.
- Variance terms. σ1²/n1 = 5.29/35 = 0.15114. σ2²/n2 = 7.29/48 = 0.15188.
- Standard error. √(0.15114 + 0.15188) = √0.30302 = 0.5505.
- Test statistic. z = 1.70 / 0.5505 = 3.09.
- Critical value. For a right-tailed test at α = 0.05, the critical z is 1.645 (see z-alpha). Since 3.09 > 1.645, reject H0.
- p-values. One-tailed p = P(Z ≥ 3.09) = 0.0010. Two-tailed p = 2 × 0.0010 = 0.0020.
- 95% confidence interval for μ1 − μ2. 1.70 ± 1.96 × 0.5505 = 1.70 ± 1.08, or 0.62 to 2.78. The interval excludes 0, which agrees with the rejection.
Interpreting the result
| Result | Reading |
|---|---|
| |z| larger than the critical value (1.645 one-tailed, 1.96 two-tailed, at α = 0.05) | Reject H0: the data are inconsistent with equal population means at that α. |
| |z| smaller than the critical value | Fail to reject H0. This is not evidence that the means are equal. |
| p ≤ α | Same decision as the critical-value method. |
| Confidence interval excludes 0 | Consistent with rejecting H0 in a two-tailed test at the matching confidence level. |
The one-tailed p-value is exactly half the two-tailed p-value for the same z. Choose the tail before looking at the data; see two-tailed hypothesis testing. To convert a z-score to a p-value by hand, see how to find a p-value from a z-score.
Assumptions and common mistakes
Assumptions
- Both samples are random samples.
- The two samples are independent of each other, drawn from independent populations.
- The population standard deviations are known.
- Both sample sizes are at least 30, or both populations are normally distributed.
Two sample z test vs two-sample t-test
The most common mistake is using a z-test with sample standard deviations plugged in as though they were known population values. If σ is unknown, use a two-sample t-test (see t-test). One source notes that a good estimate s from a large sample (n above 30) can stand in for σ, but a known σ is the textbook condition.
Other mistakes
- Paired data. If the two measurements come from the same units (before and after), the samples are not independent; test the mean of the differences instead.
- Choosing the tail after seeing the data. Decide one-tailed or two-tailed in advance.
- Treating a non-significant result as proof of no difference.
How to report a two-sample z-test (APA example)
For the example above, a two-tailed write-up in the style of an APA results sentence reads:
Group 1 (M = 15.00, n = 35) scored higher than Group 2 (M = 13.30, n = 48), z = 3.09, p = .002 (two-tailed), 95% CI of the difference [0.62, 2.78]. The population standard deviations were treated as known (σ1 = 2.3, σ2 = 2.7).
How to calculate a two sample z test in Excel, R or Python
Excel
The built-in Z.TEST function is a one-sample test, so compute the statistic directly. With the means in B1 and B2, the known standard deviations in C1 and C2 and the sample sizes in D1 and D2:
z: =(B1-B2)/SQRT(C1^2/D1+C2^2/D2)
right-tailed: =1-NORM.S.DIST(z_cell,TRUE)
two-tailed: =2*(1-NORM.S.DIST(ABS(z_cell),TRUE))R
Base R has no z.test, so compute it directly with pnorm:
se <- sqrt(2.3^2/35 + 2.7^2/48)
z <- (15 - 13.3) / se
pnorm(z, lower.tail = FALSE) # right-tailed
2 * pnorm(abs(z), lower.tail = FALSE) # two-tailedIf you have raw data, the DescTools package provides ZTest(x, y, alternative = "two.sided", mu = 0, sd_pop, conf.level = 0.95); the sd_pop argument (the known population standard deviation) has no default and must be supplied.
Python
from math import sqrt
from scipy.stats import norm
x1, s1, n1 = 15.0, 2.3, 35
x2, s2, n2 = 13.3, 2.7, 48
se = sqrt(s1**2 / n1 + s2**2 / n2)
z = (x1 - x2) / se
print(round(z, 2)) # 3.09
print(round(norm.sf(z), 4)) # 0.001 right-tailed
print(round(2 * norm.sf(abs(z)), 4)) # 0.002 two-tailedThe ztest function in statsmodels (statsmodels.stats.weightstats.ztest) estimates the variances from the samples rather than taking known σ values, so it does not reproduce the known-sigma test above.
2 sample z test FAQ
When should I use a two-sample z-test instead of a t-test?
Use the z-test when both population standard deviations are known and the samples are independent, with both n at least 30 or both populations normal. When the standard deviations must be estimated from the samples, use a t-test.
What is the null hypothesis of a two sample z test?
That the difference between the two population means equals a stated value, almost always 0 (H0: μ1 − μ2 = 0).
What is the difference between a two-sample z-test and a two-proportion z-test?
The two-sample z-test compares means of a numerical outcome. The two-proportion z-test compares the proportions of a yes/no outcome.
Can I use a two sample z test for paired data?
No. Paired data are not independent samples; test the mean of the paired differences instead.
Sources
- Berberyan, Nguyen and Swan. Statistics C1000: Introduction to Statistics, Citrus College, section 9.1, “z-Test for the Difference Between Two Means” (LibreTexts, CC BY 4.0). stats.libretexts.org. Checked 2026-10-29.
- DescTools,
ZTestreference documentation. rdrr.io. Checked 2026-10-29. - statsmodels,
statsmodels.stats.weightstats.ztest. statsmodels.org. Checked 2026-10-29. - StatsRef, “Test for a mean when the standard deviation is known”. statsref.com. Checked 2026-10-29.
