One-Sample Z-Test: When to Use It and How

What Is a One-Sample Z-Test?

Three conditions identify this test. You have one sample, you are comparing its mean to a fixed reference value, and you know the population standard deviation from historical data, a regulatory standard, or a prior study. When σ is unknown, the correct test is the one-sample t-test, which substitutes the sample standard deviation s and uses the t-distribution.

The test was formalized within the frequentist framework developed by Ronald Fisher, Karl Pearson, and Jerzy Neyman during the early 20th century. The standard normal distribution it relies on is central to normal distribution theory and the Central Limit Theorem.

When to Use a One-Sample Z-Test

Before running any test, you need to confirm the test fits your research question. The one-sample z-test is appropriate when all four of the following are true.

Decision Tree: Which Test Do You Need?

This decision tree walks through the most common scenarios. Start at the top and follow the path that matches your situation.

Four Assumptions of the One-Sample Z-Test

Every statistical test carries assumptions. Violating them does not automatically invalidate a result, but it does mean your p-values may be inaccurate. Check all four before proceeding.

The One-Sample Z-Test Formula

The formula measures how many standard errors the sample mean x̄ sits away from the null hypothesis value μ₀. The denominator σ/√n is the standard error of the mean — how much x̄ would vary across repeated samples of size n drawn from the population.

Null and Alternative Hypotheses

The null hypothesis always takes the equality form. The alternative hypothesis shape determines whether the test is one-tailed or two-tailed, which directly affects how the p-value is calculated. Choose the direction of H₁ based on your research question — before you look at the data.

Critical Values at Common Significance Levels

How to Perform a One-Sample Z-Test: 5 Steps

One-Sample Z-Test Calculator

Enter your values below. The calculator computes the z statistic, standard error, two-tailed or one-tailed p-value, and renders a reject/fail-to-reject decision. For the one-sample t-test version (unknown σ), use the t-test calculator.

Five Fully Solved One-Sample Z-Test Examples

Each example follows the SCORE framework and shows the complete arithmetic. These cover five different real-world contexts where the one-sample z-test appears most often.

Example 1 — Manufacturing Quality Control

Example 2 — Public Health Research

Example 3 — Education Assessment (One-Tailed)

Example 4 — Customer Satisfaction Survey

Example 5 — Environmental Science

Running the Test in R, Python, and Excel

Manual calculation is essential for understanding the logic. In practice, most analysts run z-tests in statistical software. The code below uses the data from Example 1 (pharmaceutical tablets: x̄ = 503.5, μ₀ = 500, σ = 8, n = 36).

One-Sample Z-Test in R

Base R does not have a built-in one-sample z-test function (it assumes σ is unknown for the t-test). The BSDA package provides z.test(), or you can compute it directly.

# Method 1: Manual calculation in R
x_bar <- 503.5      # sample mean
mu_0  <- 500        # null hypothesis mean
sigma <- 8          # known population SD
n     <- 36         # sample size

SE  <- sigma / sqrt(n)
z   <- (x_bar - mu_0) / SE
# Two-tailed p-value
p   <- 2 * pnorm(-abs(z))

cat("SE  =", round(SE, 4), "\n")
cat("z   =", round(z, 4), "\n")
cat("p   =", round(p, 4), "\n")

# Method 2: Using BSDA package
# install.packages("BSDA")
library(BSDA)
x <- c(503.5)  # or your full data vector
z.test(x, mu = 500, sigma.x = 8, alternative = "two.sided")

# Confidence interval (95%)
ci_lower <- x_bar - 1.96 * SE
ci_upper <- x_bar + 1.96 * SE
cat("95% CI: [", round(ci_lower, 3), ",", round(ci_upper, 3), "]")

One-Sample Z-Test in Python (SciPy)

The statsmodels library provides ztest(). Alternatively, you can compute the z statistic using SciPy's normal distribution.

import numpy as np
from scipy import stats
from statsmodels.stats.weightstats import ztest

# Sample data (or use summary statistics directly)
x_bar = 503.5
mu_0  = 500
sigma = 8
n     = 36

# Manual calculation
SE = sigma / np.sqrt(n)
z  = (x_bar - mu_0) / SE
p  = 2 * (1 - stats.norm.cdf(abs(z)))

print(f"SE  = {SE:.4f}")
print(f"z   = {z:.4f}")
print(f"p   = {p:.4f}")

# 95% confidence interval
ci_lower = x_bar - stats.norm.ppf(0.975) * SE
ci_upper = x_bar + stats.norm.ppf(0.975) * SE
print(f"95% CI: [{ci_lower:.3f}, {ci_upper:.3f}]")

# Effect size (Cohen's d)
cohens_d = abs(x_bar - mu_0) / sigma
print(f"Cohen's d = {cohens_d:.4f}")

One-Sample Z-Test in Excel

Excel includes NORM.S.DIST() and Z.TEST() functions for this test.

=== Cell Setup ===
A1: Sample Mean (x̄)     B1: 503.5
A2: Null Mean (μ₀)      B2: 500
A3: Population SD (σ)   B3: 8
A4: Sample Size (n)     B4: 36
A5: Alpha               B5: 0.05

=== Formulas ===
# Standard Error
A7: SE    B7: =B3/SQRT(B4)

# Z Statistic
A8: z     B8: =(B1-B2)/B7

# Two-tailed p-value
A9: p-value (two-tailed)   B9: =2*(1-NORM.S.DIST(ABS(B8),TRUE))

# One-tailed p-value (right)
A10: p-value (right-tail)   B10: =1-NORM.S.DIST(B8,TRUE)

# Critical value at α = 0.05
A11: Critical z (two-tail)  B11: =NORM.S.INV(1-B5/2)

# Z.TEST function (returns one-tailed p for right-tail)
# Paste your data in column D first
A12: Z.TEST result   B12: =Z.TEST(D2:D37, B2, B3)
# Multiply by 2 for two-tailed

One-Sample Z-Test vs One-Sample T-Test

The z-test and t-test address the same research question but under different conditions. The t-test is far more common in practice because the population σ is almost never truly known.

How to Interpret the p-Value and Decision

The p-value has a single precise definition: it is the probability of observing a test statistic as extreme as yours (or more extreme), given that the null hypothesis is true. It does not measure the probability that H₀ is true, the probability that the result occurred by chance, or the size of the effect.

Confidence Interval from a One-Sample Z-Test

Every z-test has an equivalent confidence interval. The two-tailed 95% CI for the population mean is:

A confidence interval and a hypothesis test give consistent conclusions: if μ₀ falls outside the 95% CI, the two-tailed test at α = 0.05 will reject H₀. For Example 1 (pharmaceutical tablets): CI = 503.5 ± 1.96 × 1.333 = [500.89, 506.11]. Since μ₀ = 500 falls outside this interval, rejection of H₀ is confirmed.

Effect Size: Cohen's d for the One-Sample Z-Test

Statistical significance depends on sample size — with a large enough n, even a trivially small difference becomes significant. Effect size measures the practical magnitude of the difference, independent of sample size. For the one-sample z-test, the standard effect size is Cohen's d.

For Example 1: d = (503.5 − 500)/8 = 0.4375 — a small-to-medium effect. Even with a significant p-value of 0.009, the actual deviation from specification (3.5 mg) may or may not warrant production changes depending on tolerance thresholds. Always report both p and d. See the full guide to effect size in hypothesis testing.

How to Report a One-Sample Z-Test in APA Format

APA 7th edition reporting for a one-sample z-test follows a consistent template. The reader needs the test name, sample mean, population σ, z statistic, sample size (or degrees of freedom), p-value, and effect size.

7 Common Mistakes in One-Sample Z-Tests

Where the One-Sample Z-Test Appears in Practice

Entity and Formula Glossary

Frequently Asked Questions