Enter the sample mean, population mean, population standard deviation, and sample size into the calculator to determine the Z-score.

One Sample Z-Test Formula

The following formula is used to calculate the Z-score in a One Sample Z-Test.

Z = (X̄ - μ) / (σ / √n)


  • Z is the Z-score X̄ is the sample mean μ is the population mean σ is the population standard deviation n is the sample size

To calculate the Z-score, subtract the population mean from the sample mean. Then, divide the population standard deviation by the square root of the sample size. Finally, divide the result of the first operation by the result of the second operation.

What is a One Sample Z-Test?

A One Sample Z-Test is a statistical method used to determine whether a sample mean significantly differs from a population mean. It is typically used when the population standard deviation is known and the sample size is large. The test calculates a Z-score, which represents how many standard deviations the sample mean is away from the population mean. This score is then compared to a critical value from the Z-distribution table to decide whether to reject or fail to reject the null hypothesis.

How to Calculate One Sample Z-Test?

The following steps outline how to calculate a One Sample Z-Test using the formula: Z = (X̄ – μ) / (σ / √n).

  1. First, determine the sample mean (X̄).
  2. Next, determine the population mean (μ).
  3. Next, determine the population standard deviation (σ).
  4. Next, determine the sample size (n).
  5. Next, substitute the values into the formula: Z = (X̄ – μ) / (σ / √n).
  6. Finally, calculate the Z-score.
  7. After calculating the Z-score, compare it to the critical value to determine if the null hypothesis should be rejected or not.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Sample mean (X̄) = 75

Population mean (μ) = 70

Population standard deviation (σ) = 8

Sample size (n) = 50