Z-Test: Formula, Examples, Uses, Z-Test vs T-Test

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Z-test Definition

z-test is a statistical tool used for the comparison or determination of the significance of several statistical measures, particularly the mean in a sample from a normally distributed population or between two independent samples.

  • Like t-tests, z tests are also based on normal probability distribution.
  • Z-test is the most commonly used statistical tool in research methodology, with it being used for studies where the sample size is large (n>30).
  • In the case of the z-test, the variance is usually known.
  • Z-test is more convenient than t-test as the critical value at each significance level in the confidence interval is the sample for all sample sizes.
  • A z-score is a number indicating how many standard deviations above or below the mean of the population is.
Z Test Formula

Z-test formula

For the normal population with one sample:

Z-test formula one sample

where   is the mean of the sample, and µ is the assumed mean, σ is the standard deviation, and n is the number of observations.

z-test for the difference in mean: 

z-test formula for the difference in mean

where 1 and 2 are the means of two samples, σ is the standard deviation of the samples, and n1 and n2 are the numbers of observations of two samples.

One sample z-test (one-tailed z-test)

  • One sample z-test is used to determine whether a particular population parameter, which is mostly mean, significantly different from an assumed value.
  • It helps to estimate the relationship between the mean of the sample and the assumed mean.
  • In this case, the standard normal distribution is used to calculate the critical value of the test.
  • If the z-value of the sample being tested falls into the criteria for the one-sided tets, the alternative hypothesis will be accepted instead of the null hypothesis.
  • A one-tailed test would be used when the study has to test whether the population parameter being tested is either lower than or higher than some hypothesized value.
  • A one-sample z-test assumes that data are a random sample collected from a normally distributed population that all have the same mean and same variance.
  • This hypothesis implies that the data is continuous, and the distribution is symmetric.
  • Based on the alternative hypothesis set for a study, a one-sided z-test can be either a left-sided z-test or a right-sided z-test. 
  • For instance, if our H0: µ0 = µ and Ha: µ < µ0, such a test would be a one-sided test or more precisely, a left-tailed test and there is one rejection area only on the left tail of the distribution.
  • However, if H0: µ = µ0 and Ha: µ > µ0, this is also a one-tailed test (right tail), and the rejection region is present on the right tail of the curve.

Two sample z-test (two-tailed z-test)

  • In the case of two sample z-test, two normally distributed independent samples are required.
  • A two-tailed z-test is performed to determine the relationship between the population parameters of the two samples.
  • In the case of the two-tailed z-test, the alternative hypothesis is accepted as long as the population parameter is not equal to the assumed value.
  • The two-tailed test is appropriate when we have H0: µ = µ0 and Ha: µ ≠ µ0 which may mean µ > µ0 or µ < µ0
  • Thus, in a two-tailed test, there are two rejection regions, one on each tail of the curve.

Z-test examples

If a sample of 400 male workers has a mean height of 67.47 inches, is it reasonable to regard the sample as a sample from a large population with a mean height of 67.39 inches and a standard deviation of 1.30 inches at a 5% level of significance?

Taking the null hypothesis that the mean height of the population is equal to 67.39 inches, we can write:                           

H0 : µ = 67.39

Ha: µ ≠ 67.39

= 67.47“, σ = 1.30“, n = 400

Assuming the population to be normal, we can work out the test statistic z as under:

Z = 1.231

Z-test examples

z-test applications

  • Z-test is performed in studies where the sample size is larger, and the variance is known.
  • It is also used to determine if there is a significant difference between the mean of two independent samples.
  • The z-test can also be used to compare the population proportion to an assumed proportion or to determine the difference between the population proportion of two samples.

Z-test vs T-test (8 major differences)

Basis for comparison



DefinitionThe t-test is a test in statistics that is used for testing hypotheses regarding the mean of a small sample taken population when the standard deviation of the population is not known.z-test is a statistical tool used for the comparison or determination of the significance of several statistical measures, particularly the mean in a sample from a normally distributed population or between two independent samples.
Sample sizeThe t-test is usually performed in samples of a smaller size (n≤30).z-test is generally performed in samples of a larger size (n>30).
Type of distribution of populationt-test is performed on samples distributed on the basis of t-distribution.z-tets is performed on samples that are normally distributed.
AssumptionsA t-test is not based on the assumption that all key points on the sample are independent.z-test is based on the assumption that all key points on the sample are independent.
Variance or standard deviationVariance or standard deviation is not known in the t-test.Variance or standard deviation is known in z-test.
DistributionThe sample values are to be recorded or calculated by the researcher.In a normal distribution, the average is considered 0 and the variance as 1.
Population parametersIn addition, to the mean, the t-test can also be used to compare partial or simple correlations among two samples.In addition, to mean, z-test can also be used to compare the population proportion.
Conveniencet-tests are less convenient as they have separate critical values for different sample sizes.z-test is more convenient as it has the same critical value for different sample sizes.

References and Sources

  • C.R. Kothari (1990) Research Methodology. Vishwa Prakasan. India.
  • https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/One-Sample_Z-Tests.pdf
  • https://www.wallstreetmojo.com/z-test-vs-t-test/
  • https://sites.google.com/site/fundamentalstatistics/chapter-13
  • 3% – https://www.investopedia.com/terms/z/z-test.asp
  • 2% – https://www.coursehero.com/file/61052903/Questions-statisticswpdf/
  • 2% – https://towardsdatascience.com/everything-you-need-to-know-about-hypothesis-testing-part-i-4de9abebbc8a
  • 2% – https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/One-Sample_Z-Tests.pdf
  • 1% – https://www.slideshare.net/MuhammadAnas96/ztest-with-examples
  • 1% – https://www.mathandstatistics.com/learn-stats/hypothesis-testing/two-tailed-z-test-hypothesis-test-by-hand
  • 1% – https://www.infrrr.com/proportions/difference-in-proportions-hypothesis-test-calculator
  • 1% – https://keydifferences.com/difference-between-t-test-and-z-test.html
  • 1% – https://en.wikipedia.org/wiki/Z-test
  • 1% – http://www.sci.utah.edu/~arpaiva/classes/UT_ece3530/hypothesis_testing.pdf
  • <1% – https://www.researchgate.net/post/Can-a-null-hypothesis-be-stated-as-a-difference
  • <1% – https://www.isixsigma.com/tools-templates/hypothesis-testing/making-sense-two-sample-t-test/
  • <1% – https://www.investopedia.com/terms/t/two-tailed-test.asp
  • <1% – https://www.academia.edu/24313503/BIOSTATISTICS_AND_RESEARCH_METHODS_IN_PHARMACY_Pharmacy_C479_4_quarter_credits_A_Course_for_Distance_Learning_Prepared

About Author

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Anupama Sapkota

Anupama Sapkota has a bachelor’s degree (B.Sc.) in Microbiology from St. Xavier's College, Kathmandu, Nepal. She is particularly interested in studies regarding antibiotic resistance with a focus on drug discovery.

2 thoughts on “Z-Test: Formula, Examples, Uses, Z-Test vs T-Test”

  1. The formula for Z test provided for testing the single mean is wrong. The correct formula is
    wrong. Please check and correct it. It should be Z = (𝑥̅−𝜇)/𝜎/√n

    • Hi Ramnath, Sorry for the mistake. Thank you so much for the correction. We have updated the page with correct formula.


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