Z-Test vs Chi-Square: Difference and Comparison

Z test and Chi-square are two different statistical hypotheses testing. Both tests give an alternate point of view to null value hypotheses.

Key Takeaways

  1. Statistical tests: Z-test is a hypothesis test using the standard normal distribution to compare a sample statistic to a population parameter. In contrast, the chi-square test is non-parametric, comparing observed frequencies to expected frequencies under the null hypothesis.
  2. Data type: Z-test is used for continuous data, while the chi-square test is used for categorical data.
  3. Applications: Z-test is employed for testing the mean or proportion of a single population, while the chi-square test is utilized for independence, the goodness of fit, or homogeneity tests.

Z-Test vs Chi-Square

The Z-test is used when the sample size is large and the population standard deviation is known, used to test hypotheses about the mean of a normal population. The Chi-square test is used when the sample size is small, thus used to test hypotheses about the distribution of a categorical variable.

Z test vs Chi square

Z-test is used for dealing with problems relating to large samples (n>30). It is easier to use when the standard deviation is available.

The Chi-square test was used for testing relationships between categorical values. The null hypotheses of the Chi-square say that two categorical variables in the population should be independent.


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Comparison Table

Parameter of ComparisonZ-TestChi-square
Statistic usedThe statistics used for the alternate hypothesis testing is called Z-statistic.The statistics used for null hypothesis testing is called the Chi-square statistic.
Null and Alternate valuesInvalid: The sample mean is the same as the population mean.Null: Both Variables C and D are independent.
Alternatively, it can be said that the results of the sample mean and population mean should be different.Alternative: Both variable A and variable B are not independent.
ConditionsThe standard deviation should be known. The sample size should be large enough, or else z-test may not perform well. The test statistics should follow a normal distribution.There should be a minimum of five observations at each variable level. The test can be done only if there are categorical values. The sampling method should be simple and random.
Formulaz = (x-μ)/(σ / √n)
x = sample mean.
μ = population mean.
σ / √n = standard deviation.
Χ2 = Σ(O − E)2/E
O = each Observed (actual) value
E = each Expected value
UsesDetermines if the results of two means obtained from two populations are different when the variance and data are largeIt uses categorical data in comparing two or more groups where the values are mentioned.


What is Z-Test?

A Z-test is nothing but a type of hypothesis test. The samples are distributed while conducting the test. It is used only when there is a standard deviation, and the sample data should always be extensive (n>30). 

In other words, it validates hypotheses drawn by the sample to the same population.

Conditions required to perform a Z-test:

  1.  The sample data should be greater than 30.
  2. The data points should be independent of each other; that is, there should be no similarities or overlapping.

How to run a Z-test?

  1.  First, the null (H0) and the alternative hypothesis (HA) must be stated.
  2. Then, choose the alpha level.

I am advised that Z-test should analyse the null hypothesis when the data is on a large scale, and the standard deviation is known.

z test

What is Chi-Square?

The Chi-Square test is best defined as a statistical hypothesis test. This test is used either for comparing a group with a value or multiple groups with categorical data.

The advantages of this test are the robustness of the given data. It can only be used when two categorical variables are related to some population.

The Chi-square test is a goodness-fit statistic because it measures how well the observation data fits the distributed data. It can only happen when the two given variables are independent.

Main Differences Between Z-Test and Chi-Square

  1. In Z-test, the samples are evenly distributed, whereas, in Chi-square, it should be simple and randomly selected from the given population.
  2. Both tests used different methods but were used for giving alternate hypotheses to the null value hypotheses.

  1. https://www3.nd.edu/~kyuan/papers/nest-chisq-z.pdf
  2. https://www.sciencedirect.com/science/article/pii/S0167947313003204
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