Difference Between Z-Test and Chi-Square

Edited by Diffzy | Updated on: April 30, 2023


Difference Between Z-Test and Chi-Square

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To evaluate various hypotheses, both the Z test and the Chi-square statistic are used. Both approaches provide a unique point of view when it comes to the evaluation of null value hypotheses.

Researchers utilize the Chi-Square test, a statistical tool, to look at the differences between categorical variables within the same population. Consider, for the sake of argument, that a study team wants to know whether or not education and married status are linked among Americans as a whole.

Z test: Large sample hypothesis testing. To assess the difference between actual and predicted frequencies, use a Chi-square test. The difference between the population means is what's at stake in the F test for hypotheses.

Z-Test vs Chi-Square

When contrasting the Z-test with the Chi-square test, it is essential to keep in mind that the Z-test is a statistical test that compares two populations to determine whether or not their means are different from one another. The importance of having a sample size that is large enough to calculate a standard deviation cannot be overstated. If you want to assess whether or not two categorical variables in a population are connected, you may use a different approach that's known as the Chi-square test.

Z-tests often emphasize large sample sizes (n greater than 30). When the standard deviation can be easily accessed, it makes the use of the information considerably easier. A statistical method known as an alternative hypothesis is used in contrast to the null hypothesis.

The Chi-square test is the statistical method of choice for analyzing categorical data. According to the assumptions of the Chi-square test, the two categorical variables studied should be independent of one another in the population.

Two groups may be compared with the use of the Z-test by contrasting the proportions of their respective populations. You may compare the difference in population proportions between two or more groups by using the Chi-square test, or you can compare one group to a value. Both of these comparisons are possible.

Using the Z-test, two groups may be compared by comparing their population proportions. Using the Chi-square test, you may compare the difference in population proportions between two or more groups, or you can compare one group to a value.

Comparison Table Between Z-Test and Chi-Square (in Tabular Form)

Parameter of Comparison Z-Test Chi-square
Statistic used Z-statistic is the name given to the statistics used to test the alternative hypothesis. The Chi-square statistic is used for evaluating null hypotheses.
Null and Alternate values In other words, there is no statistical significance to the sample mean. Null: C and D are not reliant on each other.
  Alternately, it is possible to argue that the sample mean and population mean should be distinct. Alternative: Variables A and B aren't independent of one another.
Conditions You need to know the standard deviation. The z-test may not work well if the sample size is too small. The results of the test should be plotted using a normal distribution as a guide. Each level of a variable should have at least five observations. Only if there are categorical values can the test be performed. Simplify and randomise the sampling process.
Formula z = (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

Uses When the variance and data are high, it's possible to tell whether the two means derived from two separate populations are different. Compares two or more groups using categorical data where values are provided.

What is Z-Test?

Only a hypothesis may be tested using a Z-test. During the testing, samples are often distributed to other people. If there is a predetermined standard deviation, it can only be used if there is a sufficient amount of sample data (n greater than 30). When extrapolated to the whole population, the findings of the sample lend credence to any theory drawn from those results. A large sample from each group may be used in conjunction with a z-test to determine whether or not the means of the two populations differ from one another. It is assumed that the statistic being tested will have a normal distribution, and as a result, it is necessary to be familiar with nuisance variables such as the standard deviation.

This hypothesis test, which makes use of the z-statistic, is sometimes referred to as the z-test. The standard deviation of the z-statistic is derived from a normal distribution. Therefore, the z-test is most accurate when there are more than 30 samples since the central limit theorem indicates that as the number of samples rises, the samples are expected to be more or less regularly distributed.

Z-tests are required to include the alternative and null hypotheses, in addition to the alpha and z-score. The next thing that has to be done is the computation of the test statistic, followed by the presentation of the results and the conclusion. A z-statistic, which is also known as a z-score, is a number that shows how much a z-test result deviates from the mean of the population. This number may also be written as "z-score."

Different kinds of z-tests include the following: the one-sample location test, the two-sample location test, the paired difference test, and the maximum likelihood estimate. Z-tests and t-tests have a tight relationship; however, t-tests are better suited for use in investigations with smaller sample sizes. Z-tests are more often used. In a t-test, it is assumed that the standard deviation is unknown, but in a z-test, it is assumed that the standard deviation is known. To establish an assumption about the standard deviation of a population, it is necessary to ensure that the sample variance is equal to the population variance.

Z-test refers to a statistical test where the distribution of the test statistic under the null hypothesis may be approximated by a normal distribution. This kind of test is known as a hypothesis test. The mean of the distribution is what Z-tests look at. The Z-test is more applicable to real-world scenarios than the Student's t-test because it assigns a single important value to each significance level in the confidence interval (for example, 1.96 for 5 percent two-tailed) (through the corresponding degrees of freedom). Analyzing a set of data using the Z test and the Student's t-test, both of which help determine the significance of the data, is one way to determine whether or not the data is relevant. The z-test, on the other hand, is seldom used in practice since it is difficult to ascertain the deviation of the population being studied.

A Z-test must be performed under certain conditions:

  • A sample size of at least 30 is required.
  • In other words, there should be no parallels or overlaps among the data points.
  • The data should be evenly dispersed at all times.
  • Racial and ethnic samples are taken at random from the general population.

What is the procedure for doing a Z-test?

  • H0 must first be presented, and then other hypotheses may follow (HA).
  • Choose an alpha level, to begin with.
  • Z's criticality may be assessed using the Z table.
  • The Z state statistic has finally been figured out. '
  • Now is the time to compare the test statistic to the crucial number z. If you accept the null hypothesis, this will lead you to the answer (H0).

In cases when the data is vast and the standard deviation is known, Z-tests should be used to analyze the null hypothesis.

What is Chi-Square?

The Chi-Square test is the most accurate representation of a statistical hypothesis test. Using this test, one may either do a comparison based on values or numerous comparisons based on categories.

The fact that this test can successfully process the data that is presented to it is a significant advantage. To make use of it, there need to be two categorical variables, and both of those variables need to be related to a population.

The Chi-square test may be used to determine how well the data fit together. For this to take place, the two variables in question must first and foremost be independent of one another.

A chi-square test is a statistical test that is used to compare the findings that were seen with the outcomes that were predicted. This test's objective is to establish if a disparity between the observed data and the predicted data is the result of random variation or whether it can be attributed to a link between the variables that are the subject of your investigation.

The Chi-square test is designed to determine the degree of probability that an observed distribution is the result of random occurrences. It is also referred to as a "goodness of fit" statistic since it evaluates how well the distribution of the data that was seen corresponds to the distribution that would be anticipated if the variables were unrelated to one another.

If the test statistic follows a chi-squared distribution by the null hypothesis, then it is acceptable to carry out chi-squared tests such as the Pearson's chi-squared test as well as variants on this test. To determine whether or not there is a statistically significant difference, the expected frequencies on a contingency table may be compared to the actual frequencies using Pearson's chi-squared test.

One frequent use of this test is to split the observations into two separate categories. If the null hypothesis turns out to be true, then the test statistic that is derived from the data will have a frequency distribution that looks like an x2. The purpose of the test is to determine how likely it is that the observed frequencies are the result of random occurrences.

When there is no correlation between the variables being tested, x2 distribution test statistics are produced. The x2 test, which is based on the observations of the two pairs, may be used to test whether or not the null hypothesis of independence between two random variables is true.

There is a wide variety of the chi-squared test, but they all have one thing in common: the sampling distribution of the test statistic (under the assumption that the null hypothesis is correct) becomes closer and closer to a chi-squared distribution as the sample sizes get larger.

The Chi-Square test is used in the field of statistics to compare the data that was seen with the data that was expected. It is also possible to apply this test to see whether or not our category data is consistent with this test. When categorical variables are analyzed, it is easier to identify whether the difference between them is the result of a relationship between them or whether it is the product of chance.

Main Differences Between Z-Test and Chi-Square in Points

  • Every one of these tests is predicated on two different statistical assumptions: Due to this limitation, they are only appropriate for usage with very large datasets.
  • The Z-test should only be used when the number of observations is more than 30 since this is the only circumstance in which it is considered suitable. Chi-square analysis, on the other hand, is the technique of choice when dealing with categorical variables that are independent of one another and belong to the same population.
  • Both the Z-test and the Chi-square are used to conclude whether or not the null hypothesis should be accepted.
  • It is recommended that a sample be randomly selected from the population that is provided for the Chi-square test rather than randomly distributed for the Z-test.
  • Both studies used separate methodologies to present alternate hypotheses to the null hypotheses they were testing.
  • Chi-squared tests, in contrast to z-tests, are better suited for analyzing qualitative data (or proportions in the case of z).
  • The standard procedure calls for utilizing the common pooled proportion as the test, which enables one to get an estimate of the variance of the difference between two proportions. In contrast to the chi-square test, an estimate may be made of the standard normal deviation (z).
  • In a Z-test, the samples are selected at random; however, in a Chi-square test, the samples should be generated at random from the population that is provided.


Both of these methods may be used in the analysis of massive datasets. They adhere to their very own customs and are constrained by their very own rules. For the Z-test findings to be meaningful, there has to be a certain amount of standard deviation. In contrast, the Chi-square test helps compare two different variables that are not related to one another. Both statistical tests are successful and yield more accurate hypotheses when applied to large datasets.


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


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"Difference Between Z-Test and Chi-Square." Diffzy.com, 2024. Sat. 15 Jun. 2024. <https://www.diffzy.com/article/difference-between-z-test-and-chi-square-441>.

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