Difference Between Z-test and T-test (With Table)

While making a critical hypothesis, observed changes in the mean are statistically significant. It is a major consideration to give a perfect analysis for a condition. These analyses are excellent for hypothesis testing or significance testing.  

There are various test statistics are performed for testing the hypotheses, such as z-test and t-test. They are also applicable in business, science, and many other disciplines. Z-test and t-test differences are the main course of discussion in this article. 

Z-test vs T-test 

The main difference between z-test and t-test is that z-test is used to determine whether the calculation of two sample means is different if the sample is large and the standard deviation is available. But t-test is used to determine how different averages data sets differ from each other if the standard deviation or variance is not known.  

Z-tests are the statistical calculations used to compare the average population to a sample. In terms of standard deviations, the z-test tells a data point is how far from the data set average. It generally compares a sample to a population that is used to deal with large samples relating to problems. They are useful if the standard deviation is known.  

T-tests are used to test hypothesis calculations. They are useful to determine in case there are significant statistics in comparison between the independent sample groups of two. Or it can be said that it asks if the comparison between the two group’s averages due to random chance occurred unlikely. 

Comparison Table Between Z-test and T-test 

Parameters of ComparisonZ-testT-test
Based onNormal distributionStudent-t distributtion
Formulaz = (x̄ – μ) / (σ / √n)t = ( x̄ – μ) / (s / √n)
Sample sizeLargeSmall
Population varianceKnownUnknown
Data sizeGreater than 30Smaller than 30

What is Z-test? 

A z-test is a test of statistics to decide whether the means of two populations are dissimilar if the sample size is large and the variance is known. To perform an accurate z-test, that test statistic has a nuisance parameter like a standard deviation be known.  

It is also a hypothesis test and a normal distribution should be followed by z-statistic. It is better to use a z-test for greater than 30 samples. This is because the number of samples gets larger under the central limit theorem and the samples are considered normally distributed.  

The alternative and null hypotheses, z-score, and alpha should be stated to conduct a zed test. Next, the conclusion and results should be stated, and the test statistic must be calculated. The population variance of the z-test is known. In a z-test, there is a normal distribution for z with variance as one and mean as zero

A z-score, or z-statistic, is a number that represents standard deviations below or above the mean population and a z-test derived score. Test which can be conducted as z-tests is two-sample location test, and maximum likelihood estimate, one sample location test and paired different test. 

What is T-test? 

A T-test is a kind of inferential statistic to determine a significant difference in the middle of two groups of means, that might be related to certain features. It is used as a tool of hypothesis testing and allows testing of an applicable assumption to a population.  

It is generally used when the data sets might have unknown variances and follow a normal distribution. To determine the statistical significance, this test looks at the degrees of freedom and the t-distribution values. It is necessary to use an analysis of variance to conduct a test with three or more means.  

T-test allows to compare the two data sets’ average values and determine if the origin is from the same population. Many different types of t-tests perform based on the type and data of analysis required. It works on a smaller size and should not be less than five but also not exceed thirty. 

Three key data values are required to calculate at t-test. It generally includes the number of data values of every group, the standard deviation of every group, and the mean difference. 

Main Differences Between Z-test and T-test 

  1. Z-test is used to determine whether the calculation of two sample means is different. But t-test is used to determine how different averages data sets differ from each other.  
  2. The data of the z-test is distributed normally just take an example, if data >30 we can assume it is. On the other hand, the t-test is not normally distributed.  
  3. Z-test data point does not affect another data point or is not related, whereas data points of t-test may be related to each other, where the value or behavior of one point may affect another.  
  4. From a broader population, data of z-test was randomly selected. On the flip side, the data of the t-test was not randomly selected.  
  5. In a z-test, there is a normal distribution for z with variance as one and mean as zero, while the t-test works on a smaller size and should not be less than five but also not exceed thirty. 

Conclusion 

It can be concluded that the z-test and t-test are two of the test statistics performed for testing the hypotheses. They require data along with a normal distribution, in simple words, sample data around the mean is distributed evenly. They are applicable in business, science, and many other disciplines.  

Z-test is used to determine whether the calculation of two sample means is different if the sample is large and the standard deviation is available. On the other hand, the t-test is used to determine how different averages data sets differ from each other if the standard deviation or variance is not known. Z-test is based on normal distribution, whereas t-test is based on student-t distribution. 

References 

  1. https://www.jstor.org/stable/2332232
  2. https://www.ncbi.nlm.nih.gov/pmc/articles/pmc4667138/
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