Statistics is a discipline that entails collection, organization, analysis, interpretation and presentation of data. The discipline is applicable in industries, scientific research and solving social problem.

Learners need to have a proper understanding of statistical population, tests, and models to excel in this course. Some of the common statistical tests are T-test and Z-test.

These statistical methods help in data analysis of various disciplines like business, science, sociology, and more. But T-test and Z-test can be confusing to some extent.

So, what is the main difference between T-test and Z-test? **The former is a type of distribution based on student t-distribution while the latter is based on the normal distribution. **

This article provides further differences between the T-test and Z-test in a tabular form for better understanding. Take the time to read through it and learn how to calculate them.

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## Comparison Table (T-test vs Z-test)

Basic Terms |
T-test |
Z-test |

Meaning | It is a type of parametric test that helps to identify the average of two sets of data and how they differ from each other when the standard deviation or variance is not given. | It is a type of hypothesis test that helps to ascertain the average of two sets of data and how they differ from each other when standard deviation or variance is given. |

Population Variance | Unknown. | Usually known. |

Sample Size | Comparatively smaller. | Comparatively larger. |

Key Assumptions | All data points are not dependent.
Sample values to be taken accurately and recorded. |
All data points are independent.
Normal distribution of Z with an average zero and variance equal to one. |

Type of Distribution | Based upon student-t distribution. | Based upon normal distribution. |

Application | The limited sample size does not exceed thirty. | For a large sample size with a known standard deviation. |

## What Is a T-test?

It is a type of hypothesis test that helps to tell the difference between two sample groups that are independent in nature. In other words, a T-test tells a distinction between the average of two groups if they have occurred due to a random chance.

Keep in mind that a T-test is ideal when dealing with problems that have a limited sample size. It is following an at-distribution procedure and the standard deviation is not known in most cases.

The degree of freedom usually affects the shape of t-distribution. The degree of freedom also signifies that the number of independent observations in a given set of observations.

### Key Assumptions of T-test

- All data points are independent.
- The sample size is small and not exceeding thirty.
- The sample values have to be taken and recorded accurately.

The statistic test formula:

Where:

**x**is the sample mean.**s**is the standard deviation.**n**is the sample size.**u**is the population means.

## What Is a Z-test?

It is a hypothesis test that ascertains if the average of two sets of data differs from each other despite the standard deviation and variance being given.

The univariate statistical analysis help to determine to what extent data points differ from their mean in a standard deviation. The hypothesis test is applicable when the population variance is known.

Besides that, Z-test is applicable to large sample size with known population variance and standard deviation. It is also based upon standard normal distribution.

### Key Assumptions of Z-test

- All sample observations are independent.
- The sample size is usually larger than thirty.
- Has a normal distribution with zero mean and one variance.

The statistical test formula:

Where:

- x is the sample mean.
- n is the sample size.
- σ is the population standard deviation.
- u is the population means.

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## Main Difference between T-test and Z-test

- Z-test sample size is large. The T-test sample size is small and not more than thirty.
- T-test both standard deviation and variance is unknown. Z-test both standard deviation and variance is known.
- The T-test is based upon t-distribution. Z-test is based upon normal distribution.
- T-test all data points are not dependent. Z-test all data points are independent.
- The T-test is ideal for not more than thirty sample sizes. Z-test is applicable on a large sample size with known population variance.

## Frequently Asked Questions

**What Is the T-test Used For?**

It is an inferential statistic that helps to determine if there is a significant difference between the means of two groups with their related features.

**Why Do We Use T-test and Z-test?**

We perform a One-Sample t-test when we want to compare a sample mean with the population mean. The difference from the Z Test is that we do not have the information on Population Variance here. We use the sample standard deviation instead of population standard deviation in this case.

**Is the Z Score and Z-test the Same?**

Not really. Z Score is the number of standard deviations of a particular value away from the mean. Z- test denotes a univariate statistical analysis used to test the hypothesis that proportions from two independent samples differ a lot.

## In Conclusion

The application of the T-test and Z-test help to tell the distinction between these two hypotheses. The main difference between the T-test and Z-test is that the former uses a small sample size while the latter uses a large sample size.

Keep in mind that the T-test is applicable when the population variance is unknown while the Z-test is ideal when the population variance is known. Always try to be careful when choosing the perfect parameters for testing the hypothesis.

**More Sources and References**

- https://bloomingtontutors.com/blog/when-to-use-the-z-test-versus-t-test
- https://www.analyticsvidhya.com/blog/2020/06/statistics-analytics-hypothesis-testing-z-test-t-test/
- https://en.wikipedia.org/wiki/Z-test
- https://www.investopedia.com/terms/t/t-test.asp
- https://www.investopedia.com/terms/z/z-test.asp