# Difference Between T-test and Z-test

T-test and z-test terms are used to statistically test the hypothesis by comparing two sample means. The two tests are parametric hypothesis testing procedures since their variables are measured on an interval scale.

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A hypothesis is a conjecture to be accepted or rejected after further observation, investigation, and scientific experimentation.

## Key Takeaways

1. T-tests are used to compare the means of two groups when the population standard deviation is unknown, while Z-tests are used when the population standard deviation is known, and the sample size is large.
2. T-tests rely on the t-distribution, which is used for smaller sample sizes and unknown population standard deviations, whereas Z-tests use the standard normal distribution.
3. In practice, t-tests are more common due to the rarity of known population standard deviations. At the same time, Z-tests are generally reserved for situations with large sample sizes and known population parameters.

## T-test vs Z-test

The Z-test is used when the population mean and the standard deviation is known, it assumes that the population is normally distributed. The t-test is used when the population standard deviation is unknown and must be estimated from the sample data. The t-test assumes that the sample is normally distributed.

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A T-test is best for problems with limited sample sizes, whereas a Z-test works best for problems with large sample sizes.

## What is T-Test?

The t-test is a parameter applied to an identity to identify how the data averages differ when the variance or standard deviation is not given. The t-test is based on the Student t-statistic, having the mean being known and the variance of the population approximated from the sample.

The population’s standard deviation is estimated by dividing the sample’s standard deviation by the population size’s square root.

## What is Z-Test?

On the other hand, the z-test is the hypothesis test that ascertains if the averages of two sets of data differ, given the variance or standard deviation.

The z-test is a univariate test based on the standard normal distribution.

## Main Differences Between T-Test and Z-Test

While the two statistical methods are commonly involved in data analysis, they primarily differ in their application, formulae structure, and assumptions. The following are the key differences between the t-test and the z-test of the hypothesis.

### Type of Distribution

Both the t-test and z-test employ the use of distributions to compare values and reach conclusions in testing the hypothesis. However, the two tests use different distribution types.

Notably, the t-test is based on the Student t-distribution. On the other hand, the z-test is based on Normal distribution.

### Population Variance

While using the t-test and z-test to test the hypothesis, the population variance plays a significant role in obtaining both the t-score and z-score. While the population variance in the z-test is known, the population variance in the t-test is unknown.

However, with the t-score calculation dependent on the population variance, we can always estimate the population variance given the standard deviation or variance of the sample mean and sample size. Notably, the population standard deviation is estimated by dividing the sample population standard deviation by the square root of the sample size.

### Sample Size

While sample sizes differ from one analysis to another, there is a practical hypothesis test for any sample size. Notably, the z-test is used in hypothesis testing when the sample size is large.

On the other hand, the t-test is used in hypothesis testing when the sample size is small. A large sample size, in this case, refers to a sample size greater than thirty, that is, n ˃ 30.

Consequently, a small sample size refers to a sample size that is less than thirty, that is, n ˂ 30, with n denoting the sample size.

### Key Assumptions

While conducting either the t-test or z-test, some assumptions are held by statisticians. Notably, in a t-test, all data points are assumed, not dependent.

Sample values to test a hypothesis are to be taken and recorded accurately. Additionally, the t-test assumes to be working with small sample size.

Notably, to apply the t-test, the sample size should not exceed thirty and not below five. Above thirty, it would be considered significant, and below five, it would be considered too small.

On the other hand, in a z-test, all samples are assumed to be independent. The sample size is also assumed to be significant.

Notably, a large sample size while conducting a hypothesis test using the z-test should have the sample size exceed thirty. Additionally, the distribution of z is assumed to be expected, with a mean of zero and a variance of one.

### Use

While both tests compare population averages, the two tests differ. The t-test helps determine the availability of statistical significance between two independent sample datasets.

The t-test is suited for testing the hypothesis of problems with a limited sample size, that is, a sample size of less than thirty and with an unknown population variance. On the other hand, the z-test is used to show the deviation of a data point from the average of a set of data.

Additionally, the z-test is used for data sets that have known the standard deviation. The data set’s sample size should also be extensive; it should exceed thirty.

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