T-test and z-test are terms common when it comes to the statistical testing of hypothesis in the comparison of two sample means. Notably, the two tests are parametric procedures of hypothesis testing since they are both their variables are measured on an interval scale.

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

**T-test vs Z-test**

The difference between T-test and Z-test is that a T-test is used to determine a statistically significant difference between two sample groups that are independent in nature, whereas Z-test is used to determine the difference between means of two populations when the variance is given.

A T-test is best with the problems that have a limited sample size, whereas Z-test works best for the problems with large sample size.

**Comparison Table Between T-Test and Z-Test**

Parameter of Comparison | T-Test | Z-Test |

Type of Distribution | Student t-distribution | Normal distribution |

Population Variance | Suitable for unknown population variance. | Suitable for known population variance. |

Sample Size | Small sample size. | Large sample size. |

Key Assumptions | All data points are assumed, not dependent. | All data points are assumed to be independent. |

Sample values are accurately collected and recorded. | Distribution of z is assumed to be normal, with a mean of zero and a variance of one. | |

Use | The sample size is small. | The sample size is large. |

For limited sample sizes, not exceeding thirty. | For large sample sizes and known standard deviation. |

**What is T-Test?**

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

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

**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 from each other being given the variance or standard deviation.

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

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

While the two statistical methods are commonly involved in the analysis of data, they largely differ from their application, formulae structure, and assumptions amongst other differences. 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 the testing of 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 both the t-test and z-test in the testing of the hypothesis, the population variance plays a major 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 from dividing the sample population standard deviation by the square root of the sample size.

**Sample Size**

While sample sizes differ from analysis to another, there is a suitable test of hypothesis 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 that is 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 be used in the test of a hypothesis are to be taken as well as recorded accurately. Additionally, the t-test assumes to be working with a 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 regarded to be large, and below five, it would be regarded to be 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 large. Notably, a large sample size while conducting a test of hypothesis using the z-test should have the sample size exceed thirty.

Additionally, the distribution of z is assumed to be normal, with a mean of zero and a variance of one.

**Use**

While both tests are used in the comparison of population averages, the two tests differ in their use. The t-test is useful in the determination of the availability of statistical significance between two independent sample datasets. The t-test is suited for the test of the hypothesis of problems with limited sample size, that is, sample size less than thirty and with the population variance unknown.

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 large; that is, it should exceed thirty.

**Frequently Asked Questions (FAQ) About T-test and Z-test**

**Is the Z score and Z-test the same?**

**Z Score** is the number of standard deviations of particular value away from the mean.

**Z- test** denotes a uni-variate statistical analysis used to test the hypothesis that proportions from two independent samples differ a lot. It determines to what extent a data point is away from its mean of the data set, in standard deviation.

**What is Z in probability distribution?**

Z denotes the normal distribution in the probability distribution. It is a normal continuous probability distribution and it is also known as Gaussian distribution.

F(z) is a normal distribution density which is called the bell curve because its shape looks like a bell.

**What does T-value mean?**

The T value measures the size of the difference relative to variation in the sample data. The greater the value of T, the greater of evidence against the null hypothesis.

**What are the 3 types of T-tests?**

The list of three types of T-tests is given below:

**One sample T-test:** we compare the mean or average of any group against the set average of the group. The value of the average can be theoretical or population.

**Independent two-sample T-test:** Used to compare the means of two different samples.

**Paired sample T-test:** Here we measure one group at two different times. We compare different means for a group under two different conditions or at two different times.

**Conclusion**

Despite being nearly similar, the T-test and Z-test differ largely from their application. The big difference remains to be the use of a T-test for small sample sizes and the z-test for larger sample sizes.

Additionally, the t-test is suitable when the population variance is unknown while testing for the hypothesis of a sample size whose population variance is known requires the z-test.

Therefore, one should be careful while choosing the perfect parameter for the test of the hypothesis.

**References**

- https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/t-test/
- https://www.ajodo.org/article/S0889-5406(15)00612-5/fulltext

Table of Contents