One of the most popular ways to test a hypothesis is the t-testing method.

Organizations and businesses make use of t-tests to compare the likelihood that a projected outcome will occur. For example, a marketer will perform a t-test to determine customer preferences by comparing sets of customers against each other. I.e. Do the customers in section A spend more on beauty than the customers in section B?  

There are different t-tests, and each type has its own distinct applications. In this article, we will dig into t-testing in survey analysis, consider the three different t-tests, and how they can be used. Let’s get started!

What are T-tests?

A t-test is one of the different types of inferential statistics that you can use to compare the values of two groups and determine if two data sets from the same population have any significant difference with the mean of the two data sets.

A t-test allows the testing of an assumption applicable to a population. If you want to conduct a test with three or more mean, you must use an analysis of variance.

What a t-test does is that it takes a sample from the two data sets, assumes a null hypothesis that the two mean are equal and establishes the problem statement.

Read: Pilot Testing: Meaning, Importance + [Question Example]

What are the Types of T-tests?

There are three types of t-tests, and the three are categorized as dependent and independent t-tests.

1. The one-sample t-test: This is used to compare the mean of a single population against a known mean. It aims to find out if the mean of one group is equal to a target value.

2. The unpaired two-sample t-test: This is used to compare the mean of two independent samples or groups and how they are equal to the target value.

3. The paired t-test: This is used to compare the mean between two related groups of samples or observations at different times. It might be one month, or one year apart.

One important property of the t-test is that large samples t-tests are often valid even with the violation of the assumption of normality. This property makes the T-test one of the most useful procedures that can be used to make inferences about the population mean.

Read: Concept Testing: Types, Methods + Examples

When to Use Each Type of T-test?

Now that you are familiar with the types of t-tests, if you’re wondering when each type of t-test can be used, know that a t-test can only be used when comparing the mean of two populations.

Make use of an ANOVA test if you want to compare more than two groups. Because the t-test is a test of difference, it will assume that your data are independent, approximately distributed, and also possess similar amounts of variance in each of the populations being compared.

So before choosing a t-test, consider these two things

1. whether the groups being compared are from a single population or not.

2. whether you should test the difference in one specific direction. Which will it be, one-sample, two-sample, or paired t-test?

When to use a Paired t-test: Use a paired t-test if the groups come from a single population.

When to use a Two-sample t-test: Use a two-sample t-test or an independent t-test if the groups come from two different.

When to use a One-sample t-test: Use a one-sample t-test if one group is being compared against a standard value.

There are other t-test methods that you can use depending on what you intend to measure. For example, use a two-tailed t-test if you want to know whether two populations are different from one another and a one-tailed t-test if you want to test if the mean of one population is greater than or less than the other population.

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How Do T-tests Work & How They are Conducted 

T-tests generally make a comparison using the mean and standard deviations of two samples. The T-test formula then is:

Where,

And the formula for standard deviation is:

Where,

x = The values given

x¯¯¯ = The mean and

n = The total number of values.

The formula for the degree of freedom is given by Degrees of Freedom= 

Where,

n1 and n2= Represents the number of records in each sample set​

One-sample t-test formula: Let us represent a set of values as X and the size of the value as n, if the mean is m and the standard deviation S. The formula for one-sample t-test will be:

This type of t-test can be used only when the data are normally distributed.

Two sample t-test: Let the two groups to be compared be represented by A and B. Remember that the mean is still m and the value is n.

So, if mAmA and mBmB represent the mean of groups A and B. nAnA and nBnB will represent the sizes of the two groups. Namely A and B, respectively. The formula is as follow:

S2S2 gives an estimate of the common variance of the two samples. The formula is as follow:

Paired sample t-test: The formula can be calculated as follow:

m and s, represent the mean and the standard deviation of d which stands for difference. n is the size of d.

The value of the t-test statistics being greater than the analytical value, implies that the difference is significant. If otherwise, then it isn’t. The extent of significance or (p-value) corresponds to the risk indicated by the calculated |t| value from the t-test table.

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What is a T-test used for/ Applications of T-testing?

A t-test is used to compare the mean of two samples, dependent or independent. T-test also determines if the mean of the sample is different from the assumed mean.

To apply the t-test three key data values are required. These values include the difference between the mean values from each of the data sets, the standard deviation of each of the groups, and the number of data values of each group.

The result of the t-test produces the t-value. This calculated t-value is then compared against a value obtained from the T-Distribution Table. The t-test compares the difference between the groups and checks whether it shows a significant difference in the study or if the difference is insignificant and random.

Read: Analysis of Variance (ANOVA): Types, Examples & Uses

How to Interpret Test Results

When you input a hypothesis, you will receive the output of your test as:

1. A data output table. It provides an explanation of what is being compared.

2. The t-value may come out negative; this is fine! In most cases, what is needed is the absolute value of the difference or the distance from 0. 

3. The degrees of freedom are related to your sample size. It shows how many ‘free’ data points are available in the test you are comparing. Your statistical test stands a chance of performing better if the degrees of freedom are greater.

The higher values of t-value, which is also called t-score, show that there’s the presence of a large difference between the two sample sets.

In essence, the smaller the t-score, the higher the similarity that exists between the two sample sets. A large t-value shows that the two groups are not the same while the small t-value score shows the similarity between the groups.

Read: What is Data Interpretation? + [Types, Method & Tools]

Limitations of T-testing and When Not to Use a T-test

You should not use a t-test to measure differences among more than two groups. If there are more than two groups being compared, a t-test will undermine the actual error.

Ensure that the data in the one sample is at least symmetric. Also, make sure that outliers being present do not distort the results.

It is important to be certain that the situation for the paired t-test is similar. Make sure that the differences in the data pairs are reasonably symmetric, and that outliers do not distort the results of these differences.

For independent samples, the data in each of the samples must be normal. Ignoring the above listed may result in the inaccuracy of results.

What this implies is that it is inappropriate or rather a misuse of this method to compare the mean among multiple groups. i.e., to carry out a comparison between more than two groups.

To use the T-test you can only compare quantitative data from one sample design. It will amount to a misuse of the t-test to perform a check for multiple samples. You can confirm if indeed the data obeyed normal distribution.

One popular method that can be explored to achieve this is to compare the mean and the standard deviation of the data. The data may not comply with normal distribution if the mean of a value is much smaller than its standard deviation. Ultimately, the t-test may also be inaccurate or inappropriate.

In this case, it is better to perform a t-test after an appropriate variable transformation.

FAQs About T-tests

• What is the sample size for t-test?

The t-test also known as the parametric test is useful for testing samples whose size is less than 30. The normal distribution and the distribution of the t-test will not be identifiable if the size of the sample is more than 30.

Hence the best size to achieve an accurate test result that is distinguishable is 30 or less. The method to be used to check this is not significantly different from the procedure used to measure for large-size samples.

However, it is best when the number of observations is below 60, or they amount to 30 or less.

A good rule of thumb is to overestimate the variance of the effect size when calculating the sample size for an independent sample t-test. Researchers practice this as it forces them to have to collect more observations of the result, which then leads to the measures of effects being more precise and accurate

• What is the difference between a z-test and a t-test?

A z-test is a statistical test that determines if there’s a difference between the mean of two populations when the size of the sample is large and the variances are known. As a hypothesis test, the z-test follows a normal distribution.

To perform an accurate z-test, the statistic is assumed to have a normal distribution and the standard deviation known, unlike the t-test that assumes a null hypothesis that the two mean are equal.

While the t-test assumes the standard deviation is unknown, the z-test assumes the standard deviation is known.

Z-test and t-test are closely related but when an experiment has a small sample size, it is best to perform a t-test.

• What is the difference between the one-sample, two-sample, and paired T-tests

The one-sample t-test seeks to make inferences regarding the mean of a population. One sample t-test is applied when only a sample is given and a hypothesis needs to be run on the sample.

The two-sample t-test on the other hand is more common than the one-sample t-test because in most cases, the comparison is made between the mean of two groups. A two-sample t-test can also be used when given one sample and a hypothesis needs to be done on the sample.

Unlike the one-sample t-test and two-sample t-test, the paired t-test is used to test two different treatments within the same sample population. The mean of the two treatments are compared

Read: Inferential Statistics: Definition, Types + Examples

• What is the difference between the F-test and t-test?

An F-test is applied to compare the two standard deviations of two samples and assume their variability, the t-test checks the hypothesis of whether the sample mean is significantly different from the given mean or if they are the same.

Also, a t-test can be paired t-test, or a normal t-test but there is only one type of the f-test, and it compares the standard deviation of two-sample data.

The degree of freedom is also different. The degree of freedom for the t-test is (n-1) where n represents the number of sample values. On the other hand, the degree of freedom for the f-test is (n1-1,n2-1) where n1 and n2 represent the numbers of observations in the one and two samples

Conclusion

This article explained in depth what a t-test is, the types of t-tests there are and how they can be used. T-test shows the significant differences between the mean of the two groups. It provides outcomes that explain if the differences measured in mean and obtained happened by chance. A t-test will tell you what you really want to know.