What is T-Test? Definition, All 3 Types, and How to Read T-Test Values [Complete Guide]

•By Leonard Cucos•en
Statistical TestsResearch Methods

What is T-Test?

A t-test is a statistical test used to compare the means of two groups or to test whether the mean of one group differs from a specified value.

Imagine you have two balloons. Each balloon represents a group of people you're interested in, such as students from Class A and Class B. Now you want to know which class has a higher average test score. The t-test is like a tool that helps you measure and compare the sizes of both balloons to see how different they are.

If the balloons are very similar in size, you might not be able to tell if they're different. But if one balloon is noticeably larger than the other, you can say the two balloons differ significantly.

What is the T-Value?

In the statistical test called t-test, the t-value is a number that tells us whether the results we observe are due to a real reason or just by chance.

In the balloon example, we're interested in whether each balloon (or data group) is significantly different in size. The t-value helps us understand this.

If the t-value calculated is very high, it indicates that seeing one balloon larger than the other—similar to comparing average scores of students in Class A and B and getting a high t-value—means we can say one class has a significantly higher average score than the other.

To check whether the size of each balloon differs significantly, we use a t-test, which is like using a measuring tool for balloon size. We start by collecting scores from all students in Class A and Class B, which is similar to measuring balloon sizes.

Next, we calculate the average score of students in each class, which is like finding the average size of the balloons. We use the t-test to compare the averages we get, similar to using a measuring tool to see if the average balloon sizes differ significantly.

How to Read T-Test Values

If the t-value from the test is very high, it means the average score of students in one class is significantly higher than the other class. This is like saying one balloon is clearly larger than the other, and you can be confident that it's not due to chance or measurement error, like wind blowing into the room making one balloon appear bigger.

On the other hand, if the t-value is low, it means the average scores of both classes don't differ significantly, similar to finding that both balloons are roughly the same size, and you can't clearly say there's a difference.

In simple terms, a t-test is a tool used to decide whether the difference we see in a dataset is significant or not. Generally, a t-test tells us whether the mean of a sample differs significantly from the population mean or from the mean of another group, using the calculated t-value and the p-value.

T-Test and P-Value

When we perform a t-test, we get a t-statistic from our data. This t-value shows the difference between the mean we observe and the hypothesized mean under the null hypothesis.

We then compare the calculated t-statistic with the p-value at the significance level we set (e.g., 0.05).

The p-value tells us:

  • If the p-value is less than the significance level we set (e.g., 0.05), it means the results we observe are unlikely to happen by chance, and we reject the null hypothesis.
  • If the p-value is higher than the significance level, it means the results we observe could happen by chance, and we don't have enough evidence to reject the null hypothesis or accept the alternative hypothesis.

In hypothesis testing with t-tests, there are two main hypotheses:

Null Hypothesis: There is no difference between the groups or populations we're studying.

  • Example: H₀: μ₁ equals μ₂ means the mean of the first group (μ₁) equals the mean of the second group (μ₂)

Alternative Hypothesis: There is a statistically significant difference between the groups or populations.

  • Example: Hₐ: μ₁ ≠ μ₂ means the mean of the first group does not equal the mean of the second group

How Many Types of T-Tests Are There?

There are 3 types of t-tests:

1. One-Sample T-Test

A one-sample t-test is a statistical tool used to compare the mean of a sample we have with a predetermined value (called the test value or population mean) to see if there's a significant difference between the two values. This can be used in various situations.

For example:

Suppose we want to test whether a new exercise program affects the height of growing children, and we have an average height value for children this age from existing data of 150 centimeters.

We randomly select 30 children who participate in this exercise program and record their heights after 6 months in the program. We find that the average height of the sample group of children in the program is 153 centimeters.

We will use a one-sample t-test to compare the average height of children in the program (153 cm) with the expected average from the general population (150 cm).

The hypotheses we should set for the one-sample t-test are:

Null Hypothesis (H₀): The average height of children in the exercise program does not differ from the expected population average, which in this case is 150 centimeters. That is, μ equals 150 centimeters.

Alternative Hypothesis (H₁): The average height of children in the exercise program differs significantly from the general population average. That is, μ ≠ 150 centimeters.

Therefore, the t-test in this situation would be a one-tailed t-test, where we're only interested in testing whether the average height of children who participated in the program is greater than 150 centimeters.

If the p-value from the test is less than 0.05, we reject the null hypothesis (H₀) and accept the alternative hypothesis (H₁). We can conclude that the exercise program has a significant effect on increasing these children's height compared to the general average height of children of the same age.

2. Paired Samples T-Test

Also called a paired t-test, this is a statistical method used to compare the means of two related datasets.

Two related datasets mean data that comes from the same sample group in two different situations or at two different time points, such as:

  • Before and after an experiment
  • Measuring patients' blood pressure before and after medication
  • Measuring the weight of the same person before and after a weight loss program
  • Measuring test scores of students before and after attending a training course

For example:

Measuring students' test scores before and after they participate in an additional education course. We want to know whether test scores changed significantly after teaching.

In this test, we have two sets of test scores:

  • The first set is test scores before receiving teaching
  • The second set is test scores after receiving teaching

We can set the null and alternative hypotheses as follows:

Null Hypothesis (H₀): Students' test scores did not change significantly after participating in the course. This means the average of test scores before and after teaching will be equal.

Alternative Hypothesis (H₁): Students' test scores changed significantly after participating in the course. This means the average test scores after teaching are higher or lower than test scores before teaching.

Therefore, testing these hypotheses will use test score data before and after teaching from the same students to check whether there's a significant change in test scores after receiving teaching.

If the p-value obtained from the test is less than the set significance level (usually 0.05), we can reject the null hypothesis and accept the alternative hypothesis that participating in the education course affected changes in students' test scores.

3. Independent Samples T-Test

Used to compare the means of two unrelated groups. In statistics, these two groups are viewed as independent from each other, meaning measuring values in one group doesn't affect measuring values in the other group.

For example:

Measuring the difference between average weights of newborn babies in two hospitals to see if they differ. We select samples of newborn babies from each hospital and measure their weights.

After that, we use an independent t-test to compare the average weights of newborn babies from both hospitals. The null and alternative hypotheses can be written as follows:

Null Hypothesis (H₀): There is no difference in average weights of newborn babies between the two hospitals. That is, the average weight of newborn babies in Hospital A and Hospital B are equal (μA equals μB).

Alternative Hypothesis (H₁): There is a difference in average weights of newborn babies between the two hospitals. That is, the average weight of newborn babies in Hospital A does not equal the average in Hospital B (μA ≠ μB).

Therefore, the independent t-test will compare the average weights of newborn babies from both hospitals, and if the p-value obtained from the test is less than the set significance level (usually 0.05), we reject the null hypothesis H₀ and accept the alternative hypothesis H₁ that there is a significant difference in the average weight of newborn babies between the two hospitals.

T-Test Formulas

T-test formulas have several forms depending on the type of test, as follows:

Independent T-Test Formula

This is the basic formula for an independent t-test, which is used to compare the means of two separate groups:

t=Xˉ1−Xˉ2s12n1+s22n2t=\frac{\bar{X}_1-\bar{X}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}

X̄₁ and X̄₂ are the means of the two groups we want to compare (calculate the mean of each group (X̄₁ and X̄₂) separately by finding the sum of data in each group and dividing by the number of data in that group)

s₁² and s₂² are the variances of each group, which indicate how much the data in each group is spread out from the mean

n₁ and n₂ are the number of data points or sample sizes in each group

One-Sample T-Test Formula

For a one-sample t-test, used to compare a sample mean with a known population mean or predetermined value, the formula is:

t=Xˉ−μ(sn)t=\frac{\bar{X}-\mu}{\left(\frac{s}{\sqrt{n}}\right)}

X̄ is the sample mean (calculate the mean (X̄) of the sample by finding the sum of all data in that sample, then dividing by the number of data points (sample size n))

μ is the predetermined population mean or the value we want to test against the sample

s is the standard deviation of the sample

n is the number of data points in the sample

The t-value obtained from this formula tells us how much the sample mean differs from the predetermined population mean. If this t-value is significantly high or low compared to the value from the t-distribution at the set significance level (e.g., 0.05), it means the sample mean differs statistically significantly from the value we want to test.

Paired-Sample T-Test Formula

The formula for a paired-sample t-test, used to compare the means of related data in two sets, which are often before and after measurements, is:

t=dˉsd/nt=\frac{\bar{d}}{s_d / \sqrt{n}}

d̄ is the mean of the differences of paired data (measured values after the experiment minus measured values before the experiment), then find the sum of these differences and divide by the number of pairs of data

sᴅ is the standard deviation of the differences of paired data

n is the number of paired data

The t-value obtained from this calculation is compared with the value from the t-distribution at the set significance level, such as 0.05 or 5%, to see whether there's a statistically significant difference between the means before and after the experiment.

Frequently Asked Questions

A t-test is a statistical test used to compare the means of two groups or to test whether the mean of one group differs from a specified value, to see if there is a statistically significant difference using the t-value and p-value to make decisions.
The t-test value itself doesn't have a fixed threshold, but you must look at the p-value together. If the p-value is less than 0.05 (or 5%), it's considered statistically significant. A very high or low t-value (both positive and negative) usually gives a p-value less than 0.05, indicating a significant difference.
Reading t-test values involves looking at 2 main values: (1) t-value - if the t-value is very high (positive or negative), it indicates a large difference (2) p-value - if p-value < 0.05, it indicates a statistically significant difference. We reject the null hypothesis (H0) and accept the alternative hypothesis (H1) that the groups are different.
There are 3 types of t-tests: (1) One-Sample t-Test - compares the mean of a sample group with a specified value (2) Paired Samples t-Test - compares the mean of the same group measured twice (before-after) (3) Independent Samples t-Test - compares the means of two independent groups.
The t-value is a statistic calculated from data, showing the difference between means relative to the variability of the data. The p-value is the probability that the results obtained occurred by chance. If p-value < 0.05, it means there's less than a 5% chance the results occurred by chance, so we conclude there's a significant difference.
Use it when you want to compare the mean of one sample group with a predetermined value, such as testing whether the average height of children in an exercise program (153 cm) differs from the general average (150 cm), or testing whether the average student score differs from a set standard.
Paired t-test is used with data from the same sample group measured twice (such as before-after experiment, pre-test-post-test scores), while independent t-test is used with data from two separate groups with no relationship (such as comparing heights of males vs. females, weights of babies from Hospital A vs. B).
There are 3 formulas by type: (1) One-Sample: t equals (X̄ minus μ) divided by (s divided by √n) (2) Independent: t equals (X̄₁ minus X̄₂) divided by √[(s₁² divided by n₁) plus (s₂² divided by n₂)] (3) Paired: t equals d̄ divided by (sᴅ divided by √n) where X̄ equals mean, μ equals population mean, s equals standard deviation, n equals sample size, d̄ equals mean of differences.
Theoretically, t-tests assume data is normally distributed. But in practice, t-tests are robust to violations of this assumption, especially when sample size is large (n > 30) according to the Central Limit Theorem. If data is clearly non-normal, you should test normality with Shapiro-Wilk test or use non-parametric tests instead, such as Mann-Whitney U test.
The null hypothesis (H₀) is the hypothesis that there is no difference between groups or means are equal. The alternative hypothesis (H₁) is the hypothesis that there is a difference between groups. If p-value < 0.05, we reject H₀ and accept H₁ that there is a significant difference.

Wrapping Up

In this article, you learned everything about t-tests comprehensively:

T-Test Definition:

  • A t-test is a statistical test to compare means of data groups
  • Uses t-value and p-value to decide whether there's a significant difference

All 3 Types of T-Tests:

  1. One-Sample t-Test - compares the mean of a sample group with a specified value
  2. Paired Samples t-Test - compares the mean of the same group measured twice (before-after)
  3. Independent Samples t-Test - compares the means of two independent groups

How to Read T-Test Values:

  • p-value < 0.05 equals statistically significant
  • Reject the null hypothesis (H₀) and accept the alternative hypothesis (H₁)

Key Points to Remember:

  • T-tests require normally distributed data (but are robust if n > 30)
  • Choose the appropriate t-test type for your data and research question
  • p-value less than 0.05 equals significant difference

You can now understand and confidently apply t-tests in your research!