In statistics, a t-test is a method used to check if there is a significant difference between the averages (means) of two groups. It helps us find out if the differences we see in the data are real or just happened by chance. This test is useful when comparing small groups of data and is often used in experiments or surveys.

There are three main types of t-tests:

1. One-sample t-test – This checks if the average of one group is equal to a known value (for example, checking if the average score of a class is equal to 75).
2. Independent t-test – This compares the averages of two separate groups (like boys vs. girls).
3. Paired t-test – This compares the averages of the same group at two different times (like before and after training).

In this mathematics article, we will learn about the paired t-test, paired t-test formula, paired t-test table, difference between paired and unpaired t-test, how to find the paired t-test and solve problems based on the paired t-test.

What is Paired T-Test

A paired t-test is a statistical method used to compare the averages (means) and standard deviations of two related sets of data. These two sets are usually from the same group of people or items, measured at two different times or under two different conditions. The goal is to check if there is a significant difference between them.

This test helps us decide if the change we observe is real or just happened by chance. For example, we might measure students' test scores before and after a training session. If the scores are different, the paired t-test tells us whether that difference is meaningful.

We calculate the differences between each pair, then check if those differences have a consistent pattern, using their mean and standard deviation. If the differences are large compared to their spread, we say there is a significant change.

Hypotheses of a Paired T-Test:

There are two possible hypotheses in a paired t-test.

The null hypothesis (H₀) states that there is no significant difference between the means of the two groups.

The alternative hypothesis (H₁) states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

To learn about the mean, median and mode, please click here.

Assumptions of a Paired T-Test:

The assumptions for a paired t-test are given below:

• The dependent variable is normally distributed.
• The observations are sampled independently.
• The dependent variable is measured on an incremental level, such as ratios or intervals.
• The independent variables must consist of two related groups or matched pairs.

Paired T-Test Formula

Paired t-test is a test which is based on the differences between the values of a single pair, i.e., one deducted from the other. In the formula for a paired t-test, this difference is denoted by ‘d’. The formula of the paired t-test is defined as the sum of the differences of each pair divided by the square root of n times the sum of the differences squared minus the sum of the squared differences, overall n−1.

The formula for the paired t-test is given by

\(t=\frac{\sum d}{\sqrt{\frac{n(\sum d^{2})-(\sum d)^{2}}{n-1}}}\).

Here, \(\sum d\) is the sum of the differences.

Paired T-Test Table

Paired T-test table enables the t-value from a t-test to be converted to a statement about significance. The table is given below:

Difference Between Paired and Unpaired T-Test

The difference between a paired t-test and unpaired t-test are tabulated below:

How to Find the Paired T-Test?

Let us take two sets of data that are related to each other, say X and Y with xᵢ ∈ X, yᵢ ∈ Y, where i = 1, 2, ..., n. Follow the steps given below to find the paired t-test.

Assume the null hypothesis that the actual mean difference is zero.

Determine the difference dᵢ = yᵢ − xᵢ between the set of observations.

Compute the mean difference.

Calculate the standard error of the mean difference, which is equal to S_d / √n, where n is the total number, and S_d is the standard deviation of the difference.

Determine the t-statistic value.

Refer to the t-distribution table and compare it with the tₙ₋₁ distribution. It gives the p-value.
 

Paired T-Test Summary

Here are a few important notes on the paired t-test:

A paired t-test is designed to compare the means of the same group or item under two separate scenarios.

There are two possible hypotheses in a paired t-test. The null hypothesis (H₀) states that there is no significant difference between the means of the two groups. The alternative hypothesis (H₁) states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

The formula for the paired t-test is given by
t = (Σ d) / √[(n(Σ d²) − (Σ d)²) / (n − 1)],
where Σ d is the sum of the differences.

The data is taken from subjects who have been measured twice.

95% confidence interval is obtained from the difference between the two sets of joined observations.

To learn about the formula of standard deviation, please click here.

Understanding the Assumptions for Paired Sample t-Test

Before using a paired sample t-test, we need to check some important conditions:

1. Type of Data (Level of Measurement):
   A paired t-test works best when the data is numerical and continuous. This means the values can be measured on a scale and can take on any number within a range — like height, weight, or income.
   On the other hand, discrete data has set values (like “low,” “medium,” and “high”) and doesn’t work as well for this test.
   Sometimes, if the discrete data (like survey ratings) is on a scale, we can still use it as an approximation of continuous data.
2. Independence of Data:
   Each data point should be collected independently, meaning one value doesn’t affect another.
   Although we can’t test this directly, we can assume independence if the data was collected fairly — for example, if it was randomly selected and not repeated from the same source.
3. Normal Distribution:
   The test assumes that the differences between paired values follow a normal (bell-shaped) distribution.
   You can check this visually using a histogram.
   In real-world situations, the data doesn’t need to be perfectly normal — if it looks roughly symmetric, it’s usually fine to use the paired t-test.
4. Outliers (Unusual Data Points):
   Outliers are values that are far away from the rest of the data. These can affect the accuracy of the test results.
   A boxplot is a good tool to spot outliers.
   Sometimes it’s okay to remove outliers, but doing so should be done carefully because it might also remove useful information.
   If there are too many outliers, it may be better to use a different test, like the Wilcoxon Signed Rank Test, which works better with non-normal data.

Properties of the Paired t-Test

1. Compares Two Related Groups
   • The paired t-test is used when you want to compare two sets of related data — for example, before and after treatment on the same group of people.
2. Based on the Difference
   • It calculates the difference between each pair of observations, then checks whether the average difference is significantly different from zero.
3. Data Must Be Paired
   • The data comes in pairs from the same subject or matched subjects — such as the same students tested before and after a class.
4. Requires Continuous Data
   • The values should be numeric and continuous, such as time, weight, temperature, or scores.
5. Normality Assumption
   • The differences between paired values should be approximately normally distributed (bell-shaped curve).
   • Slight deviations from normality are acceptable in larger samples.
6. Random Sampling
   • The pairs should be selected randomly from the population to ensure fair and unbiased results.
7. Dependent Samples
   • Unlike independent t-tests, the paired t-test assumes that the two sets of data are not independent, but dependent or linked.
8. Sensitive to Outliers
   • Extreme values (outliers) can affect results. It's good to check data visually using boxplots or histograms before applying the test.
9. One-Tailed or Two-Tailed Test
   • You can perform a one-tailed test (if you expect a specific direction of change) or a two-tailed test (if you just want to check for any difference).
10. Helps in Real-World Comparisons

• It's widely used in medical studies, education, psychology, and other fields to measure improvement or change in the same group over time.

Paired T-Test Solved Example

Example 1: What conclusion should be made with respect to an experiment when the p-value is 0.0680.0680.068 and the significance level α=0.05\alpha = 0.05α=0.05?

Solution:
Since the p-value 0.0680.0680.068 is greater than the significance level 0.050.050.05, we fail to reject the null hypothesis.
This means that there is not enough statistical evidence to support the alternative hypothesis at the 5% significance level.

Example 2: In which of the following cases would you use a paired-sample t-test?
(a). When comparing the same participant’s performance before and after training.
(b). When comparing two separate groups of people.

Solution: A T-test can be used in making observations on the same sample before and after an event.

In option (b) the data does not involve observations before and after an event for the same set of people.

Thus, the correct answer is (a): when comparing the same participant’s performance before and after training.