1111 LearnersLast updated on June 18th, 2025
Paired T-Test
A Paired T-Test is a type of statistical hypothesis test where the test is conducted to compare the average values or means of two related sets of observations. These observations are conducted randomly to ensure that any difference in results is due to the test itself and not other factors.
What is a Paired T-Test?
A Paired T-Test is a way of finding the mean difference between two tests conducted on the same group, an individual, object or unit. It helps in data analysis to compare two means (average) of different tests of the same group. This test is also called a dependent sample t-test. The tests are done twice in order to observe the changes that occurred before and after.
For example, a test is conducted to check if a new type of running shoes makes people run faster. The test is done for a group of people to run a track before and after using the new shoes. It helps to determine if the difference in times is just random, or if the new shoes really made a difference. The difference between the before and after test is a Paired T-test result.
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Paired Vs Unpaired T-Test
Both paired and unpaired (independent) t-test are both statistical tests used to compare the means of two groups, but they are used in different situations. Let’s understand them in detail.
Paired (dependent) T-test used when the same group of people or items is measured twice (before and after) of the tests. Whereas, an unpaired (independent) T-test are used when comparing two different groups that are unrelated.
In Paired T-test, the data sample is taken from groups which are measured twice. Whereas in Unpaired T-test, the data is taken from independent groups.
- In Paired T-test, the sample size must be equal while testing at both the times. But it is not necessary to have two equal group sizes in an Unpaired T-test.
- When Unpaired T-test compares the mean (average) difference of two groups of data, Paired T-test find the difference between the mean of the same group taken at different intervals.
How does Paired T-Test Work?
Here are some key takeaways on how to conduct a paired t-test. In order to conduct an accurate paired t-test, it is a must to follow these rules given below:
- The data you measure in a Paired T-test can be numbers that have decimals or fractions, like 12.5 or 7.8, instead of only whole numbers like 12 or 7.
- A random sample of data should be collected, meaning that the observations must be independent and not influenced by any external factors.
- Every sample or group must have the same subject. Only related samples or groups can use the paired t-test.
- The dependent variable data in a paired t-test should not have deviations or outliers (like the values that are very different from rest of the data).
- There should be an approximate distribution of the dependent variable.
After analyzing the rules of this test, using a formula, you can find the difference between the means of the two tests conducted. The formula of the paired t-test is given below:
t = d/sd/n
Where,
d = Mean of the difference between paired values.
sd = Standard deviation of the differences
n = Number of pairs
Steps to Use this Formula:
Step 1: Find the difference (d) between each pair of values.
Step 2: Calculate the mean of these differences (d).
Step 3: Find the standard deviation of these differences (sd).
Step 4: Divide sd by the square root of n.
Step 5: Divide d by the result from Step 4 to get the t-value.
Paired T-Test Table
The paired t-test table helps to determine the t-value into a statement that shows whether the results are statistically significant. The following table is provided:
| Two-Tailed Significance | ||||||
| Degrees of Freedom (n-1) | α = 0.20 | 0.10 |
0.05 |
0.02 | 0.01 |
0.002 |
|
1 |
3.078 |
6.314 |
12.706 |
31.821 |
63.657 |
318.3 |
| 2 |
1.886 |
2.92 |
4.303 |
6.965 |
9.925 |
22.327 |
| 3 |
1.638 |
2.353 |
3.182 |
4.541 |
5.841 |
10.214 |
| 4 |
1.533 |
2.132 |
2.776 |
3.747 |
4.604 |
7.173 |
|
5 |
1.476 |
2.015 |
2.571 |
3.305 |
4.032 |
5.893 |
|
6 |
1.44 |
1.943 |
2.447 |
3.143 |
3.707 |
5.208 |
|
7 |
1.415 |
1.895 |
2.365 |
2.998 |
3.499 |
4.785 |
|
8 |
1.397 |
1.86 |
2.306 |
2.896 |
3.355 |
4.501 |
| 9 |
1.383 |
1.833 |
2.262 |
2.821 |
3.25 |
4.297 |
|
10 |
1.372 |
1.812 |
2.228 |
2.764 |
3.169 |
4.144 |
|
11 |
1.363 |
1.796 |
2.201 |
2.718 |
3.106 |
4.025 |
|
12 |
1.356 |
1.782 |
2.179 |
2.681 |
3.055 |
3.93 |
|
13 |
1.35 |
1.771 |
2.16 |
2.65 |
3.012 |
3.852 |
| 14 |
1.345 |
1.761 |
2.145 |
2.624 |
2.977 |
3.787 |
|
15 |
1.341 |
1.753 |
2.131 |
2.602 |
2.947 |
3.733 |
Real life Applications of Paired T-Test
A Paired T-test is done in situations where we can compare before and after any tests, experiments, etc. Here are some of the real-life applications of Paired T-test:
- Medical Field: Scientists test a new medicine on rats, in order to find its efficiency before and after.
- Education: In schools, examinations are conducted to test the progress of students in learning.
- Business and Marketing: A company tests customer’s interest before and after running a new email advertisement campaign to the existing customers.
Common Mistakes of Paired T-Test and How to Avoid Them
The chance of making mistakes while conducting a paired T-test is very high, as the formula is sometimes daunting for students. Here are the top five mistakes that students might make and how to avoid them.
Mistake 1
Using a Paired T-test for Independent Data
The paired T-test is only for comparing before and after results of the same subjects (e.g., same person, same group, same object). If comparing and contrasting different groups, use an independent T-test instead.
Mistake 2
Not Checking for Deviation
Deviation can affect the results. Always plot the data to check for unusual values and decide whether to remove them or use a more robust test.
Mistake 3
Ignoring Sample Size
A very small sample size may not give reliable results, while a very large sample may find tiny differences that don’t matter. Use power analysis to determine the correct number of subjects.
Mistake 4
Using Raw Scores Instead of Differences
The test is based on differences between paired values, not the raw values themselves. Always calculate D = After - Before and use this difference in the test.
Mistake 5
Forgetting Assumptions of the Test
Make sure your data meets the requirements like data must be continuous, the difference must be normally distributed, each pair of values must be dependent (related to the same subject).
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Solved Examples for Paired T-Test
Problem 1
If the p- value in a paired t-test is 0.032, and the significance level () is 0.05, what conclusion should be made?
The null hypothesis is rejected
Explanation
Since p-value (0.032) < (0.05), we reject the null hypothesis, meaning there is a significant difference between the paired data.
Problem 2
Which of the following scenarios is suitable for a paired t-test? Measuring students’ test scores before and after a new study method. Comparing test scores of two different groups of students.
(a) Measuring student’s test scores before and after a new study method.
Explanation
A paired t-test is used when the same subjects are measured before and after an event. In (b), two separate groups are compared, which requires an independent t-test.
Problem 3
What is the key assumption for performing a paired t-test?
The difference between paired values should follow a normal distribution.
Explanation
The paired t-test assumes that the differences (not individual values) between paired observations should be normally distributed for valid results.
Problem 4
Which of the following is an example of paired data? (a) The weight of 50 different newborns in a hospital. (b) The weight of 10 babies before and after a new feeding formula.
(b) The weight of 10 babies before and after a new feeding formula.
Explanation
Paired data involves measuring the same subjects twice under different conditions, making (b) the correct choice.
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FAQs
1.When should you use a Paired T-test?
2.What is the null hypothesis in a Paired T-test?
3.What are the assumptions of a Paired T-test?
4.What is the difference between a Paired T-test and an independent T-test?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
