Inferential Statistics • Lesson 16
Paired t-Test
Learn how to compare two related measurements using the paired t-test, calculate differences, test hypotheses, and interpret results in real-world data analytics.
What You Will Learn
1. What Is a Paired t-Test?
A paired t-test is used when the two measurements being compared are naturally connected or matched.
Instead of treating the two groups as independent, we analyze the difference within each pair.
Before vs After
Same person measured before and after an intervention.
Matched Subjects
Two observations deliberately matched on important characteristics.
Repeated Measurements
The same subject or unit measured under two conditions.
Think in Pairs
Customer
A
Before
₹2,400
After
₹2,700
Difference
₹2,700 − ₹2,400 = ₹300
2. The Key Idea: Calculate Differences
The paired t-test transforms the original paired observations into a single set of differences.
Difference for each pair
dᵢ = Afterᵢ − Beforeᵢ
Once the differences are calculated, the paired t-test essentially becomes a one-sample t-test on the differences.
Example: Website Redesign
Suppose the same five websites are measured for average page-load time before and after a performance improvement.
| Website | Before | After | Difference |
|---|---|---|---|
| A | 520 | 500 | -20 |
| B | 610 | 570 | -40 |
| C | 480 | 450 | -30 |
| D | 550 | 510 | -40 |
| E | 590 | 560 | -30 |
Differences:
−20, −40, −30, −40, −30 milliseconds
3. Hypotheses for a Paired t-Test
The hypotheses are usually written in terms of the population mean difference, represented by μd.
Null Hypothesis
H₀: μd = 0
There is no average difference between the paired measurements.
Alternative Hypothesis
Hₐ: μd ≠ 0
There is an average difference between the paired measurements.
Directional alternatives
Two-sided: Hₐ: μd ≠ 0
Greater than: Hₐ: μd > 0
Less than: Hₐ: μd < 0
4. Calculate the Mean Difference
After calculating every paired difference, calculate their average.
d̄ = Σdᵢ / n
Our example
Differences: −20, −40, −30, −40, −30
Sum = −160
n = 5
d̄ = −160 / 5 = −32 ms
5. Standard Error of the Mean Difference
The standard error is calculated using the standard deviation of the paired differences.
SE = sd / √n
Here, sd is the sample standard deviation of the differences and n is the number of pairs.
6. The Paired t-Test Formula
t = d̄ / (sd / √n)
Degrees of freedom:
df = n − 1
d̄
Mean difference
sᵈ
SD of differences
n
Number of pairs
df
Degrees of freedom
Worked Example
Suppose the five paired differences have:
Mean difference
d̄ = −32
SD of differences
sᵈ ≈ 8.37
Number of pairs
n = 5
Step 1 — Standard Error
SE = 8.37 / √5 ≈ 3.74
Step 2 — t-statistic
t = −32 / 3.74 ≈ −8.56
Step 3 — Degrees of freedom
df = 5 − 1 = 4
Important:
A large absolute t-statistic indicates that the observed mean difference is large relative to its estimated standard error. Statistical significance still requires comparing the result with the appropriate p-value or critical value.
7. Why Use a Paired Test?
Pairing can reduce the effect of differences between subjects because each subject is compared with itself or with a deliberately matched counterpart.
Example: Employee Training
Suppose the same employees take a productivity test before and after training.
8. Paired vs Two-Sample t-Test
| Feature | Paired t-Test | Two-Sample t-Test |
|---|---|---|
| Relationship | Related / matched | Independent groups |
| Main analysis | Differences within pairs | Difference between group means |
| Example | Before vs After | Group A vs Group B |
9. Important Assumptions
Each observation has a meaningful partner.
Different pairs should be independent of one another.
The outcome should be quantitative so that differences can be calculated.
For small samples, the distribution of the paired differences should be reasonably close to normal.
The pairing should make practical sense for the research question.
Interactive Practice
Test your understanding before moving ahead.
CHECK YOUR UNDERSTANDING
When is a paired t-test most appropriate?
CHECK YOUR UNDERSTANDING
What does the paired t-test analyze directly?
CHECK YOUR UNDERSTANDING
What is the usual null hypothesis for a paired t-test?
CHECK YOUR UNDERSTANDING
If n = 20 paired observations, what are the degrees of freedom?
CHECK YOUR UNDERSTANDING
Which quantity is used to calculate the standard error in a paired t-test?
CHECK YOUR UNDERSTANDING
Which example represents paired data?
Fill in the Blanks
A paired t-test analyzes the ______ between each pair of observations.
For a paired t-test, degrees of freedom are n − ______.
The standard error of the mean difference is sd divided by the square root of ______.
The standard null hypothesis for a paired t-test states that the mean difference equals ______.
Data Analytics Challenge
Marketing Campaign Performance
A company measures the same 10 stores before and after a new marketing campaign. The mean increase in weekly sales is ₹1,200, and the standard deviation of the paired differences is ₹900.
Given:
- Mean difference = ₹1,200
- SD of differences = ₹900
- n = 10
Calculate the standard error:
SE = 900 / √10
SE ≈ ₹284.60
Calculate the t-statistic:
t = 1,200 / 284.60
t ≈ 4.22
Degrees of freedom:
df = 10 − 1 = 9
Final Challenge
Before vs After Customer Spending
The same 25 customers are measured before and after a loyalty program. The average increase in spending is ₹300 and the standard deviation of the paired differences is ₹500.
Calculate:
- The standard error
- The t-statistic
- The degrees of freedom
SE = 500 / √25 = ₹100
t = 300 / 100 = 3
df = 25 − 1 = 24
Lesson Summary
✓ Use a paired t-test for related or matched observations.
✓ Calculate a difference for every pair.
✓ Analyze the mean of those differences.
✓ Standard error = sᵈ / √n.
✓ t = d̄ / (sᵈ / √n).
✓ Degrees of freedom = n − 1.
✓ Statistical significance requires a p-value or critical-value comparison; the t-statistic alone is not the final decision.