Paired-samples t-test in R

Sometimes you collect data before and after an intervention – for example, stress levels before and after a mindfulness training session. Or you work with pairs, twins, or repeated measures. In such cases, you use the paired-samples t-test, also called the paired t-test.

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What does the paired t-test examine?

Definition:
The paired-samples t-test examines whether two related measurements (e.g., before and after) differ significantly in their means.
Null hypothesis: $H_0: \mu_{\text{diff}} = 0$
Alternative hypothesis: $H_1: \mu_{\text{diff}} e 0$ (two-tailed)

The mean of the differences is analyzed: For each person, the difference between the two time points is calculated, and then it is tested whether this differs from zero on average.

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Example: Stress before and after a training session

Suppose you are studying 20 people who participated in a stress-management training session. You measure their stress level before and after the session on a scale from 1–10.

# Simulate example data
set.seed(123)
stress_vorher <- rnorm(20, mean = 7, sd = 1)
stress_nachher <- stress_vorher - rnorm(20, mean = 0.8, sd = 0.5)

daten <- data.frame(stress_vorher, stress_nachher)

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Performing the test in R

# Conducting a Paired t-Test
t.test(daten$stress_vorher, daten$stress_nachher, paired = TRUE)

Important: paired = TRUE indicates that the two variables are paired observations.

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Example output (abbreviated)

Paired t-test

data:  daten$stress_vorher and daten$stress_nachher
t = 5.201, df = 19, p-value = 5.13e-05
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.662 1.391
mean of the differences: 1.027

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Interpreting the results

Statistic	Meaning
t = 5.201	Test statistic – how much the mean of the differences deviates from 0
df = 19	Degrees of freedom: number of pairs minus 1
p-value = 0.0000513	Very small p-value → the difference is significant
Confidence interval [0.66, 1.39]	With 95% confidence, the true difference lies within this range
mean of differences = 1.027	On average, stress was 1.03 points lower after the training

Conclusion: Stress levels decreased significantly after the training.

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Visualizing the differences

# Display the difference as a histogram
diff <- daten$stress_vorher - daten$stress_nachher
hist(diff, main = "Difference: stress before - after", col = "lightgray",
     xlab = "Difference values", breaks = 8)

# or boxplot
boxplot(daten$stress_vorher, daten$stress_nachher, names = c("Before", "After"),
        main = "Stress level before and after training", ylab = "Stress (1-10)",
        col = c("lightblue", "lightgreen"))

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Summary

Step	What you do
1	Prepare two paired measurements (e.g. before/after)
2	Run t.test(x, y, paired = TRUE)
3	Interpret the p-value, confidence interval, and mean difference
4	Optional: visualize with a histogram or boxplot

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Q&A for reflection

Question 1: When do you use the paired-samples t-test?
Answer: When you have two paired measurements for each case, e.g. pre–post data or repeated measures.

Question 2: What does the test specifically assess?
Answer: Whether the mean of the differences differs significantly from zero.

Question 3: How do you run it in R?
Answer: With t.test(x, y, paired = TRUE)

Question 4: What does a p-value of 0.00005 mean?
Answer: It is extremely unlikely to obtain such a difference by chance. The result is highly significant.