What is the formula for a paired t-test?

The paired t-test statistic is t = (d̄ − μ₀) / (s_d / √n), where d̄ is the mean of the paired differences, μ₀ is the hypothesized mean difference (usually 0), s_d is the standard deviation of the differences, and n is the number of pairs. The result follows a t-distribution with n − 1 degrees of freedom.

What are the assumptions of a paired samples t-test?

The paired t-test assumes each observation in one sample is paired with exactly one observation in the other sample, the data is continuous, the differences between pairs are approximately normally distributed (especially important for small samples), and there are no extreme outliers among the differences.

How is a paired t-test different from an independent samples t-test?

A paired t-test compares two related measurements, such as the same subjects measured before and after a treatment. An independent samples t-test compares two separate, unrelated groups. The paired test analyzes the differences within each pair, while the independent test compares the means of two distinct populations.

Paired T-Test Calculator | statistical-calculators.site

Results

Mean Sample 1 (X̄₁):

Mean Sample 2 (X̄₂):

Mean Difference (d̄):

Std Dev of Differences (s_d):

Number of Pairs (n):

Degrees of Freedom (df):

T-Statistic (t):

Correlation (r):

P-value:

Results at Different Significance Levels (α):

α = 0.01 (1%):

α = 0.05 (5%):

α = 0.10 (10%):

Conclusion (at α = 0.05):

Interpretation:

What is a Paired Samples T-Test?

A paired samples t-test (also known as a dependent t-test) is a statistical method used to compare two related means. It is commonly applied when measuring the same group before and after an intervention, or comparing matched pairs like twins or pre-test/post-test scores. The test helps determine whether the average difference between the paired observations is statistically significant.

The Formula

The paired t-test works by reducing two related samples into a single set of differences, then running a one-sample t-test on those differences against a hypothesized value (usually 0). The test statistic is:

$$ t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}} $$

Where $\bar{d}$ is the mean of the paired differences, $\mu_0$ is the hypothesized mean difference (typically 0), $s_d$ is the standard deviation of the differences, and $n$ is the number of pairs. The test statistic follows a t-distribution with $n - 1$ degrees of freedom.

Assumptions of the Paired T-Test

Paired observations: Each value in Sample 1 must be logically linked to exactly one value in Sample 2 (e.g., the same subject measured twice).
Continuous data: The variable being measured should be on an interval or ratio scale.
Normality of differences: The differences between pairs should be approximately normally distributed — not the raw samples themselves. This assumption matters most for small samples (n < 30); the test is fairly robust to violations with larger samples.
No extreme outliers in the differences, which can distort both the mean difference and the standard deviation.

Worked Example

Using this calculator's default data — Sample 1 (before): 10, 15, 8, 11, 13; Sample 2 (after): 12, 14, 9, 11, 15 — the differences (Sample 1 − Sample 2) are: −2, 1, −1, 0, −2. This gives a mean difference d̄ = −0.8, standard deviation of differences s_d ≈ 1.30, and with n = 5 pairs (df = 4):

$$ t = \frac{-0.8 - 0}{1.30 / \sqrt{5}} \approx -1.375 $$

With df = 4 and a two-tailed test, this gives p ≈ 0.24. Since p > 0.05, we fail to reject H₀ — there isn't enough evidence, from this small sample, to conclude a significant difference between the before and after measurements.

How to Do This in Statistical Software

Using the default example data in this calculator (Sample 1: 10, 15, 8, 11, 13; Sample 2: 12, 14, 9, 11, 15), here's how to run the same test in common statistical tools.

RStudio

R's built-in t.test() function performs the paired t-test directly with paired = TRUE:

sample1 <- c(10, 15, 8, 11, 13)
sample2 <- c(12, 14, 9, 11, 15)

t.test(sample1, sample2, paired = TRUE, mu = 0, alternative = "two.sided")

Python

scipy.stats.ttest_rel runs the same paired t-test:

from scipy import stats

sample1 = [10, 15, 8, 11, 13]
sample2 = [12, 14, 9, 11, 15]

t_stat, p_value = stats.ttest_rel(sample1, sample2)
print(t_stat, p_value)

SPSS

Analyze → Compare Means → Paired-Samples T Test. Move your two related variables into the Paired Variables box as a pair (Variable 1 and Variable 2), then click OK. SPSS outputs the mean difference, standard deviation, t-statistic, degrees of freedom, and two-tailed significance.

JASP

T-Tests → Paired Samples T-Test. Move your two variables into the Paired Variables box (they should appear side by side as a pair). JASP outputs the t-statistic, df, p-value, and effect size (Cohen's d) by default.

Paired T-Test Calculator

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