2.6
2.3022
1.029563
2.5253
4
2.6
2.3022
1.029563
2.5253
4
The Paired T-Test Calculator evaluates whether the mean difference between paired observations is significantly different from zero. Also called the dependent samples t-test, it is used when the same subjects are measured under two conditions (before/after, treatment A/treatment B) or when subjects are naturally paired (twins, matched pairs).
The paired t-test is more powerful than the independent two-sample t-test when the design involves repeated measures, because it controls for individual-level variability by analyzing the differences within each pair rather than raw group means. This makes it ideal for pre-post intervention studies, crossover clinical trials, and any within-subjects experimental design.
Enter the difference scores (d = measurement 1 - measurement 2) for each pair and specify how many pairs you have (up to 10). The calculator computes the mean difference, standard deviation of differences, standard error, t-statistic, and degrees of freedom.
The paired t-test reduces the problem to a one-sample t-test on the difference scores:
Mean Difference:
$$\bar{d} = \frac{1}{n}\sum_{i=1}^{n} d_i$$
Standard Deviation of Differences:
$$s_d = \sqrt{\frac{\sum_{i=1}^{n}(d_i - \bar{d})^2}{n - 1}}$$
Standard Error:
$$SE = \frac{s_d}{\sqrt{n}}$$
Test Statistic:
$$t = \frac{\bar{d}}{s_d / \sqrt{n}}$$
with $df = n - 1$ degrees of freedom. Under $H_0$, the mean difference is zero: $H_0: \mu_d = 0$. A large |t| provides evidence that the true mean difference is non-zero.
The Mean Difference shows the average change between the two conditions. A positive value means condition 1 produced higher values on average. The T-Statistic standardizes this difference by the standard error — compare |t| to critical values with the given df to determine significance.
The standard deviation of differences indicates consistency: a small sd means most pairs changed by a similar amount, while a large sd indicates high variability in the response.
Inputs
Results
Five participants lost an average of 2.6 kg. The t-statistic of 2.53 with 4 df suggests a meaningful effect, though the small sample means we should interpret cautiously.
Inputs
Results
Eight patients showed a mean reduction of 5.5 points. With t=-5.5 and 7 df, this is highly significant — strong evidence the treatment reduces the outcome.
Use a paired t-test when the two measurements come from the same subjects (before/after, repeated measures) or from matched pairs. Use an independent t-test when the two groups contain different, unrelated subjects. Using the wrong test can lead to incorrect conclusions.
For each pair, compute d = value_before - value_after (or condition_1 - condition_2). Be consistent with the direction. Positive differences mean the first measurement was larger. Enter these differences into the calculator.
This calculator supports up to 10 pairs for demonstration purposes. For larger datasets, compute the mean and standard deviation of your difference scores externally and use the One-Sample T-Test Calculator with those summary statistics, testing against μ₀ = 0.
Assumptions: (1) the difference scores are independent across pairs, (2) the difference scores are approximately normally distributed (important for small n), and (3) the data is measured on a continuous scale. For non-normal differences with small n, consider the Wilcoxon signed-rank test.
By analyzing differences, the paired design eliminates between-subject variability (which can be substantial). The variance of the differences is typically much smaller than the variance of the raw scores, leading to a smaller standard error and a larger t-statistic for the same effect.
Yes. While the standard null hypothesis is μ_d = 0, you could test against any hypothesized difference by subtracting it from each d_i before entering the values. However, this calculator tests the standard H₀: μ_d = 0.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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