Paired t-Test Calculator

Test the mean difference in paired measurements (before and after, or matched units) with the t statistic, p-value, a confidence interval, and the effect size dz.

Result

Deterministic calculation

Mean difference (first − second)

−0.5833

Method: Paired t-test (one-sample t on the differences)
Pairs n, SD of the differences: 6, 0.2639
t statistic, degrees of freedom: −5.414, 5
p-value (two-sided): 0.00291
95% confidence interval for the mean difference: −0.8603 to −0.3063
Effect size, dz = mean difference / SD of differences: −2.21
Means of first and second: 12.32 and 12.9

Note

The p-value is the probability, if the null hypothesis and every assumption of the test were true, of a result at least as extreme as the one observed. It is not the probability that the hypothesis is true, and it says nothing about how large or important an effect is. Judge the effect size and its interval, the design and the assumptions along with it, and do not treat a p-value on one side of 0.05 as proof of an effect or of its absence.

Note

Assumptions: each pair comes from one unit (the same subject, or matched units), pairs are independent of each other, and the differences (not the raw values) are roughly normal. The pairing is the point: it removes variation between units, which an unpaired test on the same data would leave in.

Note

A before-and-after change in one group, with no control group, cannot separate the effect of the treatment from the effect of time, regression to the mean or repeated measurement.

Note

With fewer than eight pairs the test has little power, and the normality of the differences cannot be checked.

Significant figuresScientific notation

Calculation

differences dᵢ = first − second for each pair
t = mean(d) / (SD(d) / √n) = −0.5833 / 0.1078 = −5.414, with n − 1 = 5 degrees of freedom

Formula

d_i = x_{i,1} - x_{i,2}

t = \dfrac{\bar{d}}{s_d/\sqrt{n}},\quad \nu = n - 1

d_z = \dfrac{\bar{d}}{s_d}

$d$: the difference within each pair
$n$: the number of pairs

How it works

When the same subjects or samples are measured twice, or units are matched, each pair contributes one difference, and the question is whether the mean difference is zero. A paired test is a one-sample t-test on the differences. Because it compares each unit with itself, it removes the variation between units, which makes it more sensitive than an unpaired test on the same data.

The pairing has to be real: measurements on the same unit, or units matched on a meaningful characteristic. A before-and-after design with no control group cannot separate the effect of a treatment from the effect of time or repeated measurement, whatever the p-value.

Worked example

Six samples measured before and after a treatment.

The mean difference (before − after) is −0.583 with SD 0.264.
t = −5.414 with 5 degrees of freedom; p = 0.0029.
95% CI for the mean difference: −0.860 to −0.306; dz = −2.21.

The second measurements are higher by 0.58 on average (95% CI 0.31 to 0.86); p = 0.0029 under the test's assumptions.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

Each pair is one unit measured twice, and pairs are independent of each other.
The differences (not the raw values) are roughly normally distributed.
The same measurement scale and conditions for both members of a pair.

Common mistakes

Pairing samples that are not really related, which wastes degrees of freedom.
Ignoring order effects, such as a learning or drift effect between the first and second measurement.
Using it for repeated measurements at more than two time points, which needs a different analysis.

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Paired t-Test Calculator

Result

Calculation

Formula

How it works

Worked example

Assumptions

Common mistakes

Related tools

Result

Calculation