STEM Calculators

Inferences About the Difference Between Two Population Means for Paired Samples

Statistics • Estimation and Hypothesis Testing, Two Populations

Written by STEM Calculators Team Published Updated
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This is a paired samples procedure. Convert each pair to a difference di = x1i − x2i, then run a one-sample t inference on the differences. Degrees of freedom: df = n − 1.

Summary stats for the differences

Compute from differences di. If you paste paired raw data below, these fields are filled automatically.

Key formulas (paired t)
\[ \begin{aligned} d_i &= x_{1i}-x_{2i} \\ \bar d &= \frac{1}{n}\sum_{i=1}^{n} d_i \\ s_d &= \sqrt{\frac{\sum_{i=1}^{n}(d_i-\bar d)^2}{n-1}} \\ SE &= \frac{s_d}{\sqrt{n}},\quad df=n-1 \end{aligned} \]
Optional: paste paired raw data (auto-compute d̄, sd, n)

Paste the pairs as two columns (x1, x2). You can separate by commas, tabs, or spaces. Each row must contain both values. Differences are computed as d = x1 − x2.

Paired values (2 columns)

Tip: header row is allowed; non-numeric rows are ignored.

No paired raw data computed yet.
Preview differences (first 12)

Interval form: d̄ ± t* · (sd/√n).

Ready
t model visualization (paired t, df = n − 1)
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The shaded area updates after calculation.
Differences preview
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Paste paired data to see the differences.

When this method applies

  1. Observations are paired/matched (same subjects measured twice, before/after, matched items, etc.).
  2. Analyze the differences di = x1i − x2i.
  3. The distribution of differences is approximately normal, or n is large enough for the t procedure to be reliable.
What is being estimated/tested
\[ \mu_d = \mu_1-\mu_2 \] The calculator performs a one-sample t procedure on d.
Enter values and click Calculate.
Batch mode: paste CSV (paired summary stats per row)

Each row should contain summary stats for the differences: dbar, sd, n, and optional: task (ci/ht), conf, alpha, delta0, alt (two/gt/lt).

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Frequently Asked Questions

What is a paired samples t test used for?

A paired samples t test is used when each subject contributes two related measurements, such as before and after values, or when observations are matched in pairs. The analysis focuses on the differences within pairs and tests whether the mean difference mu_d equals a hypothesized value.

How is the test statistic computed in a paired t procedure?

First compute each pair difference d_i, then summarize with dbar and s_d. The test statistic is t = (dbar - d0) / (s_d / sqrt(n)) with df = n - 1.

What assumptions does the paired t method require?

The pairs must be meaningfully matched or come from the same subjects, and the differences should be approximately normally distributed for small n. The pairs should be independent of each other across subjects.

How do I interpret a confidence interval for mu_d?

A confidence interval gives a plausible range of values for the true mean difference mu_d at the chosen confidence level. If the interval includes 0, the data are consistent with no average change (or no average difference) at that confidence level.

When should I use a paired t method instead of an independent two-sample t method?

Use the paired method when measurements are linked by subject or matching, so analyzing within-pair differences is appropriate. Use the independent two-sample method when the two samples come from unrelated groups.

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