Confidence Interval for Paired Differences (Dependent Samples)

Use this calculator for confidence intervals when comparing two related/matched samples (before-after studies, matched pairs, repeated measures).

When to Use This Calculator

  • Paired/matched samples - same subjects measured twice or matched pairs
  • Before-after studies - testing effect of treatment on same individuals
  • Dependent samples - observations are related/correlated
  • Examples: Pre-treatment vs post-treatment, left eye vs right eye, husband vs wife

How to Use

Step 1: Paste first measurement data (comma-separated)

Step 2: Paste second measurement data (same order)

Step 3: Select confidence level (typically 95%)

Step 4: Click “Calculate”

Confidence interval for Difference of means
  Sample 1 Sample 2
Enter Data (Separated by comma ,)
Confidence Level ($1-\alpha$)
Results
Number of pairs of Obs. (n):
Mean of Diff.: $\overline{d}$
Std. Dev. of D: $s_d$
Standard Error of $\overline{d}$
Degrees of Freedom:
t critical value:
Margin of Error $E$
Lower Limit
Upper Limit

Theory

$$CI = \overline{d} \pm t_{\alpha/2, n-1} \times \frac{s_d}{\sqrt{n}}$$

Where:

  • $\overline{d}$ = mean of differences
  • $s_d$ = standard deviation of differences
  • $n$ = number of pairs
  • df = n - 1

Key Advantages

Why use paired analysis?

  • Controls for individual differences - each person is their own control
  • Increased power - better chance of detecting real differences
  • Narrower confidence intervals - individual variability removed

Worked Example

Scenario: Measuring blood pressure before/after medication on 8 patients.

Data: Before: [120, 125, 118, 130, 122, 128, 125, 132], After: [115, 118, 110, 122, 115, 120, 118, 125]

Differences: [5, 7, 8, 8, 7, 8, 7, 7]

Solution:

  • $\overline{d} = 7.125$ mmHg
  • $s_d = 0.99$ mmHg
  • $SE = 0.99/\sqrt{8} = 0.35$
  • For 95% CI, df=7, $t_{0.025,7} = 2.365$
  • E = 2.365 × 0.35 = 0.83
  • CI = 7.125 ± 0.83 = [6.30, 7.96]

Interpretation: We’re 95% confident the medication reduces blood pressure by 6.3 to 8.0 mmHg.

Related: Tutorial