Paired t-test
The Paired t-test compares two related measurements from the same subjects — typically before-and-after measurements after an intervention. Example: comparing cholesterol levels before and after a 12-week diet program in the same patients.
Step 1 — Provide your paired data
Through the Data Input panel, you provide two columns of measurements — the "before" values in Measurement 1 and the "after" values in Measurement 2. Sources include:
- Manual entry — paste two columns of paired values
- Excel import — upload your file
- Sample data — choose from three built-in scenarios:
- Blood Pressure Medication Trial (Systolic BP, Diastolic BP, Heart Rate)
- Cognitive Training Intervention (Memory Score, Reaction Time, Attention Span)
- Weight Loss Program (Body Weight)
Step 2 — Calculate Results
Click Calculate Results. If your two columns have different lengths, the module applies intelligent pairing automatically (see Statistical methods used below).
Step 3 — Data Pairing Report (when needed)
If pairing was applied, a Data Pairing Report appears at the top of the results showing:
| Field | What it shows |
|---|---|
| Original Data | Number of observations originally in each measurement |
| Method | The pairing method that was used |
| Quality | A quality score for the pairing as a percentage |
| Used pairs | Number of paired observations the test actually used |
| Excluded | Number of observations that could not be paired |
A note clarifies: "The analysis proceeded automatically using the optimally paired data."
Step 4 — Read the results
The Test Results panel shows:
- Paired sample statistics — mean, SD, n for each measurement, plus the mean difference
- t-value and degrees of freedom
- p-value
- Effect Size (Cohen's d) with interpretation
- 95% Confidence Interval for the mean difference
- A single-line significance verdict: "There is a statistically significant difference between the paired samples (p < 0.05)" or "There is no statistically significant difference between the paired samples (p ≥ 0.05)"
Step 5 — Check the assumptions
An AssumptionChecks panel runs whenever raw paired data is available, evaluating five assumptions with a pass / warning / fail status each.
Step 6 — Power analysis
A PowerAnalysis panel reports your achieved statistical power plus a table of recommended sample sizes for 80%, 85%, 90%, and 95% power at your observed effect size, indicating how many additional pairs you would need to reach each target.
Step 7 — Discuss with the chatbot
A UniversalChatBot is available at the bottom for interpretation.
Statistical methods used
Test statistic
The Paired t-test reduces the two columns to a single column of differences and runs a one-sample t-test on those differences:
- Differences:
dᵢ = xᵢ − yᵢfor each paired observationi - t-statistic:
t = d̄ / (s_d / √n)whered̄is the mean of differences,s_dis the standard deviation of differences, andnis the number of pairs - Degrees of freedom:
df = n − 1 - 95% Confidence Interval:
d̄ ± t_critical(0.05, df) × (s_d / √n)
Hypothesis statement
- H₀: μ_d = 0 (no mean difference between paired measurements)
- H₁: μ_d ≠ 0 (two-tailed test for a non-zero difference)
- Significance level: α = 0.05
Intelligent data pairing
If the two columns do not have the same length, the module automatically pairs them. It tries four pairing methods:
| Method | What it does |
|---|---|
| Sequential | Pairs values in the order they appear |
| Nearest | Pairs each value with the closest value in the other column |
| Quantile | Pairs by matching quantile positions |
| Optimal | Tries all of the above and chooses the one with the highest pairing quality score (0–1) |
Effect size — Cohen's d
Computed as |d̄| / s_d (mean of differences divided by SD of differences, then taken as absolute value). Interpretation follows the standard Cohen (1988) convention:
| |Cohen's d| | Label |
|---|---|
| < 0.001 | No effect (identical means) |
| 0.001 – 0.20 | Very small effect |
| 0.20 – 0.50 | Small effect |
| 0.50 – 0.80 | Medium effect |
| ≥ 0.80 | Large effect |
Assumption tests
The AssumptionChecks panel runs five checks in parallel:
| Assumption | Test method | How it is judged |
|---|---|---|
| Adequate Sample Size | Counts paired observations | Pass if n ≥ 30, warning if 10 ≤ n < 30, fail if n < 10 |
| Normality of Differences | Shapiro-Wilk for n ≤ 50; Anderson-Darling for n > 50 | Normal if p > 0.05 |
| Outlier Detection | IQR (Tukey) method on the differences | Pass if 0 outliers; warning if 1–2; fail if 3+ |
| Independence of Observations | Design-based — assumed if data come from paired/repeated measures | Informational |
| Continuous Measurement Scale | Design-based | Informational |
If normality fails, the recommendation suggests considering the Wilcoxon Signed Rank Test.
Power analysis
Post-hoc power is calculated using the non-central t-distribution. The panel calculates the required sample size at four target power levels (80%, 85%, 90%, 95%) using the formula n = ((z_α + z_β) / d)².
Error handling
| Condition | Error message |
|---|---|
| n ≤ 1 | "At least 2 paired observations are required for t-test calculation" |
| All differences identical | "Standard deviation is zero — all differences are identical. Cannot perform t-test." |
| Invalid numeric data | "All data values must be valid finite numbers" |
| t-statistic non-finite | "Invalid t-statistic calculated. Check your data for extreme values." |