Student's t-Distribution · Dependent Samples
Paired Samples t-Test
Test whether the mean difference between two related measurements is significantly different from zero. Ideal for pre-test/post-test designs, matched pairs, and repeated measures on the same participants.
Data entry — paired observations
n = 0
n = 0
Paired requirement: Each value in Condition 1 must correspond directly to the same value in Condition 2 — same participant, same position. Both columns must have the same number of values.
Core statistical results
Critical t values
df = n − 1 · t-critical from Student's t-distribution
Confidence interval for mean difference (μd)
Difference scores (d = Pre − Post)
Pre vs Post comparison — group means with individual scores
Distribution of difference scores
Effect size
APA 7th edition reporting statement
Copy-paste template
Calculation details
Interpretation Report
Step-by-step analysis & plain-language conclusions
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Hypotheses
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Assumptions check
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Difference scores analysis
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Computed statistical values
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Statistical decision
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Confidence interval interpretation
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Effect size & practical significance
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Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How the Paired Samples t-Test Works
Complete theoretical foundation, assumptions, design considerations, effect size, and primary references.
What is the paired samples t-test?
Introduced by William Sealy Gosset (publishing under the pseudonym "Student," 1908), the paired samples t-test (also called the dependent samples t-test or repeated-measures t-test) tests whether the mean of a set of difference scores is significantly different from zero. It is used when the same participants are measured twice under two different conditions (pre/post), or when two measurements are taken from matched or related pairs.
t = d̄ / (Sd / √n)
where:
dᵢ = x₁ᵢ − x₂ᵢ (difference score for pair i)
d̄ = Σdᵢ / n (mean of difference scores)
Sd = √[Σ(dᵢ − d̄)² / (n−1)] (SD of difference scores)
SE = Sd / √n (standard error of d̄)
df = n − 1
Confidence interval for μd:
CI = d̄ ± t_crit(df, α/2) × SE
Cohen's d (effect size):
d = d̄ / Sd
When to use the paired t-test
- Pre-test / post-test designs — the same participants measured before and after an intervention.
- Matched pairs — two participants matched on relevant characteristics (e.g., age, gender, baseline scores), one in each condition.
- Crossover designs — each participant receives both treatments at different times (counterbalanced).
- Self-as-control — measuring the same body part twice (e.g., left vs right hand strength), or the same participants in two environmental conditions.
Assumptions
- Paired observations — each score in Condition 1 is logically linked to exactly one score in Condition 2.
- Continuous dependent variable — measured at interval or ratio scale.
- Normality of difference scores — the differences d = x₁ − x₂ should be approximately normally distributed. With n ≥ 30, the Central Limit Theorem makes the t-test robust. For small n, test normality with Shapiro-Wilk and inspect a histogram of d.
- No extreme outliers in the differences — outliers in d inflate or deflate the mean and SD, distorting t and Cohen's d.
- Independence of pairs — pairs are independent of each other (one participant's scores do not influence another's).
Effect size — Cohen's d (paired)
For paired t-tests, Cohen's d is calculated as d̄ / Sd (the mean difference divided by the SD of the differences). Benchmarks (Cohen, 1988):
- |d| < 0.20 — negligible
- |d| = 0.20–0.49 — small effect
- |d| = 0.50–0.79 — medium effect
- |d| ≥ 0.80 — large effect
Cohen's d for paired designs is typically larger than for independent samples because the pairing removes between-participant variance, increasing sensitivity.
Paired vs independent samples t-test
- Use paired t-test when there is a logical one-to-one pairing between scores in the two conditions — same participant, matched pair, or related measurement.
- Use independent samples t-test when the two groups are completely separate — different participants assigned to different conditions with no matching relationship.
- Advantage of pairing — removes individual differences (between-subject variance), increases statistical power, and allows detection of smaller effects with fewer participants.
- Disadvantage — must measure both conditions, which can introduce carryover effects in designs where order matters.
APA 7th reporting format
Template: "A paired samples t-test was conducted to compare [Condition 1] and [Condition 2]. There was a [significant/non-significant] difference in scores for [Condition 1] (M = [x̄₁], SD = [SD₁]) and [Condition 2] (M = [x̄₂], SD = [SD₂]); t([df]) = [t value], p = [p value], d = [Cohen's d], 95% CI [lower, upper]."
Primary references
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2019). How to design and evaluate research in education (10th ed.). McGraw-Hill.
Student [Gosset, W. S.]. (1908). The probable error of a mean. Biometrika, 6(1), 1–25.
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science. Frontiers in Psychology, 4, 863.