Paired t Test Calculator
Calculate dependent Paired $t$-test statistic ($t = \frac{\bar{d}}{s_d / \sqrt{n}}$), before/after mean differences, $df$, $p$-value, and 95% confidence intervals.
Calculate Dependent Paired t-Test (Before vs After)
Enter mean difference (d̄), SD of differences (s_d), and number of pairs (n).
Calculation Results
Calculated using Paired sample t-test equations: t = \frac{\bar{d}}{s_d / \sqrt{n}}, \quad df = n - 1
Quick Summary
The Paired t Test Calculator evaluates paired difference $t$-statistic ($t = \frac{\bar{d}}{s_d / \sqrt{n}}$), $p$-value, $df = n-1$, and 95% confidence intervals.
Formula Explanation
t = \frac{\bar{d}}{s_d / \sqrt{n}}
d_i = x_{1i} - x_{2i}, \quad \text{95\% CI} = \bar{d} \pm t^* \cdot \left(\frac{s_d}{\sqrt{n}}\right)How It Works
A paired t-test (repeated measures t-test) compares two measurements taken on the exact same subjects (e.g. Pre-Test vs Post-Test). The Paired t Test Calculator computes individual differences $d_i$, mean difference $\bar{d}$, and paired $t$-statistic.
Step-by-Step Worked Example
Practical Problem: A weight loss trial measures $n = 16$ subjects before and after diet ($\bar{d} = 3.5\text{ lbs}, s_d = 5.0\text{ lbs}$). Test if weight loss is significant ($\alpha = 0.05$, 2-tailed).
- Step 1: Calculate Standard Error of Differences ($\text{SEM}_d$): $\text{SEM}_d = 5.0 / \sqrt{16} = 5.0 / 4 = \mathbf{1.2500}$.
- Step 2: Calculate Paired t-statistic: $t = 3.5 / 1.25 = \mathbf{+2.8000}$.
- Step 3: Determine degrees of freedom: $df = 16 - 1 = \mathbf{15}$.
- Step 4: Calculate p-value: $p$-value for $t = 2.80, df = 15 \rightarrow \mathbf{0.0134}$.
- Step 5: Statistical Decision: Since $p = 0.0134 < 0.05$, reject $H_0$ (diet intervention produced a statistically significant weight loss).
Real-World Calculation Examples
Scenario 1: Weight Loss Trial (n = 16, d̄ = 3.5 lbs, sd = 5.0)
Parameters: d̄ = 3.5, sd = 5.0, n = 16
Result: t = +2.8000, p = 0.0134 (Statistically significant).
Scenario 2: Pre-Training vs Post-Training Test Scores
Parameters: n = 25 employees, d̄ = +8.2 points
Result: Significant skill improvement.
Scenario 3: Twin Study Left Eye vs Right Eye Pressure
Parameters: Matched pairs design
Result: Paired t-test evaluation.
Scenario 4: Blood Pressure Medication (Before vs 4 Weeks)
Parameters: n = 30 patients, d̄ = -6.0 mmHg
Result: Significant BP reduction.
Key Benefits of Using This Calculator
Eliminates Between-Subject Noise
Evaluates subject-level differences, dramatically increasing statistical power.
95% CI for Mean Difference
Outputs 95% confidence interval for true mean difference $\mu_d$.
One-Tailed & Two-Tailed Support
Evaluates both 1-tailed directional and 2-tailed non-directional hypothesis tests.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Paired t-test?
A Paired t-test (dependent t-test) compares two related measurements on the same subjects (e.g. Before vs After).
What is null hypothesis ($H_0$) in a paired t-test?
$H_0: \mu_d = 0$ (the mean difference between paired observations is zero).
How do I calculate a paired t-test in Excel?
Use formula =T.TEST(array1, array2, 2, 1) (Type 1 for Paired).
What non-parametric test replaces paired t-test?
The Wilcoxon Signed-Rank Test replaces paired t-test for non-normal paired data.
Why is a paired t-test more powerful than an independent 2-sample t-test?
By analyzing differences within subjects, it subtracts inter-subject variability from error variance.
What is Cohen's d for paired t-test?
$\text{Cohen's d} = \frac{|\bar{d}|}{s_d}$.
Can paired t-test be used for matched pairs?
Yes — matched pairs (such as twins or cases matched with controls) are analyzed using paired t-tests.
What are degrees of freedom for paired t-test?
$df = n - 1$, where $n$ is the number of pairs.
What assumptions are required for paired t-test?
1. Paired observations. 2. Continuous difference scores. 3. Normally distributed differences $d_i$.
What happens if a subject has missing data in one condition?
Incomplete pairs must be excluded from paired t-test analysis.