Paired (Dependent) Sample t-Test
Introduction / Background
The Paired t-Test is used to compare the means of two related samples to determine if there is a significant difference between them. It is often applied when measurements are taken on the same subjects before and after a treatment, or when two matched samples are studied.
This test assumes that the differences between paired observations are approximately normally distributed and is based on Student’s t-distribution.
Types / Variants
- One-tailed t-test: Tests if the mean difference is greater or less than zero.
- Two-tailed t-test: Tests if the mean difference is different from zero in any direction.
Formulas / Key Calculations
For immediate and precise calculations of your pre-test/post-test data, use our **highly visible** Paired T-Test Calculator tool.
Let **$\bar{d}$** = mean of differences ($x_2 - x_1$), **$s_d$** = standard deviation of differences, **$n$** = number of pairs.
t-Statistic:
$$t = \frac{\bar{d}}{(\frac{s_d}{\sqrt{n}})}$$
Degrees of freedom: $df = n - 1$
Compare calculated $t$ with critical $t$-value for the chosen significance level.
Conceptual Method of Calculation
Statistical packages like **SPSS** and R automate these steps, but understanding the conceptual method is essential for **interpreting the output** correctly. The paired t-Test is designed for measured values and should not be confused with methods like **cross-tabulation**, which are used for categorical frequency data.
- Calculate the differences $d = x_2 - x_1$ for each pair.
- Compute the mean of differences ($\bar{d}$).
- Calculate the standard deviation of differences ($s_d$).
- Compute the t-value: $t = \bar{d} / (s_d / \sqrt{n})$.
- Determine the degrees of freedom: $df = n - 1$.
- Compare the t-value with the critical t-value.
- Interpret the result:
- $t >$ critical $\rightarrow$ significant difference
- $t \leq$ critical $\rightarrow$ not significant
Illustrative Example
Suppose we measure wheat yield for the same plots before and after applying a new fertilizer:
- Before Fertilizer: [30, 32, 28, 31, 29] quintals/acre
- After Fertilizer: [32, 34, 30, 33, 31] quintals/acre
Step 1: Compute differences ($d$ = After - Before): [2, 2, 2, 2, 2]
Step 2: Compute mean difference: $\bar{d} = 2$
Step 3: Compute standard deviation of differences: $s_d = 0$ (example simplified)
Step 4: Compute t-value: $t = \bar{d} / (s_d / \sqrt{n}) \rightarrow$ If $s_d = 0$, $t$ is undefined, otherwise calculate normally.
Step 5: Compare with critical t-value ($df = 4, \alpha=0.05$, two-tailed $\approx 2.776$). Interpret significance accordingly. When writing up your findings, be sure to follow **APA format** guidelines for **academic reporting**.
Fields / Disciplines of Use
- Agriculture: Comparing yields before and after treatment
- Education: Pre-test and post-test score comparisons
- Medicine / Health: Comparing patient metrics before and after intervention
- Psychology: Measuring changes in behavior or performance within the same group
Common Mistakes / Misconceptions
- Pairs must be dependent/matched
- Assumes the differences are approximately normally distributed
- Cannot use if the pairs are independent; use **Two-Sample t-Test** instead
- Incorrectly applying the test to categorical data where a **cross-tabulation** and Chi-Square test would be appropriate.
Summary / Key Points
- Tests the difference between means of paired or matched samples
- Based on differences within each pair
- Uses Student’s t-distribution with $df = n - 1$
- Applicable in agriculture, education, health, and psychology for pre-post or matched comparisons
Related Statistical Articles
Continue your statistical study by exploring these related tests and analysis tools:
- t-Test for Means (Single and Two Samples): Analysis for Independent Samples
- Z-Test for Means: The Basic Alternative for Known Population $\sigma$
- ANOVA Calculator: For Comparing Three or More Group Means
- Descriptive Analysis: A Detail Guide to the Concept
- Parametric Tests in Statistics: A Complete Guide