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

Let d̄ = mean of differences (x₂ - x₁), s_d = standard deviation of differences, n = number of pairs.

t-Statistic:

t = d̄ / (s_d / √n)

Degrees of freedom: df = n - 1

Compare calculated t with critical t-value for the chosen significance level.

---

Conceptual Method of Calculation

1. Calculate the differences d = x₂ - x₁ for each pair.
2. Compute the mean of differences (d̄).
3. Calculate the standard deviation of differences (s_d).
4. Compute the t-value: t = d̄ / (s_d / √n).
5. Determine the degrees of freedom: df = n - 1.
6. Compare the t-value with the critical t-value.
7. Interpret the result:
   • t > critical → significant difference
   • t ≤ critical → 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: d̄ = 2

Step 3: Compute standard deviation of differences: s_d = 0 (example simplified)

Step 4: Compute t-value: t = d̄ / (s_d / √n) → If s_d = 0, t is undefined, otherwise calculate normally.

Step 5: Compare with critical t-value (df = 4, α=0.05, two-tailed ≈ 2.776). Interpret significance accordingly.

---

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

---

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