Meta-analysis often requires mean and standard deviation (SD), but studies frequently report incomplete data. This guide teaches how to reconstruct SDs using two solution types:

🧩 PART 1: WITHIN-GROUP CALCULATIONS

(One group only: e.g., baseline, post-treatment, or control)

✅ 1A. Using the Standard Error (SE) to Find SD

If a study reports:

• Mean = X
• SE = s.e.
• Sample size = n

Use:

SD = SE × √n

Example:

• SE = 1.1, n = 25→ SD = 1.1 × √25 = 5.5

✅ 1B. Using the 95% Confidence Interval (CI) to Find SD

If a study reports:

• Mean = X
• 95% CI = (Lower, Upper)
• n = sample size

Step 1: Find SE from CI:

SE = (Upper − Lower) / (2 × 1.96)

Step 2: Convert SE to SD:

SD = SE × √n

Example:

• 95% CI = (88.2, 91.4), n = 30 → SE = (91.4 − 88.2) / 3.92 = 0.816 → SD = 0.816 × √30 ≈ 4.47

🧩 PART 2: BETWEEN-GROUP CALCULATIONS

(Two groups: e.g., intervention vs control)

✅ 2A. Using the p-value of a t-test to Find SD

If a study reports:

• Mean difference (MD)
• p-value (e.g., p = 0.04)
• Group sizes: n₁ and n₂ (assumed equal unless specified)

Step 1: Convert p to t-value (using inverse t-distribution)Step 2: Rearrange the t-test formula:

t = (Mean₁ − Mean₂) / SE

Step 2:

SE = MD / t

Step 3: Derive pooled SD:

SE = √((SD₁²/n₁) + (SD₂²/n₂)) → Solve for SD

This requires iteration or software to solve exactly.

✅ 2B. Using the t-value to Find SD

If t-value is reported directly:

• t = (Mean₁ − Mean₂) / SE→ Rearranged:

SE = MD / t

Then use:

SD = SE × √(n₁ × n₂ / (n₁ + n₂))

Example:

• MD = 2.5, t = 2.3, n₁ = n₂ = 40 → SE = 2.5 / 2.3 ≈ 1.087 → SD = 1.087 × √(40 × 40 / 80) ≈ 6.13

✅ 2C. Using Confidence Intervals of Mean Difference to Find SD

If the study gives:

• MD = 3.0
• 95% CI = (1.5 to 4.5)
• n₁ = n₂ = 50

Step 1: Calculate SE of the difference:

SE = (Upper − Lower) / (2 × 1.96) = (4.5 − 1.5) / 3.92 ≈ 0.765

Step 2: Use pooled SE formula to back-calculate SD:

SE = √((SD²/n₁) + (SD²/n₂)) → SE = SD × √(2/n) → SD = SE × √(n/2)

→ SD = 0.765 × √(50/2) = 0.765 × 5 = 3.83

✅ 2D. Using SE of the Mean Difference to Find SD

If you already have:

• SE of the MD
• n₁ and n₂

Use:

SD = SE × √(n₁ × n₂ / (n₁ + n₂))

This gives pooled SD under the assumption of equal variance.

🧭 Best Practice Tips

• Always check group sizes and whether equal-variance assumption is valid.
• If group sizes differ, adjust pooled SD formula accordingly.
• Avoid using median/IQR to estimate SD unless necessary (approximations less robust).