Can You Convert Median and IQR to Mean and SD?

✅ Short Answer:

Yes, approximately—but only under strong assumptions, such as a symmetric or normal distribution. Several methods exist for estimating mean and standard deviation (SD) from median, IQR, min, and max, particularly useful in meta-analyses.

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🧠 Assumptions

To convert, we assume:

• The variable is approximately normally distributed or not severely skewed

• The sample size (n) is available
  

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📌 Common Formulas (Wan et al., 2014 / Luo et al., 2018)

1. Estimate Mean

If only median (M), IQR (Q1, Q3) and sample size (n) are known:

Mean ≈ (Q1 + Median + Q3) / 3

2. Estimate Standard Deviation (SD)

From IQR:

SD ≈ (Q3 - Q1) / 1.35

With min (a) and max (b) and median: If n ≥ 25:

SD ≈ (b − a) / 4

Or more accurate:

SD ≈ (Q3 − Q1) / 1.35

🩺 Example

Given:

• Median = 60

• Q1 = 50

• Q3 = 70

• n = 50

Estimate:

Mean ≈ (50 + 60 + 70)/3 = 60 SD ≈ (70 - 50)/1.35 ≈ 14.81

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⚠️ Limitations

• Only valid for symmetric or normal-like distributions

• Can underestimate variability in skewed data

• Prefer original data or bootstrapping if available
  

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📊 In Stata

If converting reported summary data (not raw):

stata: gen mean = (q1 + median + q3)/3 gen sd = (q3 - q1)/1.35