Wan's Estimator

Wan's Estimator for Meta-Analysis

Scenario 1: Min, Median, Max, n

Min:
Median:
Max:
Sample Size (n):

Scenario 2: Min, Q1, Median, Q3, Max, n

Min:
Q1:
Median:
Q3:
Max:
Sample Size (n):

Scenario 3: Q1, Median, Q3, n

Q1:
Median:
Q3:
Sample Size (n):

🎯 Purpose

In meta-analyses, especially of clinical studies, researchers often encounter studies reporting medians, ranges, and quartiles instead of means and SDs. Wan et al. (2014) developed a validated approach to estimate the sample mean and SD from incomplete summary data, allowing inclusion in meta-analysis.

Wan’s Calculator automates three estimation scenarios depending on available statistics.

🧩 Scenario 1: Minimum, Median, Maximum, Sample Size

📌 Required Inputs:

Minimum (a)
Median (m)
Maximum (b)
Sample size (n)

📘 Estimation Formulas:

Mean

Estimated Mean: $Estimated Mean \approx \frac{a + 2 m + b}{4}$

Standard Deviation (SD)

Estimated SD: $Estimated SD = \frac{b - a}{C (n)}$

Where

a: Minimum
b: Maximum
m: Median
n: Sample size
C(n): Adjustment factor based on sample size (Wan et al.)

For n < 25: use specific lookup values
For n ≥ 25: use normal approximation

Example (n = 20):

Min = 40, Median = 60, Max = 100 →Mean ≈ (40 + 2×60 + 100) / 4 = 65 →SD ≈ (100 − 40) / 4.06 = 14.78

Note: The calculator automates lookup or computes C(n) using default functions.

🧩 Scenario 2: Minimum, Q1, Median, Q3, Maximum, Sample Size

📌 Required Inputs:

Minimum (a)
First quartile (Q1)
Median (m)
Third quartile (Q3)
Maximum (b)
Sample size (n)

📘 Estimation Formulas:

Mean

Estimated Mean: Estimated Mean ≈ $\frac{a + 2 Q_{1} + 2 m + 2 Q_{3} + b}{8}$

Standard Deviation (SD)

Estimated SD: $\sqrt{\frac{{(b - a)}^{2} + {(Q_{3} - Q_{1})}^{2}}{2 C (n)}}$

Where

a: Minimum
b: Maximum
Q1: First quartile
Q3: Third quartile
m: Median
n: Sample size
C(n): Sample size–based adjustment factor

Example (n = 40):

a = 30, Q1 = 45, m = 50, Q3 = 55, b = 70 →Mean ≈ (30 + 2×45 + 2×50 + 2×55 + 70)/8 = 50 →SD ≈ √[ ((70−30)² + (55−45)²) / (2 × 4.21) ] ≈ 12.02

🧩 Scenario 3: Q1, Median, Q3, Sample Size

📌 Required Inputs:

First quartile (Q1)
Median (m)
Third quartile (Q3)
Sample size (n)

📘 Estimation Formulas:

Mean

Estimated Mean: $Estimated Mean \approx \frac{Q_{1} + m + Q_{3}}{3}$

Standard Deviation (SD)

Estimated SD: $Estimated SD = \frac{Q_{3} - Q_{1}}{A (n)}$

Where

Q1: First quartile
m: Median
Q3: Third quartile
n: Sample size
A(n): Quartile-based adjustment factor

Where A(n) is another adjustment factor depending on n:

For n ≥ 25, A(n) ≈ 1.35

Example (n = 50):

Q1 = 40, m = 50, Q3 = 60 →Mean ≈ (40 + 50 + 60)/3 = 50 →SD ≈ (60 − 40)/1.35 ≈ 14.81

🧠 Key Notes

Wan’s method assumes that data follow roughly symmetric distributions.
Use Scenario 2 when all five summary statistics are available (most precise).
The calculator handles small and large n with different correction constants.
Always specify which estimation method you used when reporting meta-analysis inputs.

Estimating Mean and SD Using Wan’s Method for Meta-Analysis

Wan's Estimator for Meta-Analysis

Scenario 1: Min, Median, Max, n

Scenario 2: Min, Q1, Median, Q3, Max, n

Scenario 3: Q1, Median, Q3, n

🎯 Purpose

🧩 Scenario 1: Minimum, Median, Maximum, Sample Size

📌 Required Inputs:

📘 Estimation Formulas:

Mean

Standard Deviation (SD)

Where

🧩 Scenario 2: Minimum, Q1, Median, Q3, Maximum, Sample Size

📌 Required Inputs:

📘 Estimation Formulas:

Mean

Standard Deviation (SD)

Where

🧩 Scenario 3: Q1, Median, Q3, Sample Size

📌 Required Inputs:

📘 Estimation Formulas:

Mean

Standard Deviation (SD)

Where

🧠 Key Notes

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