Calculator inputs
Example data table
| Study | Treatment mean | Treatment SD | Treatment n | Control mean | Control SD | Control n |
|---|---|---|---|---|---|---|
| Study A | 18.4 | 4.2 | 42 | 15.9 | 4.6 | 40 |
| Study B | 22.1 | 5.0 | 36 | 19.4 | 5.4 | 34 |
| Study C | 16.8 | 3.8 | 50 | 14.7 | 4.1 | 48 |
| Study D | 20.5 | 4.9 | 30 | 17.6 | 5.1 | 29 |
Formula used
This calculator assumes each study reports a treatment mean and a control mean on the same measurement scale.
Study mean difference: MDᵢ = MeanTreatmentᵢ − MeanControlᵢ
Within-study variance: Varᵢ = (SDTreatmentᵢ² / nTreatmentᵢ) + (SDControlᵢ² / nControlᵢ)
Fixed-effect weight: wᵢ = 1 / Varᵢ
Fixed pooled WMD: WMD = Σ(wᵢ × MDᵢ) / Σwᵢ
Standard error of pooled WMD: SE = √(1 / Σwᵢ)
Confidence interval: WMD ± z × SE
Heterogeneity: Q = Σ[wᵢ × (MDᵢ − WMDfixed)²], I² = max(0, (Q − df) / Q) × 100
DerSimonian–Laird tau²: τ² = max(0, (Q − df) / (Σwᵢ − Σwᵢ² / Σwᵢ))
Random-effects weight: w*ᵢ = 1 / (Varᵢ + τ²)
Use weighted mean difference only when all studies report the outcome in the same natural units.
How to use this calculator
- Enter a study label plus treatment and control summary statistics for every study.
- Provide means, standard deviations, and sample sizes on the same outcome scale.
- Choose the confidence level and select the model you want highlighted first.
- Press the calculate button to show the pooled result above the form.
- Review fixed and random estimates, then inspect Q, I², and τ².
- Use CSV or PDF export buttons to download the study table and summary.
Frequently asked questions
1. What does weighted mean difference measure?
It measures the pooled average difference between treatment and control means, while weighting each study by precision. More precise studies influence the pooled estimate more strongly.
2. When should I use weighted mean difference instead of standardized mean difference?
Use weighted mean difference when all studies report the same outcome in identical units, such as millimeters of mercury or kilograms. Use standardized mean difference when scales differ.
3. Why does this page show both fixed and random effects?
Fixed effect assumes one common true effect across studies. Random effects allows true effects to vary across studies and usually produces wider confidence intervals when heterogeneity exists.
4. What does I² tell me?
I² estimates the percentage of total variation caused by between-study heterogeneity rather than sampling error. Higher values suggest stronger inconsistency across study findings.
5. What is tau-squared?
Tau-squared is the estimated between-study variance in a random-effects model. It helps determine how much extra uncertainty should be added beyond within-study sampling error.
6. Can I include studies with different sample sizes?
Yes. Different sample sizes are expected. Studies with larger samples or lower variability usually receive larger weights because their mean differences are estimated more precisely.
7. Why might my pooled estimate be negative?
A negative pooled value means the treatment mean is lower than the control mean, based on the direction defined here as treatment minus control.
8. What are the main assumptions behind this calculator?
The calculator assumes independent studies, comparable outcome definitions, valid summary statistics, and consistent measurement units. Interpretation should also consider study quality and clinical relevance.