Pooled Mean and SD Calculator

Pooled Mean and Standard Deviation Calculator

Enter subgroup data below (leave blank rows empty):

🎯 Purpose

In clinical research and meta-analysis, it's common to have results from subgroups (e.g., age strata, treatment arms, centers). When each subgroup reports its mean, standard deviation (SD), and sample size, we need to aggregate these into a total (pooled) mean and SD to summarize overall effect or baseline characteristics.

This method is precise and accounts for both the group sizes and spread of data (variance), rather than simply averaging group means.

🧩 When to Use

You are synthesizing subgroups (e.g., males and females) into one group summary.
You are pooling data across centers or study sites.
The overall mean and SD are not directly reported, but you have per-group mean, SD, and n.

📌 Required Inputs

Subgroup Parameters

${\overline{X}}_{i}$ : Mean of group i
${SD}_{i}$ : Standard deviation of group i
$n_{i}$ : Sample size of group i

These are entered into light blue cells in the spreadsheet.

🧮 Step-by-Step Calculations

✅ 1. Total Sample Size

$N = \sum_{i = 1}^{k} n_{i}$

✅ 2. Pooled (Overall) Mean

${\overline{X}}_{total} = \frac{\sum_{i = 1}^{k} n_{i} {\overline{X}}_{i}}{\sum_{i = 1}^{k} n_{i}}$

Each group's mean is weighted by its sample size.

✅ 3. Pooled (Overall) Standard Deviation

Pooled Variance:

${Var}_{pooled} = \frac{\sum_{i = 1}^{k} (n_{i} - 1) \cdot {SD}_{i}^{2} + \sum_{i = 1}^{k} n_{i} {({\overline{X}}_{i} - {\overline{X}}_{total})}^{2}}{N - 1}$

Pooled Standard Deviation:

${SD}_{pooled} = \sqrt{{Var}_{pooled}}$

This formula correctly includes both the within-group variation (weighted SDs of each group) and the between-group variation (differences between group means and the overall mean), all scaled by the total degrees of freedom $ N - 1 $.

🔢 Example

Suppose you have 2 groups:

Group A: n = 30, mean = 100, SD = 15
Group B: n = 20, mean = 110, SD = 20

Step 1: Total Sample Size

$N = 30 + 20 = 50$

Step 2: Pooled Mean

${\overline{X}}_{total} = \frac{30 \times 100 + 20 \times 110}{50} = \frac{3000 + 2200}{50} = 104$

Step 3: Variance Components

Within-Group Variance

$(30 - 1) \times 15^{2} + (20 - 1) \times 20^{2} = 29 \times 225 + 19 \times 400 = 6525 + 7600 = 14125$

Between-Group Variance

$30 \times {(100 - 104)}^{2} + 20 \times {(110 - 104)}^{2} = 30 \times 16 + 20 \times 36 = 480 + 720 = 1200$

Total Variance

$Var = \frac{14125 + 1200}{49} \approx 313.78$

Pooled Standard Deviation

${SD}_{pooled} = \sqrt{313.78} \approx 17.72$

🧭 Usage Notes

The spreadsheet automates all calculations. Just fill in light blue boxes with subgroup data.
Leave unused rows blank; the sheet ignores them.
Useful in descriptive tables, forest plots, or protocol summaries.

Calculating Pooled Mean and SD from Subgroup Data: Meta-Analysis Guide