Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the combined mean and variance calculator works
SScombined = Σ[(nᵢ−1)sᵢ²+nᵢ(x̄ᵢ−x̄combined)²]; s²combined = SScombined/(Σnᵢ−1).
The combined variance describes the union of observations; it is generally larger than a within-group pooled variance when group means differ.
Worked example
Combined Mean and Variance example
Two groups of 10 with means 4 and 6 combine to mean 5, while total variance includes both within- and between-group spread.
SScombined = Σ[(nᵢ−1)sᵢ²+nᵢ(x̄ᵢ−x̄combined)²]; s²combined = SScombined/(Σnᵢ−1).
Supported inputs
Precision and limits
Model and design
Each supplied standard deviation must be a sample SD for its complete group summary, and groups must not overlap.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
The combined variance describes the union of observations; it is generally larger than a within-group pooled variance when group means differ.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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