Calculate Pooled Variance
Online calculator to compute the pooled variance of two data series
Pooled Variance Calculator
The Pooled Variance
The pooled variance (also called combined variance) is a method for estimating the variance of different populations, when the mean can be different, but the variance is assumed to be equal.
Pooled Variance Concept
The pooled variance combines the variances of two samples.
It is weighted according to sample sizes.
● Sample X ● Sample Y ● Pooled Variance Sp²
What is Pooled Variance?
The pooled variance (also combined or composite variance) is an important statistical concept:
- Definition: Weighted average of the variances of two or more samples
- Assumption: Population variances are equal (homoscedasticity)
- Weighting: By degrees of freedom (n-1) of individual samples
- Application: t-test for independent samples, ANOVA
- Prerequisite: Both populations have equal variance
- Advantage: Better estimation by combining information
Calculating Pooled Variance
The calculation is performed in several steps:
Steps
- Sample: Uses n-1 and m-1 (Bessel's correction)
- Population: Uses n and m (without correction)
- Larger Values: Higher spread in data
- Usage: Standard error calculation for t-tests
Interpretation
Applications of Pooled Variance
The pooled variance is used in many statistical procedures:
Statistical Tests
- t-test for independent samples
- Analysis of variance (ANOVA)
- Confidence intervals for differences
- Statistical process control
Practical Applications
- Clinical trials: Comparing treatment groups
- Quality control: Comparing production batches
- A/B Testing: Comparing variants
- Market research: Comparing target groups
Formulas for Pooled Variance
Pooled Variance (Sample)
Used for samples with Bessel's correction (n-1, m-1)
Sample Variance
Sample variance with Bessel's correction
Symbol Explanations
| \(S_p^2\) | Pooled variance |
| \(S_x^2\) | Variance of sample X |
| \(S_y^2\) | Variance of sample Y |
| \(n\) | Number of values in X |
| \(m\) | Number of values in Y |
| \(\overline{x}\) | Sample mean |
Example Calculation for Pooled Variance
Given
Calculate: Pooled variance for samples X and Y
1. Calculate Means
Arithmetic mean for both data series
2. Calculate Variance of X
Sum of squared deviations divided by (n-1)
3. Calculate Variance of Y
Same calculation as for X
4. Pooled Variance
Weighted average of both variances
5. Complete Result
The pooled variance estimates the common variance of both populations
Mathematical Foundations of Pooled Variance
The pooled variance is a fundamental concept in inferential statistics, used when combining information from multiple samples.
Prerequisites and Assumptions
Certain conditions must be met for correct application of pooled variance:
- Homoscedasticity: The population variances σ₁² and σ₂² are equal
- Independence: The two samples are independent of each other
- Normal Distribution: Ideally, data are normally distributed (for small samples)
- Random Samples: Data were randomly drawn from populations
- Interval Scale: Data lie on interval or ratio scale
Interpretation and Significance
Pooled variance has important statistical interpretation:
Weighting
Larger samples automatically receive more weight in calculation, as they provide more precise estimates of population variance.
Efficiency
By combining information from both samples, we get a more precise estimate of common variance than from individual samples.
Degrees of Freedom
The sum n+m-2 in denominator corresponds to combined degrees of freedom of both samples (n-1 for X, m-1 for Y).
Usage in t-Test
Pooled variance is essential for t-test with independent samples under assumption of equal variances.
Sample vs. Population
The calculator computes both variants of pooled variance:
Sample Variance
Uses Bessel's correction (n-1, m-1) in numerator and (n+m-2) in denominator. This is an unbiased estimator of population variance and is used for inferential statistics.
Population Variance
Uses n and m without correction. This describes variance in present data without inference to larger population. Less frequently used.
Advantages and Disadvantages
Advantages
- Precision: Better estimation through more data points
- Efficiency: Optimal when population variances are equal
- Standard Method: Widely used and established
- Mathematical Elegance: Simple, intuitive formula
Limitations
- Assumption of Equal Variances: Can lead to errors with heteroscedasticity
- Sensitivity: Sensitive to violation of prerequisites
- Sample Size: With very different n, m, weighting can be problematic
- Alternative Methods: Welch's test preferable for unequal variances
Summary
Pooled variance is an important tool in comparative statistics that enables precise estimation of common variance of two populations. However, its correct application requires meeting certain prerequisites, particularly homoscedasticity. In practice, the assumption of equal variances should be checked with appropriate tests (e.g., Levene's test, F-test) before using pooled variance.
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