Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Pooled sample variance

Definition: Let $x_i = \left\lbrace x_{11}, \ldots, x_{1n_i} \right\rbrace$ for $i = 1,\ldots,k$ be samples from a random variable $X$ whose expected value, but not variance potentially depends on sample index $i$. Then, the pooled sample variance of $x = \left\lbrace x_1, \ldots, x_k \right\rbrace$ is given by

\[\label{eq:var-pool} s^2_{1...k} = \frac{1}{n-k} \sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2\]

where $\bar{x}_i$ is the sample mean of the $i$-th sample:

\[\label{eq:mean-samp} \bar{x}_i = \frac{1}{n_i} \sum_{j=1}^{n_i} x_{ij} \; .\]
 
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Metadata: ID: D212 | shortcut: var-pool | author: JoramSoch | date: 2025-01-10, 17:40.