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A Bernoulli trial is a random experiment with exactly two possible outcomes: success (valued at 1 with probability $p$) and failure (valued at 0 with probability $q = 1 - p$).
The population variance $\sigma^2$ for a Bernoulli distribution is defined as:
$$\sigma^2 = p \times (1 - p)$$
To find the unbiased sample variance ($s^2$) utilizing Bessel's correction across $n$ observations, the formula scales the proportion variance by the factor $\frac{n}{n-1}$:
$$s^2 = \frac{n}{n-1} \times p \times (1 - p)$$
Statistical variance measures how far a set of numbers is spread out from their average value. In the context of binary data models—such as coin flips, conversion tracking, or pass/fail testing—the underlying structure follows a Bernoulli process. Analyzing this variance helps researchers quantify uncertainty, construct reliable confidence intervals, and execute robust hypothesis tests across multiple scientific and commercial domains.
When working with samples rather than entire populations, correcting for degrees of freedom becomes essential. Bessel's correction divides by $n-1$ instead of $n$ to eliminate bias in the estimation of population variance. Without this adjustment, sample estimates would systematically underestimate the true variability inherent in the wider population.
What is the difference between population and sample variance in Bernoulli trials? Population variance looks at the absolute probability parameters $p$ and $q$, whereas sample variance accounts for finite sample scaling via Bessel's correction factor.
Why must the sample size be at least 2? Division by $n-1$ in the denominator causes undefined mathematical operations or infinite variance metrics when $n$ equals 1.
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