---
tag: 6B05T6XHY0
kind: definition
name: Simple random sample
msc: [62D05]
defines: Y0PTCJBPEA
uses: [098PHYBTV9, 0GBZM0MT0F, 0NFPJVJJHX]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_random_sample
---

## Definition

Let $U = \set{1, \dots, N}$ be a finite population of $N$ units, and let $1 \le n \le N$.
Let $S \subseteq U$ be a random subset, defined on a [[probability space|0NFPJVJJHX]].
Then $S$ is a *simple random sample* of size $n$ from $U$ if for every subset $s \subseteq U$ with $\abs{s} = n$,
$$P(S = s) = \binom{N}{n}^{-1}.$$
Here $\binom{N}{n}$ is the [[binomial coefficient|0GBZM0MT0F]], the number of [[combinations|098PHYBTV9]] of $n$ units out of $N$.

## Remarks

Equivalently, one draws $n$ units one after the other, each uniformly at random among the units not yet drawn.
Each unit is in $S$ with probability $n/N$.
For values $y_1, \dots, y_N$ attached to the units, the sample mean $\bar y_S = \frac{1}{n}\sum_{i \in S} y_i$ is unbiased for the population mean $\bar Y$.
Its variance is $\big(1 - \frac{n}{N}\big) \frac{S_y^2}{n}$, where $S_y^2 = \frac{1}{N-1} \sum_{i=1}^{N} (y_i - \bar Y)^2$.
Sampling *with replacement*, where $n$ independent uniform draws are made, is a different design.

Example: $N = 4$ and $n = 2$, with $S$ uniform on the $\binom{4}{2} = 6$ pairs.
Each pair has probability $1/6$, as required, and unit $1$ lies in $3$ of the pairs, so it is chosen with probability $1/2 = n/N$.

Non-example: with $N = 4$, choose $r \in \set{1, 2}$ uniformly and take $S = \set{r, r + 2}$.
Every unit is chosen with probability $1/2$, but the pair $\set{1, 2}$ has probability $0$ instead of $1/6$.
So equal inclusion probabilities are not enough for a simple random sample.