simple random sample

A sample $S$ of size $n$ from a population $U$ of size $N$ is called a simple random sample if

1. it is a sample without replacement, and
2. the probability of picking this sample is equal to the probability of picking any other sample of size $n$ from the same population $U$ .

From the first part of the definition, there are $\binom{N}{n}$ samples of $n$ items from a population of $N$ items. From the second part of the definition, the probability of any sample of size $n$ in $U$ is a constant. Therefore, the probability of picking a particular simple random sample of size $n$ from a population of size $N$ is $\binom{N}{n}^{-1}$ .

Remarks Suppose $x_1,x_2,\ldots,x_n$ are values representing the items sampled in a simple random sample of size $n$ .

• The sample mean $\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$ is an unbiased estimator of the true population mean $\mu$ .
• The sample variance $s^2=\frac{1}{n-1} \sum_{i=1}^{n}(x_i-\overline{x})^2$ is an unbiased estimator of $S^2$ , where $(\frac{N-1}{N})S^2=\sigma^2$ is the true variance of the population given by $$\sigma^2:=\frac{1}{N}\sum_{i=1}^{N}(x_i-\overline{x})^2.$$
• The variance of the sample mean $\overline{x}$ from the true mean $\mu$ is $$\left(\frac{N-n}{nN}\right) S^2.$$ The larger the sample size, the smaller the deviation from the true population mean. When $n=1$ , the variance is the same as the true population variance.