# simple random sample

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

1. 1.

it is a sample without replacement, and

2. 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.

Title simple random sample SimpleRandomSample 2013-03-22 15:13:01 2013-03-22 15:13:01 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 62D05