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Homesimple random sample
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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 $s^{2}=\frac{1}{n1}\sum_{{i=1}}^{{n}}(x_{i}\overline{x})^{2}$ is an unbiased estimator of $S^{2}$, where $(\frac{N1}{N})S^{2}=\sigma^{2}$ 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
$\left(\frac{Nn}{nN}\right)S^{2}.$ 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.
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