Standard deviation
Standard deviation of the mean

Often, we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by:

σ mean = 1 N σ {\displaystyle \sigma _{\text{mean}}={\frac {1}{\sqrt {N}}}\sigma }

where N is the number of observations in the sample used to estimate the mean. This can easily be proven with (see basic properties of the variance): var (X) ≡ σ X 2 var (X 1 + X 2) ≡ var (X 1) + var (X 2) {\displaystyle {\begin{aligned}\operatorname {var} (X)&\equiv \sigma _{X}^{2}\\\operatorname {var} (X_{1}+X_{2})&\equiv \operatorname {var} (X_{1})+\operatorname {var} (X_{2})\\\end{aligned}}}

(Statistical independence is assumed.) var (c X 1) ≡ c 2 var (X 1) {\displaystyle \operatorname {var} (cX_{1})\equiv c^{2}\,\operatorname {var} (X_{1})}

hence var (mean) = var (1 N ∑ i = 1 N X i) = 1 N 2 var (∑ i = 1 N X i) = 1 N 2 ∑ i = 1 N var (X i) = N N 2 var (X) = 1 N var (X). {\displaystyle {\begin{aligned}\operatorname {var} ({\text{mean}})&=\operatorname {var} \left({\frac {1}{N}}\sum _{i=1}^{N}X_{i}\right)={\frac {1}{N^{2}}}\operatorname {var} \left(\sum _{i=1}^{N}X_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{N}\operatorname {var} (X_{i})={\frac {N}{N^{2}}}\operatorname {var} (X)={\frac {1}{N}}\operatorname {var} (X).\end{aligned}}}

Resulting in: σ mean = σ N. {\displaystyle \sigma _{\text{mean}}={\frac {\sigma }{\sqrt {N}}}.}

In order to estimate the standard deviation of the mean σ it is necessary to know the standard deviation of the entire population σ beforehand. However, in most applications this parameter is unknown. For example, if a series of 10 measurements of a previously unknown quantity is performed in a laboratory, it is possible to calculate the resulting sample mean and sample standard deviation, but it is impossible to calculate the standard deviation of the mean. However, one can estimate the standard deviation of the entire population from the sample, and thus obtain an estimate for the standard error of the mean.
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