Standard error refers to the standard deviation of its sampling distribution of the particular data set or in other words, it can be said that it is the estimation of the standard deviation of the given data set. The sampling distribution of a population mean is created by the repeated recording and sampling of the means obtained from the given data set. This forms a distribution of multiple means and moreover this distribution has its unique variance and mean. Mathematically, the variance of the sampling distribution found is equal to the variance of the population divided by the sample size. This is because as the sample size enhances, sample means group more closely around the population mean. Therefore, the association between the standard deviation and standard error is such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample population.
So, let the items that you pick be represented by the random variables Xi,1≤i≤nXi,1≤i≤n, each of them identically distributed with variance σ2. They are independently sampled, so the variance of the sum is just the sum of the variances.
Next we divide by n. We know in general that Var (kY)= k2Var(Y)Var(kY)=k2Var(Y), so putting k=1/n we have
Finally take the square root to get the standard deviation σ/√n. When the population standard deviation isn't available the sample standard deviation s is used as an estimate, giving s/√n.
However, if the Xi are normally distributed, this is easy, because then the sampling distribution is also normally distributed. It can be said that 68% of samples of the mean will lie within 1 standard error of the true mean, 95% will be within 2 standard errors. In case of large sample, the central limit theorem can be applied.
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