The Standard Error of the Sample Mean

The mean of a group of numbers is a linear combination of their weighted values. The mean of a group of numbers gives information regarding their central location. The weight of any given value is proportional to the number of times that value appears relative to the number of observations. The general formula for the mean is given in (1) where x is a discreet random variable and n is the number of observations included in the estimate.:

(1)

  

    \[  ∑_(i =1)^(i=n)▒〖(x_i-μ)〗^2 =∑_(i =1)^(i=n)▒〖〖(x_i-x)〗^2+ n〖(x - μ)〗^2 〗 \]

The sample mean provides a good (unbiased) estimate of the population mean, however, the sample mean will not be identical to the population mean and may differ from sample to sample. The expected variation of the sample mean about the population mean from sample to sample is called the standard error of the sample mean. It turns out that this variation is fairly predictable from the sample size and sample variance.

Since the mean is a linear combination, the variance of the mean can be calculated by summing the variances of the individual terms, assuming that the values of all the terms are uncorrelated (which they must be if the variable is independent and identically distributed). According to the Bienaymé formula, the variance of the sum of uncorrelated random variables is equal to the sum of their variances. This is presented in equation (2).

(2)

 

When trying to determine the variance of a variable x multiplied by a constant c, the following relation holds: Var (cx) =  C2Var (x). This means that equation (2) can be re-written as:

(3)

Each term in equation (3) should be thought of as representing the variance of a single observation in a certain position in a string of observations. Since the variance of every observation will be the same if they all come from the same population (and each observation is independent), the variance of each term will be the same. Additionally, the number of terms does not affect the variance of any individual term. The result of this is that the variance of each term will be equal to the population variance  and as a result, equation (3) can be further simplified as in equation (4):

(4)

To find the Standard Error of the Sample Mean, you find the square root the right-most term in equation (4) and get:

(5)

The standard error of the sample mean is a measure of the amount of variability that can be expected in a sample of a given size from a certain population. It serves as the denominator in t-tests and its square is the denominator in a one-way analysis of variance.