# Standard Error of the Mean

The standard error of the mean is a method used to evaluate the standard deviation of a sampling distribution. It is also called the standard deviation of the mean and is abbreviated as SEM. For instance, usually, the population mean estimated value is the sample mean, in a sample space. But, if we pick another sample from the same population, it may give a different value.

Hence, a population of the sampled means will occur, having its different variance and mean. Standard error of mean could be said as the standard deviation of such a sample means comprising all the possible samples drawn from the same given population. SEM represents an estimate of standard deviation, which has been calculated from the sample.

## Formula

The formula for standard error of the mean is equal to the ratio of the standard deviation to the root of sample size.

SEM = SD/√N

Where ‘SD’ is the standard deviation and N is the number of observations.

### How to calculate standard error of mean?

The standard error of the mean (SEM) shows us how the mean varies with different experiments, evaluating the same quantity. Thus, if the result of random variations is essential, then the SEM will have a higher value. But, if there is no change recognised in the data points after repeated attempts, then the value of the standard error of the mean will be zero.

Let us solve an example to calculate the standard error of mean.

Example: Find the standard error of mean of given observations,

x= 10, 20,30,40,50

Solution: Given,

x= 10, 20,30,40,50

Number of observations, n = 5

Hence, Mean = Total of observations/Number of Observations

Mean = (10+20+30+40+50)/5

Mean = 150/5 = 30

By the formula of standard error, we know;

SEM = SD/√N

Now, we need to find the standard deviation here.

By the formula of standard deviation, we get;

$SD = \sqrt{(1/N-1)\times ((x_{1}-x_{m})^{2})+(x_{2}-x_{m})^{2})+….+(x_{n}-x_{m})^{2})} \\ SD = \sqrt{(1/5-1)\times ((10-30)^{2})+(20-30)^{2})+(30-30)^{2}+(40-30)^{2}+(50-30)^{2}} \\ SD = \sqrt{1/4((-20)^2+(-10)^2+(0)^2+(10)^2+(20)^2} \\ SD = \sqrt{1/4(400+100+0+100+400)} \\ SD = \sqrt{250} \\ SD = 15.811$

Therefore, putting the values of standard deviation and root of number of observations, we get;

Standard error of mean, SEM = SD/√N

SEM = 15.811/√5

SEM = 15.8114/2.2361

SEM = 7.0711