# Standard Error

The standard error of the mean (SEM) can be expressed as:

![{\displaystyle {\sigma }\_{\bar {x\}} ={\frac {\sigma }{\sqrt {n\}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9dac77577c2717cbb973388e4d6563915705742)

where σ is the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation) of the population.

n is the size (number of observations) of the sample.

The **standard error** of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the **standard deviation** of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.

the standard error provides an estimate of the precision of a parameter (such as a mean, proportion, odds ratio, survival probability, etc) and is used when one wants to make inferences about data from a sample (eg, the sort of sample in a given study) to some relevant population \[[4](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3148365/#CR4)]. When the standard error relates to a mean it is called the standard error of the mean; otherwise only the term standard error is used. For instance, in the previous example we know that average size of the tumor in the sample is 7.4 cm, but what we really would like to know is the average size of the tumor in the entire population of interest (ie, all patients with such tumors, not just those in the study). We can take the sample mean as our best estimate of what is true in that relevant population but we know that if we collect data on another sample, the mean will vary according to what is called the “sampling distribution.”
