How to Compute Standard Deviation of the Mean
In statistics, the standard deviation of the mean-commonly referred to as the Standard Error of the Mean (SEM)-measures how accurately a sample mean represents the true population mean.
While the standard sample standard deviation ($s$) measures the variability of individual observations within a single sample dataset, the standard error ($SE_{\bar{x}}$) measures the variability of sample means if you were to repeat the experiment multiple times.
This guide explains the formula, provides a step-by-step worked example, and demonstrates how to calculate it using our Standard Deviation Calculator.
The Formula for Standard Deviation of the Mean
The standard error of the mean is calculated by dividing the sample standard deviation ($s$) by the square root of the total sample size ($n$):
$$\text{SEM} = \frac{s}{\sqrt{n}}$$
Where:
- $s$ = Sample standard deviation
- $n$ = Total number of observations in the sample
- $\sqrt{n}$ = Square root of the sample size
Step-by-Step Calculation Example
Suppose a laboratory measures test scores across a sample of $n = 25$ students, finding a sample mean of $\bar{x} = 85$ and a sample standard deviation of $s = 10$.
Step 1: Calculate the square root of the sample size ($n$)
$$\sqrt{n} = \sqrt{25} = 5$$
Step 2: Divide sample standard deviation ($s$) by $\sqrt{n}$
$$\text{SEM} = \frac{10}{5} = 2.0$$
Interpretation: The estimated average distance between our sample mean (85) and the true overall population mean is 2.0 points.
Standard Deviation vs Standard Error
| Metric | Standard Deviation ($s$) | Standard Error ($\text{SEM}$) |
|---|---|---|
| What it Measures | Spread of individual data points | Uncertainty of the sample mean |
| Formula | $\sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ | $\frac{s}{\sqrt{n}}$ |
| Effect of Sample Size | Remains stable as $n$ grows | Decreases as $n$ grows |
Automate Your Statistical Calculations
Calculate sample standard deviation, population standard deviation, variance, and sample means instantly using our free Standard Deviation Calculator. For more tools, visit our Math & Education Category.
For a complete beginner's step-by-step statistical guide, see our article on Standard Deviation Step-by-Step.
