Standard Error of the Mean
Standard deviation of the sample mean.
Formula
Variables
Example
With σ = 10 and n = 100, the standard error of the mean is 10/√100 = 1.
How it works
The standard error of the mean tells you how much the sample average would typically vary from one random sample to another if you repeated the sampling process many times. It is smaller than the population standard deviation because averaging n independent observations reduces random fluctuation by a factor of the square root of n. This comes from the fact that variances add for independent variables, so the variance of the mean is σ²/n and the standard deviation of that mean is σ/√n.
When to use it
Use this when you want the precision of a sample mean, especially in inference, confidence intervals, and hypothesis tests. It is common in experimental science, survey analysis, quality control, and any setting where a mean is estimated from repeated or sampled measurements.
Worked example, step by step
Suppose the population standard deviation is σ = 18 and the sample size is n = 81. SE = σ / √n SE = 18 / √81 SE = 18 / 9 SE = 2 So, the standard error of the mean is 2.
Common mistakes to avoid
- Using the sample standard deviation s in the formula without noting that this changes the interpretation
- Forgetting to take the square root of n and dividing by n instead
- Confusing the standard error with the standard deviation of individual data values
Did You Know?
The standard error shrinks as √n — to double your precision you need four times the data.
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Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
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Bayes' Theorem
IntermediateUpdates a probability given new evidence.