Standard Error Calculator Formula
Understand the math behind the standard error calculator. Each variable explained with a worked example.
Formulas Used
Standard Error
se = sigma / sqrt(n)Variance of the Mean
variance_of_mean = pow(sigma, 2) / nRelative SE (as % of SD)
relative_se = (1 / sqrt(n)) * 100Variables
| Variable | Description | Default |
|---|---|---|
sigma | Standard Deviation | 15 |
n | Sample Size | 25 |
How It Works
How to Calculate the Standard Error
Formula
SE = sigma / sqrt(n)
The standard error of the mean measures the precision of the sample mean as an estimator of the population mean. It is the standard deviation of the sampling distribution of the mean. As sample size increases, SE decreases, making estimates more precise.
Worked Example
Population SD = 15, sample size = 25. What is the standard error?
- 01SE = sigma / sqrt(n)
- 02SE = 15 / sqrt(25)
- 03SE = 15 / 5 = 3
- 04Variance of mean = 15^2 / 25 = 225 / 25 = 9
Frequently Asked Questions
What is the difference between standard deviation and standard error?
Standard deviation measures the spread of individual observations. Standard error measures the spread of sample means. SE is always smaller than SD (by a factor of sqrt(n)) because averaging reduces variability.
Why does SE decrease with larger samples?
Larger samples average out individual variability more effectively. The mean of 100 observations is more stable than the mean of 4 observations, hence the sqrt(n) in the denominator.
Can I use sample SD instead of population SD?
Yes. In practice, the population SD is rarely known, so we use the sample SD (s) as an estimate. The result is an estimated standard error, and we typically use the t-distribution instead of z for inference.
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