What Is Standard Deviation From Mean and Sample Size?
People often search for the standard deviation from the mean and sample size, but those two numbers alone do not fix the spread. You need one more measure of variability, and papers usually report one of a few.
Choose what you know: the standard error of the mean, a confidence interval, the coefficient of variation or the sum of squares. The calculator returns the standard deviation, variance, standard error and, where possible, the CV, with each step shown.
The sections below give the conversion formulas, a worked example and answers on which one to use.
How to Use the Standard Deviation From Mean and Sample Size Calculator
- Choose what you know.
- Enter n and the summary values.
- Read the SD and the working.
- Check the standard error and CV.
Standard Deviation From Mean and Sample Size Formula
From the standard error
From a confidence interval
From the CV
Standard Deviation From Mean and Sample Size Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| SEM | Standard error of the mean | |
| n | Sample size | |
| t* | Critical t value | |
| CV | Coefficient of variation |
Standard Deviation From Mean and Sample Size Worked Example
A study reports a standard error of 1.8 with n = 25. Then SD = 1.8 × √25 = 1.8 × 5 = 9. With a mean of 50, the coefficient of variation is 9 ÷ 50 = 18 %.
With the example values loaded in the calculator above, the steps are:
- SD = SEM × √n = 1.8 × √25 = 9.
Result: Standard deviation = 9.
How to Read These Results
The mean and sample size alone do not fix the standard deviation. You need one more piece of information, and each option links to the SD in a simple way. With a standard error of 1.8 and n = 25, the SD is 1.8 × √25 = 9.
With the mean also known, the coefficient of variation follows: 9 ÷ 50 = 18%. If you start from a confidence interval, the SD comes from half its width, divided by the critical value, times √n. Use the t value for small samples.
Where Standard Deviation From Mean and Sample Size Is Used
Reading a paper
Recover the SD when a study reports only a standard error.
Confidence intervals
Work back from the limits of an interval to the SD.
From a CV
Find the SD from a coefficient of variation and the mean.
From sums
Compute the SD from n and Σx².
Common Standard Deviation From Mean and Sample Size Mistakes to Avoid
- Check whether a paper reports SD or SE. Confusing them changes the SD by a factor of √n.
- For small samples, use the t critical value when you back out the SD from a confidence interval.
- The mean and n alone are not enough to find the SD.
- A CV given as a percentage must be divided by 100 before you multiply it by the mean.
Standard Deviation From Mean and Sample Size Terms and Definitions
- SEM
- Standard error of the mean, SD divided by √n.
Standard Deviation From Mean and Sample Size FAQs
Can I find the standard deviation from the mean and sample size?
Not from those two alone. You need one more measure of spread, such as the standard error, a confidence interval, a CV or Σx². The calculator accepts each of these.
How do I get SD from the standard error?
Multiply the standard error by the square root of n. With SE = 1.8 and n = 25 you get SD = 1.8 × 5 = 9.
How do I get SD from a confidence interval?
Take half the width of the interval to get the margin of error, divide it by the critical value to get the standard error, then multiply by √n.
How do I get SD from the CV?
Multiply the CV, as a decimal, by the mean. A CV of 18% and a mean of 50 give an SD of 0.18 × 50 = 9.