Standard Deviation From Mean and Sample Size Calculator

From SEMSD = SEM × √n

From CI√n × width ÷ (2t*)

From CV|mean| × CV

From Σx²√((Σx² − n·mean²)/(n−1))

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

s=SEM\times\sqrt n

From a confidence interval

s=\frac{\sqrt n\,(U-L)}{2\,t^*}

From the CV

s=|\bar x|\times 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.

Where Standard Deviation From Mean and Sample Size Is Used

Reading papers

Convert an SEM to an SD.

Meta-analysis

Recover SDs from confidence intervals.

Planning

Use the SD to size a new study.

Homework

Work back from summary statistics.

Common Standard Deviation From Mean and Sample Size Mistakes to Avoid

Check whether a paper reports SD or SEM.
Use the right df for t*.
The CV needs a ratio-scale mean above zero.

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 also need a measure of spread such as the SEM, a confidence interval, a CV or Σx².

How do I get SD from the standard error?

Multiply the SEM by the square root of n.

How do I get SD from a confidence interval?

Divide the interval width by twice the t critical value, then multiply by √n.

How do I get SD from the CV?

Multiply the mean by the CV as a decimal.