Standard Deviation Calculator

Sample and population standard deviation, variance, SEM, coefficient of variation, quartiles and skewness from raw data, with every step of the calculation shown.

The Standard Deviation Calculator computes the sample standard deviation s = √[Σ(x − x̄)² / (n − 1)] or the population standard deviation σ = √[Σ(x − x̄)² / n] from a pasted data set, together with the mean, median, mode, variance, standard error of the mean, coefficient of variation, quartiles, skewness and kurtosis.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Paste your numbers.
Choose sample (n − 1) or population (n).
Read the table; expand the calculation steps to check the arithmetic.
Copy the results into your lab report.

What you enter

A list of numeric values

What you get

Mean, median, mode
Sample and population SD and variance
SEM and CV
Quartiles, IQR, range, skewness, kurtosis

Formula

s = √[ Σ(xᵢ − x̄)² / (n − 1) ]   |   σ = √[ Σ(xᵢ − μ)² / n ]   |   SEM = s / √n

Deviations from the mean are squared, summed and divided by n − 1 (sample, Bessel's correction) or n (population); the square root returns the result to the original units. The standard error scales the sample SD by √n to describe the precision of the mean.

Example: replicate absorbance readings

Input: 4.2 5.1 3.8 6.0 5.5 4.9 5.3 4.4 6.2 5.0

Result: Mean 5.04 · sample SD 0.7763 · variance 0.6027 · SEM 0.2455 · CV 15.4%

Σ(x − x̄)² = 5.424; divided by n − 1 = 9 gives 0.6027; the square root is 0.7763.

Questions

Should I use sample or population standard deviation?

Almost always sample (n − 1). Only use the population formula when your data set is the complete population rather than a sample from it.

What is the difference between standard deviation and standard error?

SD describes spread in the data; SEM (= SD/√n) describes the precision of the estimated mean and shrinks as n grows. Error bars on a mean should say which one they show.

How is the coefficient of variation used?

CV = SD/mean × 100 % lets you compare variability between measurements with different units or magnitudes — assay repeatability is usually reported this way.

Limitations

Use sample (n − 1) unless your data are the entire population.
Standard deviation is sensitive to outliers; check the IQR for skewed data.

References

Altman DG, Bland JM (2005). Standard deviations and standard errors. BMJ 331:903.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.