Standard Deviation Calculator
Compute sample and population standard deviation, mean, variance, and sum of squares for any list.
Parameters & Inputs
Summary & Breakdown
Dataset Summary Statistics
About the Standard Deviation Calculator
Standard deviation is the primary statistical metric used to measure data dispersion€”how tightly clustered or widely spread observation values are around their arithmetic mean.
A low standard deviation indicates that data points cluster tightly near the average, while a high standard deviation reflects greater dispersion and volatility.
This calculator computes both sample standard deviation ($s$ using Bessel's $n-1$ correction) and population standard deviation ($\sigma$ using $N$), alongside mean and variance.
How the Calculations Work
Calculates the arithmetic mean, computes each value's squared deviation from the mean, sums the squares (SS), divides by degrees of freedom (n-1 for sample, N for population), and takes the square root.
s = sqrt(sum((x_i - mean)^2) / (n - 1)) | sigma = sqrt(sum((x_i - mean)^2) / N)
Variables & Definitions
- n: Total count of observations in sample
- mean: Arithmetic average of all numbers
- n - 1: Bessel's correction compensating for sample variance underestimation
For dataset [10, 12, 23, 23, 16, 23, 21, 16]: Count n = 8. Sum = 144. Mean = 18.0. Sum of squared differences = 192.0. Sample variance = 192 / 7 = 27.43. Sample standard deviation s = sqrt(27.43) = 5.24.
Key Terms Explained
Bessel's Correction
Dividing by n - 1 rather than n when computing sample variance to eliminate downward estimation bias.
Variance
The average of the squared deviations from the mean; equals standard deviation squared.
Normal Distribution (68-95-99.7)
In bell curves, 68.2% of data falls within 1 standard deviation, 95.4% within 2, and 99.7% within 3.
Standard Error of the Mean (SEM)
Sample standard deviation divided by the square root of n (s / sqrt(n)).
Practical Tips & Pitfalls to Avoid
Use sample standard deviation for subsets
Unless you measured every single individual in the entire population, use sample standard deviation (s).
Outliers inflate standard deviation heavily
Because differences are squared, extreme outliers have a disproportionate expanding effect on variance.
Separate numbers with commas or spaces
Paste raw numbers separated by commas, tabs, spaces, or line breaks.
Interpret alongside the mean
A standard deviation of 5 has vastly different significance on a mean of 10 versus a mean of 1,000.
Frequently Asked Questions
Population standard deviation divides by N and applies only when you have data for every member of a population. Sample standard deviation divides by n - 1 to correct for sample bias.
No. Because deviations are squared before taking the principal square root, standard deviation is always greater than or equal to zero.
A standard deviation of 0 means every single observation in the dataset has the exact same identical value.
Standard deviation is simply the square root of variance, returning dispersion back into the original units of measurement.
In any Gaussian normal distribution, approximately 68% of observations lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.