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Standard Deviation Calculator

Calculate the standard deviation and variance of a dataset from summary statistics. Supports both population and sample standard deviation with Bessel's correction for unbiased estimation.

Reviewed by Christopher FloiedPublished Updated

This free online standard deviation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

The sum of all data values in your dataset.

The sum of each data value squared.

Total number of data points.

Results

Mean

70

Population Std Dev (sigma)

18.7083

Sample Std Dev (s)

20.9165

Population Variance

350

Sample Variance

437.5

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Standard Deviation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Standard Deviation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Standard Deviation Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
  • •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
  • •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
  • •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.

About Standard Deviation Calculator

The standard deviation calculator measures the amount of dispersion or spread in a dataset. A low standard deviation means data points cluster near the mean, while a high standard deviation indicates wide spread. Standard deviation is the most commonly used measure of variability in statistics, finance, quality control, and the natural sciences. This calculator computes both the population standard deviation (when your data represents the entire population) and the sample standard deviation (when your data is a sample from a larger population, using Bessel's correction with n-1 in the denominator). It also reports the corresponding variances and the mean.

The Math Behind It

Standard deviation quantifies how much individual data points deviate from the mean. The population standard deviation sigma is defined as the square root of the average squared deviation from the mean. The sample standard deviation s uses (n-1) instead of n in the denominator, known as Bessel's correction, to produce an unbiased estimate of the population variance. The square of the standard deviation is the variance, which has the convenient property of being additive for independent random variables: Var(X+Y) = Var(X) + Var(Y). Standard deviation has the same units as the original data, making it more interpretable than variance. In a normal distribution, the empirical rule states that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Chebyshev's inequality provides a weaker bound for any distribution: at least 1 - 1/k^2 of data lies within k standard deviations of the mean. In finance, standard deviation of returns is the primary measure of investment risk (volatility). In manufacturing, reducing standard deviation directly improves process capability indices (Cp, Cpk).

Formula Reference

Population Standard Deviation

sigma = sqrt(sum((x_i - mu)^2) / N)

Variables: sigma = population std dev; x_i = each value; mu = population mean; N = population size

Sample Standard Deviation

s = sqrt(sum((x_i - x_bar)^2) / (n - 1))

Variables: s = sample std dev; x_bar = sample mean; n = sample size; n-1 = Bessel's correction

Worked Examples

Example 1: Test scores

Five test scores: 60, 70, 70, 80, 70. Find the sample standard deviation.

Step 1:Sum = 350, Sum of squares = 3600 + 4900 + 4900 + 6400 + 4900 = 24700, n = 5.
Step 2:Mean = 350/5 = 70.
Step 3:Sample variance = (24700 - 350^2/5) / (5-1) = (24700 - 24500)/4 = 50.
Step 4:Sample std dev = sqrt(50) = 7.07.

The sample standard deviation is approximately 7.07 points.

Example 2: Manufacturing measurements

Bolt diameters (mm): 10.1, 10.0, 9.9, 10.0, 10.0. Calculate population standard deviation.

Step 1:Sum = 50.0, Sum of squares = 500.02, n = 5.
Step 2:Mean = 10.0.
Step 3:Population variance = 500.02/5 - 100 = 0.004.
Step 4:Population std dev = sqrt(0.004) = 0.0632 mm.

The population standard deviation is 0.063 mm, indicating tight dimensional control.

Common Mistakes & Tips

  • !Using the population formula (dividing by n) when working with a sample, which underestimates variability.
  • !Confusing standard deviation with standard error. Standard error = s/sqrt(n) and measures uncertainty in the mean, not data spread.
  • !Forgetting that standard deviation cannot be negative; if you get a negative value, check your computation.

Related Concepts

Variance

Variance is the square of standard deviation and is important for mathematical derivations and ANOVA.

Z-Score

Z-scores use standard deviation to standardize data points for comparison across different scales.

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Frequently Asked Questions

When do I use population vs. sample standard deviation?

Use population standard deviation when your data includes every member of the population. Use sample standard deviation when your data is a subset drawn from a larger population, which is the more common scenario in practice.

What is Bessel's correction?

Bessel's correction divides by (n-1) instead of n when computing sample variance. This corrects the bias that arises because the sample mean is used in place of the unknown population mean.

Can standard deviation be zero?

Yes, a standard deviation of zero means all data values are identical. There is no spread at all.