Skip to main content

Standard Deviation Calculator

Quick answer

This calculator computes sample or population standard deviation and variance step by step, with the mean, sum of squares, and a full worked example.

Figures are estimates from this calculator’s standard formula — adjust the inputs in the tool below for your exact number.

How Does the Standard Deviation Calculator Work?

Enter your numbers and choose sample or population mode; the calculator computes the mean, the squared deviations, their sum, the variance, and finally the square root of the variance. Worked example for the population 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40/8 = 5; the squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32; the population variance is 32/8 = 4 and the population standard deviation is the square root of 4 = 2. For a sample you would divide by n - 1 = 7 instead, giving a variance of 4.57 and a standard deviation of about 2.14.

  1. 1

    Enter your numbers

    Separate values with commas, spaces, or line breaks. Decimals and negatives are fine.

  2. 2

    Pick sample or population

    Choose population (divide by n) when you have every value in the group, and sample (divide by n - 1) when your data is a sample used to estimate a larger population.

  3. 3

    Read the full breakdown

    The result shows the mean, count, sum of squared deviations, variance, and standard deviation, so you can verify every step by hand.

FAQs

What is the formula for standard deviation?

For a population, SD = square root of (the sum of each squared deviation from the mean divided by n). For a sample, divide by n - 1 instead. The calculator applies the exact formula and shows the intermediate sum of squares.

When do I use n versus n - 1?

Divide by n when your data is the entire population; divide by n - 1 (Bessel's correction) when it is a sample. Using n - 1 removes the downward bias a sample has when estimating the population variance.

What is the difference between variance and standard deviation?

Variance is the average of the squared deviations; standard deviation is its square root. Standard deviation is easier to interpret because it is in the same units as the original data, while variance is in squared units.

What does a standard deviation of 2 mean?

It means a typical value sits about 2 units away from the mean. In a roughly normal distribution, about 68 percent of values fall within one standard deviation of the mean and about 95 percent within two.

Can I enter large datasets?

Yes, the calculator accepts long pasted lists, and every intermediate quantity (mean, squared deviations sum, variance) is recomputed on each run so you can audit the result.

Related tools

Pick another tool to continue your calculation.

σxy

Covariance Calculator

Calculate covariance between two variables.

📈

Binomial Distribution Calculator

Calculate binomial distribution probabilities.

🔔

Normal Distribution Calculator

Calculate normal distribution probabilities from z-scores: P(Z <= 1.96) = 0.975, and an IQ of 130 with mean 100 and SD 15 gives z = 2, so only 2.28% score higher. Includes the 68-95-99.7 rule.

🐟

Poisson Distribution Calculator

Calculate Poisson distribution probabilities for rare events.

📉

Skewness and Kurtosis Calculator

Calculate skewness and kurtosis coefficients.

📉

Kurtosis Calculator

Calculate kurtosis for 5 values.