Standard Deviation Calculator
Calculate standard deviation, variance, and mean for any data set. Get both sample and population statistics instantly, with every step shown.
How to Use This Calculator
- Enter your numbers — Type or paste a list of values separated by commas, spaces, or new lines
- Choose sample or population — Pick sample if your numbers are a subset of a larger group, population if they’re the entire group
- Click Calculate — Instantly see the standard deviation, variance, mean, and full step-by-step breakdown
Enter Numbers
Data Type
📐 Sample vs. Population Standard Deviation
- Population (σ): Use when your numbers represent the entire group you care about — e.g. every student's height in a small classroom. Divide the sum of squared differences by
N. - Sample (s): Use when your numbers are a subset used to estimate a larger population — e.g. 50 survey respondents out of 10,000 customers. Divide by
n − 1(Bessel's correction) instead ofn, which corrects for the bias of estimating from a smaller group. - Sample standard deviation is always slightly larger than population standard deviation for the same data, since dividing by a smaller number increases the result.
🧮 Formula Reference
Sum ÷ Count
Σ(x − x̄)² ÷ N
Σ(x − x̄)² ÷ (n − 1)
√Variance
A low standard deviation means values are clustered close to the mean; a high standard deviation means they're spread out.
Sample standard deviation requires at least 2 values (dividing by n − 1 is undefined for a single value).
📊 Real-World Examples
- Test scores: A low standard deviation means most students scored close to the class average.
- Investment returns: Standard deviation of monthly returns is a common measure of volatility/risk.
- Manufacturing quality control: Low standard deviation in part measurements means consistent production.
- Weather: Comparing the standard deviation of daily temperatures between two cities shows which has more variable weather.
Frequently Asked Questions
What is standard deviation in simple terms?
Standard deviation measures how spread out a set of numbers is from the average. A low standard deviation means values are clustered close to the mean; a high standard deviation means they are more spread out. It's the square root of the variance.
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squared differences by N (the full group size). Sample standard deviation divides by n − 1 instead (Bessel's correction), which corrects for the bias of estimating a population's spread from only a subset. Use sample when your data is part of a larger group; use population when your data is the entire group.
How do I calculate standard deviation by hand?
1) Find the mean of your numbers. 2) Subtract the mean from each number and square the result. 3) Add up all the squared differences. 4) Divide by N (population) or n − 1 (sample) to get the variance. 5) Take the square root of the variance — that's your standard deviation.
Why do you divide by n − 1 for a sample instead of n?
Dividing by n − 1 (called Bessel's correction) corrects for the fact that a sample's mean is itself an estimate, which tends to slightly understate true variability. Dividing by the smaller number n − 1 makes the sample variance a better (unbiased) estimate of the population variance.
Can standard deviation be negative?
No. Standard deviation is always zero or positive because it's calculated from squared differences and a square root. A standard deviation of 0 means every value in the data set is identical.
What is a 'good' standard deviation?
There's no universal answer — it depends on the units and context of your data. A useful way to judge it is relative to the mean (the coefficient of variation) or by comparing it to similar data sets. For example, a standard deviation of 5 points is small for SAT scores but large for a 10-point quiz.