What is a Variance & Standard Deviation Calculator?
This calculator computes the variance and standard deviation of a data set to measure how spread out the values are.
How it's calculated
Variance = Σ(value − mean)² ÷ n
Standard deviation = √variance
For a sample, divide by n−1 instead of n.
Step-by-step example
Standard deviation of 2, 4, 6, 8
- Step 1 Find the mean: (2+4+6+8)÷4 = 5
- Step 2 Sum of squared deviations: (−3)²+(−1)²+1²+3² = 9+1+1+9 = 20
- Step 3 Variance = 20÷4 = 5, standard deviation = √5 ≈ 2.24
Interpretation: On average, values sit about 2.24 units away from the mean of 5.
Why square the deviations?
Simply adding up deviations from the mean always gives 0, since positive and negative differences cancel out. Squaring removes the sign problem, and it also weights larger deviations more heavily, making outliers more noticeable. Since squaring changes the units (cm becomes cm²), taking the square root at the end converts back to the original units, giving standard deviation.
Good to know
- Two groups with the same average can be completely different if their standard deviations differ — which is why the average alone never tells the full story.
- In a normal distribution, the 68-95-99.7 rule holds: about 68% of values fall within 1 standard deviation of the mean.
- This is where Six Sigma comes from — a quality target of defects occurring beyond 6 standard deviations from the mean, corresponding to roughly 3.4 defects per million.
- Dividing by n−1 instead of n for sample data corrects a known bias: using n alone tends to systematically underestimate the true population variance.
- In finance, volatility is literally the standard deviation of returns — for the same expected return, lower standard deviation is considered a more stable investment.
Frequently asked questions
Q. Should I look at variance or standard deviation?
Standard deviation — it's in the same units as the original data, making it easier to interpret.
Q. When do I divide by n vs. n−1?
Use n for a complete population (population variance); use n−1 for a sample drawn from a larger population (sample variance).
Q. What does a standard deviation of 0 mean?
All values are identical to the mean — there's no spread in the data at all.