Standard Deviation Calculator — Variance
Calculate population or sample standard deviation, variance, and mean. Shows all statistical calculations step-by-step.
Formula
σ = √(Σ(x - μ)² / N)
Standard deviation is the square root of the variance, representing the average distance from the mean.
Worked Examples
Example 1: Test Scores
Problem:Scores: 80, 85, 90
Solution:Mean = 85. Variance = ((5² + 0² + 5²)/3) = 50/3 = 16.67. SD = √16.67 ≈ 4.08.
Result:Mean: 85, SD: 4.08
Frequently Asked Questions
What is Standard Deviation?
It is a statistic that measures the dispersion of a dataset relative to its mean. A low SD indicates data points tend to be close to the mean; a high SD indicates high variation.
What is Variance?
Variance is the average of the squared differences from the mean. Standard Deviation is simply the square root of Variance.
Can Standard Deviation be negative?
No. Since it involves squaring differences and then taking a square root, it must always be non-negative. It can be 0 if all data points are identical.
What affects Standard Deviation the most?
Outliers (extreme values) can heavily influence the SD because their distance from the mean is squared, disproportionately increasing the result.
Why is SD preferred over Variance?
SD is expressed in the same units as the original data (e.g., dollars, inches), whereas Variance is in squared units, making SD easier to interpret.
What is a "Z-Score"?
A Z-score tells you how many standard deviations a specific data point is from the mean. Z = (X - Mean) / SD.
How is SD used in finance?
It measures volatility (risk). An investment with a high standard deviation has returns that vary widely from the average, implying higher risk.
Why are the deviations squared instead of using absolute values?
Squaring makes every deviation positive, weights large departures from the mean more heavily than small ones, and produces a smooth function that calculus can minimise, which is the basis of least squares regression. Variances of independent variables also add, so the squared units combine cleanly. Mean absolute deviation is a legitimate alternative and resists outliers better, but it lacks those algebraic properties.
How does standard deviation differ from the standard error of the mean?
Standard deviation describes the spread of individual observations. The standard error describes the spread of the sample mean itself and equals the sample standard deviation divided by the square root of n. With s = 10 and n = 100, the standard error is 10 divided by 10, that is 1. Collecting a larger sample shrinks the standard error but leaves the underlying standard deviation unchanged.
When is a robust spread measure better than standard deviation?
Because each deviation is squared, a single extreme value can dominate the result, so a heavy-tailed or contaminated dataset can report a standard deviation that describes no typical observation at all. In that case the interquartile range or the median absolute deviation summarises the bulk of the data more faithfully. Report the robust measure alongside the mean and standard deviation rather than in place of them.
Background & Theory
Standard Deviation (σ or s):
A measure of the amount of variation or dispersion of a set of values.
- Low SD: Values are close to the mean (expected value).
- High SD: Values are spread out over a wider range.
Formulas:
- Population SD (σ): √(Σ(x - μ)² / N)
- Sample SD (s): √(Σ(x - x̄)² / (n - 1))
Variance:
The average of the squared differences from the Mean. SD is the square root of Variance.
Empirical Rule (68-95-99.7):
In a normal distribution:
- 68% of data falls within 1 SD of the mean.
- 95% falls within 2 SDs.
- 99.7% falls within 3 SDs.