A numerical walkthrough
For values 0,1,2 with probabilities .25,.5,.25, μ=1, variance=.5, and standard deviation≈.7071. This Discrete Variance example can be compared with distribution center.
Math calculator
Discrete Variance Calculator
Measure probability-weighted squared spread around a discrete mean. Formula and variance and deviation remain visible in one place for independent verification.
Discrete Variance inputs
Calculation inputs
Practical meaning
Variance distinguishes distributions with the same mean but different uncertainty, risk, or concentration.
Mathematical foundation
Discrete variance is Σ(x−μ)²P(x), where μ is expected value. Standard deviation is its nonnegative square root.
Reading variance and deviation correctly starts with the mathematical structure described here.
Calculating it by hand
Find expected value, square every deviation from it, weight those squares, and add.
Using the Discrete Variance result
Probabilities must align and sum to one. Variance has squared units, so standard deviation is easier to compare on the original scale.
An impossible variance and deviation sign or magnitude usually points to field assignment before it points to rounding.
Reading sensitivity and precision
The input labels—possible values and corresponding probabilities—encode the model used on this page. Write those labels beside source values when transferring a problem from paper or a spreadsheet. That small step catches transposed quantities and mixed units before they become a polished-looking variance and deviation.
How the result responds
Adding a constant to every value leaves variance unchanged; scaling by c multiplies variance by c². A one-field trial makes this relationship visible without reworking the entire example.
Expected value identifies center; variance describes dispersion around it. The distinction determines whether this page fits the original question.
Recording the answer
Choose rounding after considering how the result will be used. Comparison may need only a few significant digits, while a later multi-step calculation benefits from carrying more. In either case, retain the page's formula with the variance and deviation so the underlying definition remains visible.
If the number moves into a spreadsheet, give its cell a variance and deviation heading and retain the source values possible values and corresponding probabilities nearby. Context matters more than extra displayed digits.
Reproducing Discrete Variance
A reproducible Discrete Variance record contains Possible values, Corresponding probabilities, the selected calculation mode, and the final rounding rule. Keep decimals on the zero-to-one scale during probability arithmetic and label any percentage conversion separately. Exact counting results should remain integers even when they are very large.
For Discrete Variance, for an independent check, simplify the setup by removing one stage or fixing one object. The smaller result should relate to the original through a known factor, coefficient, or complement rather than through coincidence.
Questions about Discrete Variance
Can variance be negative?
No.
What does zero variance mean?
All probability is at one value.
Why square deviations?
To prevent cancellation.
What units does deviation use?
The original value units.