Find Variance in Data and Probability Distributions

Identify whether the inputs are data or probabilities

The word variance occurs in both descriptive statistics and probability. A raw list treats each listed observation with equal weight, including duplicates. A discrete random variable instead comes with outcomes and their probabilities, which need not be equal. The same squared-distance idea underlies both cases, but the weighting must match the object being described. Writing the intended mean and weights first prevents a probability problem from being accidentally turned into an unweighted data-summary exercise.

Calculate variance for an equally weighted list

For the observations 1, 3 and 5, the mean is 3. Squared deviations are 4, 0 and 4. Their average gives population variance 8/3≈2.666667. The usual sample variance divides their sum by two and is 4. Enter this list in the local calculator to see both conventions. Neither variance equals the range, which is 5−1=4 by a different definition; the matching sample variance and range in this particular example are a numerical coincidence.

Use expectation for a discrete random variable

Suppose X takes values 0, 1 and 2 with probabilities 1/4, 1/2 and 1/4. Its expectation is 0(1/4)+1(1/2)+2(1/4)=1. Its variance is (0−1)²(1/4)+(1−1)²(1/2)+(2−1)²(1/4)=1/2. The probabilities sum to one and supply the weights. No n−1 correction applies to this fully specified distribution. The question is about its mathematical variance, not an estimate from a random sample of observations.

Represent rational weights carefully for a tool check

The small distribution above can be represented by the equally weighted list 0, 1, 1, 2 because those frequencies match its probabilities exactly. The calculator's population variance is then 0.5, which verifies the distribution calculation. Its sample variance, 2/3, has a different interpretation and is not the variance of the specified probability distribution. Expanding rational weights is practical only for modest integer frequencies; arbitrary probabilities should be handled with a direct weighted expectation rather than forced into a large list.

Derive the alternative expectation formula

Expanding the square gives E[(X−μ)²]=E(X²−2μX+μ²). Because μ=E(X) is a constant, this becomes E(X²)−2μ²+μ²=E(X²)−μ². For the example, E(X²)=0²/4+1²/2+2²/4=1.5, so subtracting the squared mean one gives variance 0.5. The identity assumes the relevant expectations exist. It is an algebraic relation, not a guarantee that subtracting two almost equal large floating-point numbers will preserve all useful precision.

Apply shifts and scale factors

For Y=aX+b, the expectation becomes aE(X)+b and the variance becomes a²Var(X). Adding a constant shifts every value and its mean together, leaving all deviations unchanged. Multiplication scales every deviation by a before it is squared. With the probability example and Y=3X+10, the mean is 13 and variance is 9×0.5=4.5. A negative multiplier also squares, so variance cannot become negative simply because the ordering or sign of values is reversed.

Know when a variance is zero or undefined

A finite distribution has variance zero when all probability lies at one value; a constant raw list likewise has population variance zero. Sample variance needs at least two observations because its denominator is n−1. In more general probability models, a finite variance need not exist if the second moment is not finite. The local raw-list calculator does not investigate such theoretical distribution properties. A displayed numerical result should therefore be interpreted within the finite data supplied to it.

State the quantity and the convention

Report whether the answer describes a listed population, estimates a wider population using sample variance, or belongs to a specified probability distribution. Include squared units when the variable has a measurement unit. Standard deviation is the nonnegative square root and has the original unit, while standard error refers to uncertainty in an estimator and is a separate quantity. These labels matter more than reporting additional decimal places, because a precisely rounded answer to the wrong variance definition remains the wrong answer.

Related guides

Calculate Variance: Population and Sample Examples

Follow two variance calculations from mean to squared deviations, verify n versus n−1, and check the result using an independent identity.

Find Standard Deviation with a Checked Worked Example

Calculate squared deviations, choose the sample or population denominator, take the square root, and interpret standard deviation in original units.

Calculate Standard Deviation from Grouped Frequencies

Use class midpoints and frequency weights to estimate grouped standard deviation, choose n or n−1, and explain the lost within-class detail.