Variance Calculator: A Simple Statistical Tool

Worked example: 10, 20, 30, 40, 50

Step 1 — mean: (10+20+30+40+50)/5 = 150/5 = 30

Step 2 — squared differences from the mean:

(10−30)² = 400
(20−30)² = 100
(30−30)² = 0
(40−30)² = 100
(50−30)² = 400
Sum = 1,000

Step 3 — divide by the appropriate count. This is where population and sample variance diverge:

Population variance = 1,000 / 5  = 200.0000   (divide by N)
Sample variance     = 1,000 / 4  = 250.0000   (divide by N−1)

Step 4 — standard deviation is the square root of variance:

Population std. dev. = √200 ≈ 14.1421
Sample std. dev.     = √250 ≈ 15.8114

Why sample variance divides by N−1, not N

When you're calculating variance for an entire population, dividing by N gives the true variance. But when your data is only a *sample* meant to estimate a larger population's variance, dividing by N gives a value that's systematically too small (called downward bias) — dividing by N−1 instead (known as Bessel's correction) corrects for that bias. Use population variance when your numbers are the entire group you care about; use sample variance when they're a subset representing a bigger group.

Why the tool refuses a single number

Variance measures spread around a mean — with only one data point, that point equals the mean and the spread is undefined (sample variance would divide by zero, since N−1 = 0). The tool requires at least two values for this reason.