Variance calculator

Paste numbers, choose a data type, and see how far the values spread around their mean.

Decimal places

Population variance ฯƒยฒ

4

Mean ฮผ5
Population standard deviation ฯƒ2
Sum of squared deviations32
Divisor N8
Number of values8

Contents

What the variance calculator does

Squared distances of observations from the middle of a number line

Variance measures how widely numbers spread around their mean. Paste the observations, choose whether they form the entire population or a sample, and the calculator shows the appropriate variance.

Decide what your list represents

A complete group uses the population formula. Each distance from the mean is squared, the squares are added, and that sum is divided by the number of observations.

ฯƒ2=โˆ‘i=1N(xiโˆ’ฮผ)2N\sigma^2=\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}

Here xix_i is an observation, ฮผ\mu its population mean, and NN the count. If your list stands in for a larger group, choose sample. The denominator becomes nโˆ’1n-1 after estimating the mean from those same observations.

s2=โˆ‘i=1n(xiโˆ’xห‰)2nโˆ’1s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}

Do not choose sample merely because a list is short. The question is whether the listed items are every item you intend to describe. A population of one has variance zero; a sample needs at least two numbers.

Check the arithmetic on familiar lists

Eight completed measurements. For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5 and the squared deviations sum to 32. As a complete population, variance is 32 / 8 = 4. This describes only that finished set.

Eight observations sampled from a longer run. The same numbers still have mean 5 and squared-deviation sum 32. Sample variance is 32 / 7, about 4.571. The larger answer comes from the changed denominator, not from changed data.

Equal readings. Values 7, 7 and 7 have mean 7, so every deviation is zero. Both modes return variance 0. No spread was detected at the precision of these readings.

Units and limits matter

Variance uses squared units. If the input is in seconds, the result is in square seconds. For a spread in the original unit, see standard deviation, the square root of variance. A large value has no universal cutoff: compare it with another list using the same unit and context.

Separate values with spaces, newlines or semicolons. A decimal comma without a following space is a decimal mark, so 1,5 is one value; 1, 5 is two. Do not group thousands. The display can be rounded to 0โ€“10 decimal places; calculation retains machine precision. The limit is 10,000 values and 200,000 characters. Extreme magnitudes that cannot be calculated reliably are rejected.

Questions before using the result

The correct divisor and unit depend on the question behind the data. These short answers help avoid a plausible-looking but misleading number.

When should I use sample variance?

Use it when the listed observations represent a larger population and you want the n minus 1 correction. Use population variance to describe the complete group in the list.

Can variance be negative?

No. Squared deviations and their average cannot be negative.

Why is the unit squared?

Every difference from the mean is squared before averaging. Measurements in metres therefore give a variance in square metres.

Does rounding change the calculation?

No. It changes only the displayed number of decimal places. The raw calculation uses the entered values.

Should I remove an outlier?

First check whether it is a real observation or a data error. Removing a genuine extreme value changes the group being described and can lower variance substantially.

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Calcopedia (en) โ†’ Variance