Variance Calculator

What Is the Variance Calculator?

Variance measures the average squared deviation from the mean -- the same underlying spread measure as standard deviation, but on a squared scale.

How to Use This Tool

Enter your data set, then choose sample or population. Variance appears instantly.

The Formulas

Calculating Variance by Hand

The formulas above are easier to follow with a small data set. Take the hypothetical values 1, 2, 3.

• The mean is (1 + 2 + 3) / 3 = 2.
• The deviations from the mean are -1, 0, and 1. Squaring gives 1, 0, and 1.
• The sum of squared deviations is 2.

Squaring is what makes variance useful and slightly awkward at the same time. It stops positive and negative deviations from cancelling out, and it gives larger deviations more weight, but it also means the answer is in squared units. If your data is in dollars, the variance is in dollars squared, which is hard to picture. Taking the square root gives you the standard deviation back in ordinary units.

A Larger Example

Consider the default data set 2, 4, 4, 4, 5, 5, 7, 9, with a mean of 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, and 16, which add up to 32. The population variance is 32 / 8 = 4, and the sample variance is 32 / 7, about 4.5714. Switching between sample and population in the tool shows exactly this difference for the same numbers.

Useful Properties of Variance

• Adding a constant changes nothing. If every value in the data set is shifted by the same amount, such as converting to a different starting point, the variance stays the same because the spread is unchanged.
• Multiplying scales the variance by the square. If you multiply every value by 10, the variance becomes 100 times larger. This is why unit changes have a big effect on variance.
• It is never negative. Variance is zero only when all values are identical.
• Independent variances add. For two independent sources of variation, the variance of their sum is the sum of their variances. This property is a big reason statisticians work with variance in formal calculations even though standard deviation is easier to read.

Sample or Population?

Pick Sample when your list is a portion of a larger group and you want to estimate that group's variance. The n - 1 denominator compensates for the fact that the sample mean is calculated from the same data, which would otherwise make the spread look a little too small. Pick Population only when your list includes every member of the group. With large data sets the difference is small, but with a handful of values it is significant, as the 0.6667 versus 1 above shows.

Common Mistakes

• Reporting variance when readers expect standard deviation. Variance in squared units is hard to interpret. State which one you are giving.
• Using the wrong option. A mismatched choice changes the answer by a factor of n / (n - 1).
• Mistyped data. Because deviations are squared, one wrong digit can have an outsized effect on the result.
• Comparing variances of data on different scales. A variance measured in meters squared cannot be compared directly with one measured in centimeters squared without converting first.

Variance describes spread in the data you supplied. It does not by itself tell you why the data varies or whether a difference is statistically significant.

Frequently Asked Questions