What Is Variance?

The short definition

Variance is the average of the squared distances between each value and the mean. If every value is the same, every distance is zero and the variance is zero. The further the values wander from their average, the bigger the variance gets. It can never be negative.

The mean tells you where the middle of your data is. Variance tells you how much you can trust that middle to describe any single value. Two data sets can share the same mean and still behave very differently.

Same average, different story

Take two small classes that sat the same test. Both have an average score of 74.

In class A, every student scored within 4 points of the average. In class B, scores range from a fail to near-perfect. Reporting only "the average was 74" hides that difference completely. Variance is what exposes it. Class B's deviations (−20, −10, 0, 10, 20) are five times larger than class A's (−4, −2, 0, 2, 4), and because they are squared, the variance is 5² = 25 times larger.

You can check both rows yourself: pasteclass B's scores into the calculatorwith population mode selected and follow the steps.

Why the distances are squared

If you simply averaged the raw distances from the mean, the positives and negatives would cancel out and you would always get zero. Squaring fixes that, since every squared value is positive. It also gives extra weight to values far from the mean: a point 10 units away contributes 100, while a point 1 unit away contributes just 1. That makes variance sensitive to outliers, which is useful when extreme values matter and misleading when a single typo dominates the result.

The side effect is that variance comes out in squared units. Test scores give "points squared", prices give "dollars squared". That is hard to picture, which is why people usually quote the standard deviation, the square root of the variance, when describing data in words.

How to interpret a variance

There is no universal "good" or "bad" value. A few practical rules help:

• Compare it with something. A variance is most meaningful next to another variance measured in the same units: this month vs last month, machine A vs machine B.
• Convert it to a standard deviation before explaining it to anyone. "Scores typically land about 14 points from the average" is clearer than "the variance is 200".
• Scale it to the mean when sizes differ. A standard deviation of 5 is large for values around 10 and tiny for values around 10,000. The coefficient of variation (standard deviation ÷ mean) puts both on the same footing.
• Look for outliers when it seems too big. One mistyped value, such as 740 instead of 74, can multiply the variance many times over.

Where variance is used

Education and testing

Teachers and exam boards use variance to see whether a test separated students well or whether everyone landed in the same narrow band. Very low variance can mean the test was too easy or too hard for the group.

Investing and finance

The variance of an investment's returns is a standard measure of risk. Suppose fund X returns 1% every month for four months, and fund Y returns 4%, −2%, 3% and −1%. Both average 1% a month. Fund X has a variance of zero. Fund Y has a sample variance of about 8.67 (percent squared), a standard deviation near 2.94 percentage points, so its month-to-month results are far less predictable.

Manufacturing and quality control

A bolt that is supposed to be 10 mm long is useless if half the batch is 9.5 mm and half is 10.5 mm, even though the average is perfect. Factories track the variance of measurements to catch machines that are drifting before they start producing rejects.

Science and experiments

Most statistical tests, from t-tests to ANOVA (literally "analysis of variance"), compare how much variation exists between groups with how much exists within them. Without a measure of spread, you cannot tell a real effect from ordinary noise.

Useful properties

• Adding a constant changes nothing. If every student in class A gets 5 bonus points, the mean rises to 79 but the variance stays 8. The spread has not changed.
• Multiplying scales it by the square. Doubling every score multiplies the variance by 4 (to 32) and the standard deviation by 2.
• Independent variances add. For two independent quantities,Var(X + Y) = Var(X) + Var(Y). Standard deviations do not add this way, which is one reason statisticians work with variance internally.

Next steps

To work one out by hand, follow how to calculate variance. To decide whether to divide by n or n − 1, readsample vs population variance. Or skip the arithmetic and use the variance calculator, which shows every step.