AP Stats Variance Calculator

Measure how spread out your data is. Enter a list of numbers and this AP Stats Variance Calculator gives the mean, sample variance, population variance, and both standard deviations, so you can see exactly which denominator your problem needs.

What Is Variance?

Variance measures how spread out values are around the mean, as the average of the squared distances from the mean. Squaring the distances stops positive and negative differences from cancelling out, but it also puts variance in squared units, points squared, for example. Taking the square root of the variance gives the standard deviation, which is in the original units.

How to Use This Calculator

1. Type your data values separated by commas, in any order. 2. Press Calculate Variance. 3. Read the mean first, then choose sample variance (divide by n − 1) if your data are a sample, or population variance (divide by n) if your data are the whole population.

How to Calculate Variance

Formulas: mean x̄ = Σx / n. Sample variance s² = Σ(x − x̄)² / (n − 1). Population variance σ² = Σ(x − x̄)² / n. Standard deviation is the square root of variance: s = √s² and σ = √σ².

Worked example: For 12, 15, 18, 22, 23, the mean is 18. Squared deviations are 36, 9, 0, 16, and 25, totalling 86. Sample variance = 86 / 4 = 21.5 and sample SD ≈ 4.64. Population variance = 86 / 5 = 17.2 and population SD ≈ 4.15. Try these values in the calculator to see the same result.

Tips and Common Mistakes

The most common error is using the wrong denominator. AP Statistics almost always asks for the sample standard deviation, s, which divides by n − 1. Variance is in squared units, so report standard deviation when you describe spread in the original units. Outliers increase variance a lot, so check for them before trusting a single spread number.

Frequently Asked Questions

What is the difference between sample and population variance?

Sample variance divides by n − 1 and estimates a population variance from a sample. Population variance divides by n and is used only when the data are the entire population.

Why divide by n - 1 for a sample?

Dividing by n − 1 corrects for using the sample mean instead of the true population mean, and it makes s² an unbiased estimate of σ² on average.

Why is variance in squared units?

Variance averages squared deviations from the mean, so its units are squared. Take the square root to get standard deviation, which is back in the original units.

Can variance be negative?

No. Squared deviations are never negative, so variance and standard deviation are always zero or positive. Zero means all values are identical.

What does a larger variance tell me?

A larger variance means the values are more spread out around the mean. It does not by itself tell you whether the data are skewed or contain outliers.

What is the TI-84 equivalent?

On a TI-84, use STAT, CALC, 1-Var Stats. Sx is the sample standard deviation and σx is the population standard deviation; square them for the variances here.

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