Variance is a measure of spread: the average of the squared distances of the data values from their mean. Find each value’s distance from the mean, square it, add the squares, and divide by n for a population (or n − 1 for a sample). Larger variance means more spread. Its square root is the standard deviation.
In Spanish
varianza
Also called
population variance, sample variance, mean squared deviation, sigma squared
How to say it
VAIR-ee-unss
Grade level
Statistics
Algebra 1StatisticsCCSS HSS-ID.A.2SATACTASVABAP
Variance in a picture
The deviations add to 0, so variance squares them first. The far value 13 contributes 36 of the 58.
Formal definition and notation
For a population of N values with mean μ, the population variance is σ2 = Σ (x − μ)2N. For a sample of n values with mean x, the sample variance is s2 = Σ (x − x)2n − 1; dividing by n − 1 instead of n makes s2 a better estimate of the population variance. Variance is never negative, it is 0 only when all values are equal, and its units are the square of the data‘s units.
Symbol
Read it as
Meaning
σ2
sigma squared
population variance
s2
s squared
sample variance, divided by n − 1
x − x
x minus x bar
the deviation of one value from the mean
Key facts
Squaring makes every deviation positive, so values below the mean do not cancel values above it (the plain deviations always add to 0).
Adding the same number to every value does not change the variance; multiplying every value by k multiplies the variance by k2.
Squaring gives large deviations extra weight, so one outlier can raise the variance a lot.
Use n for a whole population and n − 1 for a sample; calculators label these σ and s.
Variance formulas
Variance formulas
σ2 = Σ (x − μ)2N and s2 = Σ (x − x)2n − 1
μ, x
the population mean and the sample mean
N, n
the number of values
Σ
add the squared deviations of every value
Steps: (1) find the mean, (2) subtract it from each value, (3) square each difference, (4) add, (5) divide by N or by n − 1.
Try it: square the distances
Interactive
Change the data and watch the variance: the squared distances from the mean are added, then divided by n (population) or by n − 1 (sample). Spread the dots apart to make it grow.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Variance examples, step by step
Example 1: A small population
Warm-up
Find the population variance of 2, 4, 6.
Mean: 2 + 4 + 63 = 4.
Deviations: −2, 0, 2. Squares: 4, 0, 4.
Sum: 8. Divide by N = 3: 83 ≈ 2.67.
Answer: The population variance is 83, about 2.67.
Example 2: Sample variance
Core
Five students in a sample read 10, 12, 15, 11 and 17 books this year. Find the sample variance.
Mean: 655 = 13.
Deviations: −3, −1, 2, −2, 4.
Squares: 9, 1, 4, 4, 16, with sum 34.
It is a sample, so divide by n − 1 = 4: 344 = 8.5.
Answer: The sample variance is 8.5 books squared.
Example 3: How a change to the data affects variance
Test-style
A data set has variance 4. Each value is multiplied by 3 and then 10 is added. What is the variance of the new data set?
Adding 10 to every value shifts the data but does not change any distance from the mean.
Multiplying by 3 makes every deviation 3 times as large.
Squared deviations grow by 32 = 9, so the variance becomes 9 × 4.
Answer: The new variance is 36.
Common mistakes
✗ MistakeAdding the deviations without squaring: for 3, 5, 6, 8, 13 the sum is −4 − 2 − 1 + 1 + 6 = 0, so the variance is 0.
✓ InsteadSquare first: 16 + 4 + 1 + 1 + 36 = 58, then divide: 11.6 for the population.
Deviations from the mean always add to 0. Squaring stops the positives and negatives from cancelling.
✗ MistakeUsing n for a sample: 34 ÷ 5 = 6.8 for the book data.
✓ InsteadA sample divides by n − 1: 34 ÷ 4 = 8.5.
Dividing a sample by n tends to underestimate the population’s spread; n − 1 corrects for that.
✗ MistakeReporting variance in the data’s units, such as “11.6 points.”
✓ InsteadVariance is in squared units (11.6 points squared); the standard deviation √11.6 ≈ 3.41 is in points.
Squaring the deviations squares their units.
✗ MistakeSquaring the values instead of the deviations: 32 + 52 + …
✓ InsteadSquare each distance from the mean: (3 − 7)2 + (5 − 7)2 + …
Variance measures distance from the center, so subtract the mean before squaring.
Answer: B. Every value equals the mean, 5, so every deviation is 0 and the variance is 0. 5 is the mean, 20 is the sum, and 1 has no basis.
Question 2 of 3
The standard deviation of a data set is 6. What is its variance?
Show answer and explanation
Answer: D. Variance is the standard deviation squared: 62 = 36. 3 halves it, 12 doubles it, and √6 takes a root instead of squaring.
Question 3 of 3
What is the population variance of 1, 3, 5?
Show answer and explanation
Answer: A. Mean 3, squared deviations 4, 0, 4, sum 8, divided by N = 3: 83. 4 is the sample variance (divide by 2), 8 forgets to divide, and 2 is the sample standard deviation, the square root of 4.
Variance: frequently asked questions
Why do we square the deviations in variance?
Because the plain deviations from the mean always add up to 0, so their average would always be 0. Squaring makes every deviation positive so they cannot cancel. It also gives extra weight to values far from the mean, and it makes the algebra of variance work neatly, for example the variances of independent variables add.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. Both measure spread, but variance is in squared units (dollars squared) while standard deviation is in the original units (dollars), so standard deviation is easier to interpret. A variance of 25 means a standard deviation of 5.
When do you divide by n − 1?
Divide by n − 1 when your data are a sample from a larger population and you want to estimate the population’s variance. Divide by N when the data are the whole population, such as every student in one class when that class is all you care about. With large data sets the two answers are very close.
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