Population Variance

Population variance quantifies how spread data points are around the mean across an entire population, used instead of sample variance when you have all data.

σ2=1N∑(xi−μ)2\sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2

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What each symbol means

What Population Variance takes
σ\sigma
μ\mu
xix_i
NN
Population Variance →
SymbolMeaning
σ\sigmaThe symbol for standard deviation, but σ² in this formula represents population variance; confusing σ\sigma with σ2\sigma^2 means you have forgotten to square the standard deviation.
μ\muThe population mean (average), which must be calculated first; using the sample mean instead will introduce bias into your variance estimate.
xix_iThe ii-th individual value in your population; the subscript ii just labels which data point you are referring to, and forgetting even one value changes your result.
NNThe total count of all data points in the population; using N−1N-1 instead (which is for sample variance) produces an incorrectly inflated variance.

When to use it

Use population variance when analyzing a complete dataset for a defined group, such as all employees in a company or all daily sales for a specific week.

Level

Usually taught in: Statistics

Worked examples

1. Variance of small integers

Problem

Find the population variance of the dataset: 2, 4, 6, 8, 10.
  1. μ=2+4+6+8+105=305=6\mu = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6

    Calculate the mean by summing all values and dividing by the count.

  2. (2−6)2=16,(4−6)2=4,(6−6)2=0,(8−6)2=4,(10−6)2=16(2-6)^2 = 16, \quad (4-6)^2 = 4, \quad (6-6)^2 = 0, \quad (8-6)^2 = 4, \quad (10-6)^2 = 16

    Find the squared distance of each value from the mean.

  3. 16+4+0+4+16=4016 + 4 + 0 + 4 + 16 = 40

    Add up all the squared deviations.

  4. σ2=405=8\sigma^2 = \frac{40}{5} = 8

    Divide the sum by the total number of values N=5N=5.

Answer: σ2=8\sigma^2 = 8

We calculated the mean first, then found how far each point deviates from that mean (squared to keep all values positive), summed those squared distances, and divided by the total count, which gives the average squared deviation.

2. Variance with negatives and symmetric data

Problem

Calculate the population variance of: −3,−1,1,3-3, -1, 1, 3.
  1. μ=−3+(−1)+1+34=04=0\mu = \frac{-3 + (-1) + 1 + 3}{4} = \frac{0}{4} = 0

    Sum all values and divide by the count to find the mean.

  2. (−3−0)2=9,(−1−0)2=1,(1−0)2=1,(3−0)2=9(-3-0)^2 = 9, \quad (-1-0)^2 = 1, \quad (1-0)^2 = 1, \quad (3-0)^2 = 9

    Square each deviation; when the mean is zero, each squared deviation is simply the value squared.

  3. 9+1+1+9=209 + 1 + 1 + 9 = 20

    Sum all the squared deviations together.

  4. σ2=204=5\sigma^2 = \frac{20}{4} = 5

    Divide by N=4N=4 to calculate the population variance.

Answer: σ2=5\sigma^2 = 5

This dataset is symmetric around zero, so the mean falls at the center; the calculation works the same way, but the zero mean makes each squared deviation simply the square of each data value itself.

3. Daily customer counts at a coffee shop

Problem

A coffee shop tracks customers served each day for one week: 49, 42, 49, 56, 49, 42, 56. Calculate the population variance for this week.
  1. μ=49+42+49+56+49+42+567=3437=49\mu = \frac{49 + 42 + 49 + 56 + 49 + 42 + 56}{7} = \frac{343}{7} = 49

    Sum all daily counts and divide by 7 to find the mean.

  2. (49−49)2=0,(42−49)2=49,(49−49)2=0,(56−49)2=49,(42−49)2=49,(56−49)2=49(49-49)^2 = 0, \quad (42-49)^2 = 49, \quad (49-49)^2 = 0, \quad (56-49)^2 = 49, \quad (42-49)^2 = 49, \quad (56-49)^2 = 49

    Calculate the squared deviation for each day; days equal to the mean contribute zero.

  3. 0+49+0+49+0+49+49=1960 + 49 + 0 + 49 + 0 + 49 + 49 = 196

    Add up all the squared deviations to find the total squared spread.

  4. σ2=1967=28\sigma^2 = \frac{196}{7} = 28

    Divide by the total number of days N=7N=7 to get the average squared deviation.

Answer: σ2=28\sigma^2 = 28

This real-world scenario shows how variance captures day-to-day variability; a variance of 28 indicates that the daily customer count deviates from the average of 49 by a substantial squared amount, revealing meaningful fluctuation in business volume throughout the week.

Common mistakes

Where Population Variance usually goes wrong
Answer came out wrong
Writing σ\sigma instead of σ2\sigma^2 as your final answer.
Make sure to square the standard deviation, or use the variance formula with squared deviations (xi−μ)2(x_i - \mu)^2 throughout.
Using N−1N-1 instead of NN in the denominator.
For population variance, always divide by NN; only use N−1N-1 when computing sample variance s2s^2.
Forgetting to square the deviations and summing (xi−μ)(x_i - \mu) instead of (xi−μ)2(x_i - \mu)^2.
Square each deviation (xi−μ)2(x_i - \mu)^2 before summing so that positive and negative deviations both contribute positively.
The mistakeWhy it is wrongThe fix
Writing σ\sigma instead of σ2\sigma^2 as your final answer.You have calculated standard deviation, not variance; you forgot to square the standard deviation.Make sure to square the standard deviation, or use the variance formula with squared deviations (xi−μ)2(x_i - \mu)^2 throughout.
Using N−1N-1 instead of NN in the denominator.That denominator is for sample variance; using N−1N-1 makes your population variance incorrectly too large.For population variance, always divide by NN; only use N−1N-1 when computing sample variance s2s^2.
Forgetting to square the deviations and summing (xi−μ)(x_i - \mu) instead of (xi−μ)2(x_i - \mu)^2.Unsquared deviations always sum to zero, so the variance becomes meaningless and always equals zero.Square each deviation (xi−μ)2(x_i - \mu)^2 before summing so that positive and negative deviations both contribute positively.

Tips and when to use something else

  • Population variance requires the complete dataset; if you only have a sample, use sample variance (s2s^2) with N−1N-1 in the denominator instead to avoid bias.
  • Variance is always non-negative because you square all deviations; if you get a negative number, you have made an arithmetic error.
  • The units of variance are squared (if data is in dollars, variance is in dollars²); take the square root to get standard deviation (σ\sigma), which returns to your original units.
  • Interpret variance relative to your dataset's scale—large variance means data spreads far from the mean, while small variance means data clusters tightly.

Related formulas

Arithmetic MeanArithmetic Mean is the average value found by adding all numbers in a dataset and dividing by how many numbers there are.Sample VarianceSample variance measures how spread out data points are around the mean, using n-1 instead of n to account for sample uncertainty.Standard DeviationStandard deviation measures how spread out data points are from the average—use it to understand variability in a dataset.Normal DistributionThe normal distribution describes the probability density of a continuous variable, used to model measurements and natural phenomena.Z-ScoreThe Z-Score tells you how many standard deviations a data point is from the mean, revealing whether it is typical or unusual.

Frequently asked questions

When should I use population variance instead of sample variance?
Use population variance when you have data for every member of the group you care about; use sample variance when you only have a subset and want to estimate the full population's variance.
Why do we square the deviations in the variance formula?
Squaring makes all deviations positive so they do not cancel each other out, and it emphasizes larger deviations, giving you a meaningful measure of spread rather than always getting zero.
What does a variance of 0 mean?
A variance of 0 means every single value in the population is identical to the mean; there is no variation in the data at all.
Can I compare variances from two different datasets directly?
Only if both datasets use the same units and similar scales; variance is sensitive to the measurement scale, so the same data measured in millimeters versus kilometers will have vastly different variances.

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Reviewed 2026-09-18