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Sample Variance (s²)
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Mean (x̄)
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Std. Deviation
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Count (n)
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Sum (Σx)
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Minimum
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Maximum
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Range
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Sum of Squares
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Coeff. of Variation
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What Is a Population Variance Tool?
A variance finder is an online tool that determines how spread out numbers are in a data set by computing the average of squared deviations from the mean. Variance — denoted σ² for populations and s² for samples — captures the degree of dispersion in squared units. A variance of zero means every value is identical, while larger values indicate wider spread. This tool lets you paste or type any set of numbers and instantly find both population and sample variance, along with standard deviation, mean, and a complete step-by-step solution.
Formula
Variance is the foundation of most spread measures in statistics.
Population Variance
σ² = Σ(xᵢ − μ)² / N
σ²Population variance
xᵢEach data value
μPopulation mean
NTotal number of values in the population
Sample Variance
s² = Σ(xᵢ − x̄)² / (n − 1)
s²Sample variance
x̄Sample mean
n − 1Degrees of freedom (Bessel's correction)
How to Find Variance Step by Step
Calculate the mean.Sum all data values and divide by the count.
Find deviations.Subtract the mean from each value: (xᵢ − x̄).
Square each deviation.Compute (xᵢ − x̄)² to make all values positive.
Sum the squares.Add all squared deviations: Σ(xᵢ − x̄)².
Divide to get variance.Divide by N for population variance or by n − 1 for sample variance.
Finding the variance of any data set is quick and effortless:
Enter your numbers into the input box — separate values with commas, spaces, or new lines.
Results appear instantly: both population and sample variance, plus standard deviation and mean.
Expand the step-by-step section to see every deviation and squared deviation.
Copy the results for use in reports, homework, or further analysis.
Benefits of Using a Variance Finder
Saves time on repetitive squaring and summing calculations.
Reduces the risk of arithmetic mistakes that are common with large data sets.
Provides both population and sample variance so you can choose the correct one.
Displays complementary statistics (SD, mean, range) in a single view.
Works with decimal, negative, and large numbers without difficulty.
Where Variance Is Commonly Used
Data Science
Feature selection and outlier detection rely on variance to identify useful signals.
Finance
Portfolio variance quantifies the total risk of combined assets.
Quality Assurance
Process variance determines if manufacturing stays within tolerance limits.
Psychology
Variance in survey responses reveals how much opinions differ within a group.
Engineering
Structural engineers use variance in load testing to set safety margins.
Meteorology
Temperature variance helps forecast extreme weather events.
Factors That Affect Variance in Your Data
Understanding why your variance is high or low is just as important as computing it. Several factors can significantly influence the result:
Sample size
Very small samples tend to produce unstable variance estimates. As sample size grows, the calculated variance converges toward the true population value.
Outliers
Because deviations are squared, a single extreme value contributes disproportionately. Always investigate outliers before interpreting variance.
Measurement precision
Instruments with low resolution (e.g., rounding to whole numbers) can artificially reduce variance by masking real differences.
Data transformation
Log or square-root transformations can reduce variance in skewed data, sometimes making analysis more valid.
Population homogeneity
If you mix distinct subgroups (e.g., adults and children), the combined variance will be inflated compared to within-group variance.
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To find variance: 1. Calculate the mean (average). 2. Find each value's deviation from the mean. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide by n (population) or n−1 (sample). Our variance finder does this automatically — just enter your numbers separated by commas.
What is the variance of 5, 10, 15, 20, 25?
For 5, 10, 15, 20, 25: Mean = 75/5 = 15. Deviations: −10, −5, 0, 5, 10. Squared: 100, 25, 0, 25, 100. Sum of squares = 250. Population variance (σ²) = 250/5 = 50. Sample variance (s²) = 250/4 = 62.5. Standard deviation: σ ≈ 7.071, s ≈ 7.906.
Can I find variance from standard deviation?
Yes — variance equals standard deviation squared. If you know the standard deviation (SD), simply square it: Variance = SD². For example, if SD = 4, then variance = 4² = 16. Conversely, standard deviation = √variance. This relationship holds for both sample and population measures.
How do I find variance in a frequency table?
For frequency data: 1. Find weighted mean: x̄ = Σ(fᵢ · xᵢ) / Σfᵢ. 2. For each value, compute fᵢ · (xᵢ − x̄)². 3. Sum all: Σ[fᵢ · (xᵢ − x̄)²]. 4. Divide by N (population) or N−1 (sample), where N = Σfᵢ. Our finder accepts individual data points — enter repeated values and it handles the rest.
What does a variance of zero mean?
A variance of zero means all values in the data set are identical. There is no spread — every data point equals the mean. For example, the data set {5, 5, 5, 5} has variance = 0. Any non-zero variance indicates that at least some values differ from the mean.
How does sample size affect variance?
Sample size (n) affects variance in two ways: 1. The calculated variance itself doesn't systematically increase or decrease with n — it converges toward the true population variance. 2. The reliability of your variance estimate improves with larger samples. Small samples can produce wildly different variance estimates; large samples give stable, consistent results. This is why sample variance uses n−1 to correct for small-sample bias.