Variance Calculator
Calculate population variance and sample variance simultaneously. Variance = SD² and measures the average squared deviation from the mean. Also shows SD, quartiles, and outliers.
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How It Works
The results panel shows both simultaneously. Pop variance divides by N (all values you have). Sample variance divides by n−1 (Bessel's correction). For n=5, sample variance is 25% larger than population variance — a meaningful difference. For n=100, the difference is about 1%. Both values show labeled clearly in the results.
Pop var: σ² = Σ(x−μ)²/N | Sample var: s² = Σ(x−x̄)²/(n−1)[2,4,6]: Pop var=2.667, Sample var=4.000Variance = SD². SD = √variance. Report SD when communicating with non-statisticians — it is in original units and directly interpretable. Use variance in formulas that need it (ANOVA, regression). The calculator shows both: sample SD and sample variance, population SD and population variance, all from the same single calculation.
Var = SD² | SD = √Var | both in results panelSD=2, Var=4 | SD=3, Var=9 | SD=1.5, Var=2.25If X and Y are independent random variables: Var(X+Y) = Var(X) + Var(Y). This additive property is why variance is preferred in theory. Standard deviations do not add directly. Example: if measurement error has variance 4 and instrument variance is 9, total variance = 13, total SD = √13 ≈ 3.6 (not 2+3=5).
Var(X+Y) = Var(X) + Var(Y) for independent X,YVar=4 + Var=9 → Var=13, SD=3.606Switch to the Frequency Table tab. Enter each unique value and its count. The variance calculation internally expands to the full dataset: each value×frequency contributes to the sum of squared deviations. More efficient than entering the same value hundreds of times for survey or exam score data.
Var = Σ(f × (x−x̄)²) / Σ(f) or /(Σf−1)Scores: 70×5, 80×12, 90×8 → variance computed over all 25Because deviations are squared, a single outlier can dramatically inflate variance. In Data 1,2,3,4,100: mean=22, squared deviations=(441,400,361,324,6084), variance ≈ 1522. Without 100: mean=2.5, variance=1.25. The outlier multiplied variance by 1217×. This is why robust alternatives like IQR are used for skewed data.
Outlier impact: (outlier−mean)² >> (normal_value−mean)²1,2,3,4,100: var≈1522 vs 1,2,3,4: var=1.25Quick Reference
Verify these results in the calculator above.
Population
[2,4,6]
Pop var: 2.67
Sample
[2,4,6]
Samp var: 4.00
Zero var
[5,5,5,5]
Var: 0
Sample
[1,2,3,4,5]
Samp var: 2.5
SD to Var
SD=3 → Var
9
Var to SD
Var=16 → SD
4
Classic
[2,4,4,4,5,5,7,9]
Pop var: 4
Outlier effect
[1,2,3,4,100]
Pop var: ≈1522
Tips & Shortcuts
The results show population variance and sample variance side by side — no need to calculate twice. Excel VAR() matches sample variance; VARP() matches population variance.
Variance units are squared: if data is in kg, variance is in kg². If units seem strange, convert to SD (√variance) which restores original units.
For financial risk analysis, variance of returns is the core measure. Portfolio variance combines individual asset variances weighted by their portfolio shares and correlations.
Very large variance relative to the mean (high CV²) suggests the data may need log transformation for statistical analysis.
The Frequency Table tab computes variance efficiently for grouped data — enter value and count instead of repeating each value.
Common Mistakes
Confusing variance with standard deviation
Variance = SD². If SD = 3, variance = 9 — not 3. Report SD for communication; use variance in formulas. The results panel clearly labels each.
Using population variance for sample data
Sample variance (n−1) corrects for the bias of underestimating true population variance. Using n instead of n−1 underestimates variance by a factor of n/(n−1) — significant for small samples.
Expecting variance to be in original units
Variance is in squared units. Heights in cm → variance in cm². This is mathematically necessary (squaring the deviations) but makes direct interpretation hard. Use SD for interpretation, variance for calculations.
Adding standard deviations instead of variances
You cannot add SDs of independent quantities. Var(X+Y) = Var(X) + Var(Y), so SD(X+Y) = √(SD(X)² + SD(Y)²), not SD(X) + SD(Y). Adding SDs overestimates the combined spread.
Entering numbers with commas as thousand separators
The calculator uses commas as value separators. Enter 1000, not 1,000. Remove all thousand-separator commas before pasting data.
How It Works
The results panel shows both simultaneously. Population variance (σ²) divides by N — only for complete populations. Sample variance (s²) divides by n−1 (Bessel's correction) — for sample data. For n=5, sample variance is 25% larger than population variance. For n=100 the difference is 1%. Use sample variance by default.
Pop: σ² = Σ(x−μ)²/N | Sample: s² = Σ(x−x̄)²/(n−1)[2,4,6]: Pop var=2.667, Sample var=4.000Variance = SD². Report SD for communication — it is in original units and interpretable. Use variance in formulas (ANOVA, regression) because variances of independent variables add: Var(X+Y) = Var(X) + Var(Y). The calculator shows both: sample SD and sample variance, population SD and population variance, from one calculation.
Var = SD² | SD = √Var | Var(X+Y) = Var(X) + Var(Y)SD=2, Var=4 | SD=3, Var=9 | SD=1.5, Var=2.25Switch to the Frequency Table tab. Enter each unique value and its count. The variance calculation expands internally — each value×frequency contributes to the sum of squared deviations. More efficient than entering the same value hundreds of times for exam score or survey data.
Var = Σ(f×(x−x̄)²) / (Σf or Σf−1)Scores 70×5, 80×12, 90×8 → variance over all 25 valuesSquared deviations make outliers disproportionately influential. In data [1,2,3,4,100]: mean=22, squared deviations sum=9250, variance≈1522. Without 100: mean=2.5, variance=1.25. The outlier inflated variance by over 1200×. This is why IQR-based spread measures are preferred for skewed data.
Outlier (value−mean)² >> normal (value−mean)²Data [1,2,3,4,100]: variance≈1522 vs [1,2,3,4]: variance=1.25CV = SD / |mean| × 100% = √Var / |mean| × 100%. It expresses variance as a proportion of the mean, enabling cross-dataset comparison with different units. Investment A: variance=25 on mean return 5% → CV=100% (very high risk). Investment B: variance=1 on mean return 5% → CV=20% (moderate). Lower CV = more consistent.
CV% = (√Var / |mean|) × 100 = (SD / |mean|) × 100Var=25, mean=5: CV=(5/5)×100=100% (high variability)Quick Reference
Verify these in the calculator above.
Population
[2,4,6] pop var
2.667
Sample
[2,4,6] samp var
4.000
SD to Var
SD=3 → variance
9
Var to SD
Var=16 → SD
4
Zero var
[5,5,5,5] variance
0
Sample
[1,2,3,4,5] samp var
2.5
Additive
Var(X+Y) =
Var(X)+Var(Y)
Classic
[2,4,4,4,5,5,7,9]
Pop var: 4
Tips & Shortcuts
The results show population variance and sample variance side by side — Excel VAR() matches sample variance; VARP() matches population variance.
Variance units are squared: if data is in kg, variance is in kg². Convert to SD (√variance) to restore original units for interpretation.
For financial risk, portfolio variance combines individual asset variances weighted by portfolio shares and correlations.
Very large variance relative to the mean (high CV²) suggests data may benefit from log transformation before statistical analysis.
The Frequency Table tab computes variance efficiently for grouped data — enter value and count instead of repeating each value.
Frequently Asked Questions
Variance measures spread as the average squared deviation from the mean. Population variance σ² = Σ(x−μ)²/N. Sample variance s² = Σ(x−x̄)²/(n−1). The squaring means larger deviations contribute disproportionately — variance is sensitive to outliers.
Standard deviation = √variance. Variance is in squared units (e.g. kg²), making it hard to interpret directly. SD is in original units (kg), making it interpretable. Variance is preferred in theoretical statistics because variances of independent variables add: Var(X+Y) = Var(X) + Var(Y).
Sample variance (n−1 denominator) for data that is a sample from a larger population. Population variance (n denominator) only when you have every value in the entire population. Most real-world use cases require sample variance. Excel VAR() = sample; VARP() = population.
High variance means values are widely spread from the mean — high variability. Low variance means values cluster tightly near the mean. Variance of zero means all values are identical. In finance, high variance = high risk; in quality control, low variance = consistent manufacturing.
1) Find the mean. 2) Subtract mean from each value. 3) Square each difference. 4) Sum all squared differences. 5) Divide by N (population) or n−1 (sample). Example: [2,4,6]: mean=4, diffs=(-2,0,2), squares=(4,0,4), sum=8, pop var=8/3≈2.67, sample var=8/2=4.
No. Variance is always non-negative because it sums squared values. Variance = 0 only when all values are identical. If a calculator returns negative variance, there is a floating-point rounding error or a software bug.
For a normal distribution N(μ, σ²), variance σ² completely determines the spread alongside the mean μ. The distribution formula e^(−(x−μ)²/2σ²) shows variance appears directly in the exponent. Larger variance → flatter, wider bell curve. Smaller variance → taller, narrower curve.
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