Variance Calculator — Population and Sample Variance | SoftZaR

Value	Deviation (x − mean)	Squared Deviation
12	-2.5	6.25
15	0.5	0.25
14	-0.5	0.25
10	-4.5	20.25
19	4.5	20.25
17	2.5	6.25
13	-1.5	2.25
16	1.5	2.25

Value

Deviation (x − mean)

Squared Deviation

-2.5

6.25

0.5

0.25

-0.5

0.25

-4.5

20.25

4.5

20.25

2.5

6.25

-1.5

2.25

1.5

2.25

What This Tool Does

This calculator computes variance from a numeric dataset and provides supporting spread metrics such as mean, standard deviation, range, and coefficient of variation. It supports both population and sample formulas so you can match your analysis context. A row-level deviation table is included to show how each observation contributes to total dispersion.

Variance Formulas

Population Variance

σ² = Σ(xᵢ − μ)² / N

Sample Variance

s² = Σ(xᵢ − x̄)² / (n − 1)

xᵢ: each data value
μ or x̄: mean
N or n: number of observations

Worked Example

For values [4, 6, 8], mean = 6.

Deviations: -2, 0, +2
Squared deviations: 4, 0, 4
Sum squared deviations = 8
Population variance = 8/3 = 2.667
Sample variance = 8/2 = 4.000

Interpreting Results

Higher variance indicates wider spread and less consistency around the mean.
Lower variance indicates tighter clustering and greater consistency.
Standard deviation is easier to interpret in original units, while variance is algebraically convenient.
Compare coefficient of variation when datasets use different scales.

When to Use Population vs Sample Mode

Use population mode when the dataset includes every item in the target group. Use sample mode when data are a subset used to estimate broader population behavior. Sample mode uses n−1 to reduce estimation bias and is standard in inferential statistics.

Sources and References

Rice, J. A. Mathematical Statistics and Data Analysis.
Montgomery, D. C., & Runger, G. C. Applied Statistics and Probability for Engineers.
NIST/SEMATECH e-Handbook of Statistical Methods — Measures of Variation.

Frequently Asked Questions

What is variance in simple terms?

Variance measures how spread out values are around the mean. If values cluster tightly, variance is low. If values are widely scattered, variance is high. It is calculated from squared deviations from the mean, so larger gaps contribute disproportionately. Variance is foundational for standard deviation, confidence intervals, regression diagnostics, and many machine-learning methods.

What is the difference between sample and population variance?

Population variance divides by N because you have the full population. Sample variance divides by N−1 (Bessel correction) to reduce downward bias when estimating population variance from a sample. If you collected all units in your target group, use population mode. If your data are only a subset, use sample mode for unbiased estimation.

Why are deviations squared in variance?

Squaring ensures positive and negative deviations do not cancel out, and it gives more weight to larger differences from the mean. This makes variance sensitive to outliers, which can be useful for detecting instability. Because squaring changes units, standard deviation (the square root of variance) is often preferred for direct interpretation in original units.

Can variance be negative?

No. Variance is always zero or positive because it is based on squared values. A variance of zero means all values are identical. If software returns a negative variance, it is usually a numerical precision issue or a calculation bug. In practical analytics, negative variance should be treated as invalid and investigated.

How does variance relate to standard deviation?

Standard deviation is the square root of variance. Variance is useful algebraically in formulas and model fitting, while standard deviation is easier to interpret because it matches the units of the original data. Reporting both can be helpful: variance for statistical calculations and standard deviation for communication and decision making.

When can variance be misleading?

Variance can be distorted by extreme outliers and by data with strong skewness or heavy tails. It also does not capture distribution shape, so two datasets may share variance but look very different. Pair variance with median, percentiles, and visualization (histogram or boxplot) for more complete understanding of variability.