Variance Calculator
Enter a data set to instantly find variance, standard deviation, mean, count, and sum of squares — with the complete step-by-step work shown.
Variance Calculator
Variance Meaning in Statistics
Variance measures how far individual data points lie from the arithmetic mean. When variance is low, values cluster tightly around the center; when it is high, values spread across a wider range. Alongside standard deviation, variance is one of the most widely used measures of dispersion in descriptive statistics, probability theory, and data science.
How to Calculate Variance
Walk through this example with the data set 7, 12, 3, 9, 15, 6:
- Find the mean: (7 + 12 + 3 + 9 + 15 + 6) / 6 = 52 / 6 ≈ 8.6667
- Subtract the mean and square: (7 − 8.667)² = 2.779, (12 − 8.667)² = 11.109, (3 − 8.667)² = 32.149, (9 − 8.667)² = 0.111, (15 − 8.667)² = 40.109, (6 − 8.667)² = 7.111
- Sum of squares: 2.779 + 11.109 + 32.149 + 0.111 + 40.109 + 7.111 = 93.333
- Divide: Sample variance (s²) = 93.333 / 5 = 18.667. Population variance (σ²) = 93.333 / 6 = 15.556
Enter these numbers into the calculator above to verify each step.
Sample Variance Formula vs. Population Formula
The two formulas differ only in the denominator. Population variance (σ²) divides the sum of squared differences by n, because the data set contains every member of the group. Sample variance (s²) divides by n − 1, applying Bessel's correction to avoid underestimating the true spread.
- Population: σ² = Σ(xᵢ − μ)² / n
- Sample: s² = Σ(xᵢ − x̄)² / (n − 1)
In practice, most real-world data sets are samples rather than complete populations, so sample variance is the more common choice.
Variance and Standard Deviation
Standard deviation is the square root of variance. While variance expresses dispersion in squared units (e.g., cm²), standard deviation returns the result to the original unit of measurement (e.g., cm), making it more intuitive for interpretation. To convert back, simply square the standard deviation: if s = 4.32, then s² = 18.66.
Variance in Programming Languages
Most languages and spreadsheet tools include built-in functions for variance:
- Python:
statistics.variance(data)for sample variance;statistics.pvariance(data)for population. NumPy usesnumpy.var(data, ddof=1). - Excel / Google Sheets:
VAR.S()for sample variance,VAR.P()for population variance. - JavaScript: No native function — compute manually with
reduce(), or use a library likesimple-statistics. - R:
var(x)returns sample variance by default.
Practical Uses of Variance
In finance, variance measures portfolio risk — higher variance means more volatile returns. In manufacturing, keeping variance low ensures consistent product quality. In machine learning, the bias-variance tradeoff guides model complexity: too much variance leads to overfitting, while too little indicates underfitting. Variance also underpins ANOVA, hypothesis testing, and confidence intervals across the sciences.
Frequently Asked Questions
What is variance in statistics? ▼
Variance quantifies the spread of data points around their mean. It is computed by averaging the squared deviations from the mean. A small variance indicates tightly clustered values, while a large variance signals widely dispersed data.
How do you find the variance of a data set? ▼
Calculate the mean, subtract it from each value and square the result, sum all the squared differences, then divide by n (population) or n − 1 (sample). The calculator above automates every step.
What is the difference between sample and population variance? ▼
Population variance (σ²) divides the sum of squared deviations by n. Sample variance (s²) divides by n − 1 using Bessel's correction, which prevents the sample from systematically underestimating the true population variance.
How do you convert variance to standard deviation? ▼
Take the square root of the variance. For example, if s² = 18.667, then s = √18.667 ≈ 4.32. Conversely, squaring the standard deviation gives you the variance.
Can variance ever be negative? ▼
No. Because variance sums squared differences, the result is always zero or positive. A variance of zero means every value in the data set is identical to the mean.
How is variance calculated in Python and Excel? ▼
In Python, use statistics.variance() for sample variance or numpy.var(data, ddof=1). In Excel, VAR.S() returns sample variance and VAR.P() returns population variance.
Why does sample variance divide by n − 1 instead of n? ▼
Dividing by n − 1 (Bessel's correction) compensates for the fact that a sample mean is always closer to its own data points than the true population mean. Without this adjustment, sample variance would consistently underestimate the population variance.
How many data points do I need to calculate variance? ▼
At least two. For sample variance, the formula divides by n − 1, which requires n ≥ 2 to avoid division by zero. More data points generally produce a more reliable estimate of the true variance.