What Is Variance?
Variance measures how spread out the values in a data set are around the mean. A low variance means the values cluster closely around the mean; a high variance means they are more dispersed. Variance is defined as the average of the squared differences from the mean.
There are two versions: population variance (σ²), which divides by n, and sample variance (s²), which divides by n−1 (Bessel's correction). The sample formula produces an unbiased estimate of the population variance when working with a subset of data.
Key Formulas
- Mean (μ or x̄) = Σx / n
- Population variance (σ²) = Σ(xᵢ − μ)² / n
- Sample variance (s²) = Σ(xᵢ − x̄)² / (n − 1)
- Standard deviation = √variance (same formula, just square-rooted)
- Median = middle value when sorted (or average of two middle values for even n)
Population Variance vs. Sample Variance
The choice between population and sample variance depends on whether your data represents the entire population or a sample drawn from a larger population.
- Use population variance (σ²) when you have data for every member of the group (e.g., test scores for an entire class, heights of all students in a specific school).
- Use sample variance (s²) when your data is a random sample drawn from a larger group (e.g., survey responses from 200 people representing millions).
- In practice, sample variance is the correct choice for almost all real-world data analysis.
- As sample size grows, s² converges to σ² — for large samples the difference is negligible.
Standard Deviation Interpretation
For a normal (bell-curve) distribution, approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This is known as the empirical rule or 68-95-99.7 rule.
Frequently Asked Questions
What is the difference between variance and standard deviation?
Standard deviation is simply the square root of variance. Variance is expressed in squared units of the original data (e.g., kg² if measuring weight), which makes it awkward to interpret. Standard deviation is in the same units as the original data (e.g., kg), making it easier to relate to the data set. Both measure spread; standard deviation is more commonly reported.
What does a variance of zero mean?
A variance of zero means all values in the data set are identical — there is no spread whatsoever. Every value equals the mean. This is mathematically valid but practically unusual in real data.
How do I calculate mode for a continuous data set?
For discrete data with exact repeated values, mode is the most frequently occurring value. For continuous data, there may be no repeated values, so mode is often reported as a range or found using a frequency histogram. This calculator reports mode only when two or more values appear more than once in the exact form entered.