Variance Calculator

Variance is a measure of how spread out the data points are from the mean (average). It measures the average squared distance from the mean. Low variance means data points are close to the mean, while high variance means they're more spread out. Variance is crucial for understanding data distribution and variability.

Population variance (σ²) divides by n (total count) and is used when you have data for an entire population. Sample variance (s²) divides by n-1 (count minus 1) and is used when you have a sample from a larger population. The n-1 correction (Bessel's correction) provides an unbiased estimate of the population variance.

Variance is calculated by: 1) Finding the mean of all values, 2) Subtracting the mean from each value and squaring the differences, 3) Finding the average of these squared differences. For population: σ² = Σ(x - μ)²/n. For sample: s² = Σ(x - x̄)²/(n-1).

Standard deviation is the square root of variance. It measures the same spread as variance but in the same units as the original data. Population standard deviation = √σ², sample standard deviation = √s². Standard deviation is often preferred because it's easier to interpret.

Use population variance when you have data for an entire population (e.g., all students in a school). Use sample variance when you have a subset of data from a larger population (e.g., 100 students from a school of 1000). In practice, sample variance is more common since we rarely have complete population data.

High variance indicates that data points are widely spread out from the mean, suggesting high variability or dispersion. This could mean the data is diverse, has outliers, or comes from a population with high variability. Low variance suggests data points are clustered close to the mean.

No, variance cannot be negative. Since variance involves squaring differences from the mean, all terms in the calculation are non-negative, making the final result always positive or zero. Zero variance means all values are identical.

The sum of squares (SS) is the sum of all squared differences from the mean: SS = Σ(x - x̄)². It's a key intermediate step in calculating variance and is used in many statistical tests. It measures the total squared deviation from the mean.

The squared differences show how far each data point is from the mean, squared. Larger values indicate points further from the mean. These values are used to calculate variance and standard deviation. They help you see which data points contribute most to the overall variability.

Both variance and range measure spread, but differently. Range is simply max - min, while variance considers all data points and their distances from the mean. Variance is more robust because it's not affected by just two extreme values like range is.

Use 2-4 decimal places for variance, depending on your data precision and context. For most practical purposes, 2-3 decimal places are sufficient. The calculator shows up to 4 decimal places but you can round as needed for your specific use case.

Variance is used in quality control (measuring product consistency), finance (risk assessment), research (comparing groups), weather forecasting (temperature variability), and many other fields where understanding data spread is important.