What Is Variance?
Variance is a fundamental concept in statistics that measures the spread or dispersion of a set of data points around their mean (average). A low variance indicates that the data points tend to be very close to the mean, while a high variance indicates that the data points are spread out over a wider range. This section provides a visual introduction to how two datasets can have the same average but vastly different spreads.
Mean: Variance: Std. Deviation:
Calculate Variance Step-by-Step
The best way to understand variance is to calculate it yourself. Use the interactive tool below to see how changing data points affects the mean, deviations, and the final variance value. This tool walks you through the entire process, from finding the average to summing the squared differences, providing a hands-on feel for the formula.
1. Enter Your Data
2. Choose Calculation Type
3. See The Results
Calculation Breakdown
| Data Point (x) | Deviation (x – μ) | Squared Deviation (x – μ)² |
|---|
Why Square the Deviations?
A common question is why we square the differences from the mean instead of just taking their absolute value. While the absolute value approach (called Mean Absolute Deviation) exists, squaring is preferred in statistics for two critical reasons. This section explores these justifications, explaining the mathematical advantages that make variance a cornerstone of statistical analysis.
Deviations Always Sum to Zero
If we simply summed the raw deviations from the mean, the positive and negative values would always cancel each other out, resulting in a total of zero. This would tell us nothing about the data’s spread. Squaring each deviation makes every value positive, ensuring that they accumulate to a meaningful total that reflects the magnitude of the spread, regardless of direction.
Using the data from the calculator, the sum of deviations is:
0
Population vs. Sample Variance
The formula for variance changes slightly depending on whether you have data for the entire population or just a sample of it. This distinction is crucial for accurate statistical inference. A population includes every member of a group, while a sample is just a subset. Below, we explain why we use a different denominator (N vs. n-1) and demonstrate its impact with a mini-simulation.
Population Variance (σ²)
When you have data for every single member of the group you are studying (e.g., the test scores of every student in a class), you use the population variance formula. Here, you divide the sum of squared deviations by the total number of data points, N.
Sample Variance (s²)
When you only have a sample of data and want to estimate the variance of the larger population, you use the sample variance formula. Here, you divide by the number of samples minus one, n-1. This is known as “Bessel’s Correction.”
Why n-1? A Simulation
A sample’s variance tends to be smaller than the true population’s variance. Dividing by n-1 instead of n corrects for this underestimation, giving us a more accurate estimate. Click the button below to draw a random sample from a predefined population and see how the two calculations compare to the true population variance.