Variance
Average squared deviation from the mean.
Formula
Variables
Example
For 2, 4, 6 (mean 4): σ² = (4+0+4)/3 ≈ 2.67.
How it works
Variance measures how spread out a set of values is by looking at how far each value lies from the mean, then averaging those departures after squaring them. Squaring is important because it makes all deviations positive and gives extra weight to values that are unusually far from the center. Dividing by the number of data points turns the total squared spread into a typical squared distance from the mean.
When to use it
Use variance when you want a precise numerical measure of variability in a dataset, especially in statistics, data analysis, and science. It is also the starting point for standard deviation, which is often preferred when you want spread reported in the original units.
Worked example, step by step
Data: 5, 7, 8, 10 Step 1: Find the mean. \bar{x} = (5 + 7 + 8 + 10) / 4 = 30 / 4 = 7.5 Step 2: Subtract the mean from each value. 5 - 7.5 = -2.5, 7 - 7.5 = -0.5, 8 - 7.5 = 0.5, 10 - 7.5 = 2.5 Step 3: Square each deviation. (-2.5)^2 = 6.25, (-0.5)^2 = 0.25, (0.5)^2 = 0.25, (2.5)^2 = 6.25 Step 4: Add the squared deviations. 6.25 + 0.25 + 0.25 + 6.25 = 13 Step 5: Divide by n = 4. σ² = 13 / 4 = 3.25 Final answer: the variance is 3.25.
Common mistakes to avoid
- Using the wrong mean, which makes every deviation incorrect
- Forgetting to square the deviations before averaging
- Mixing up variance with standard deviation and taking the square root too early
Did You Know?
Variance squares the deviations so positives and negatives can't cancel out.
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More in Statistics & Probability
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BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.
Combinations
IntermediateNumber of ways to choose r items from n (order irrelevant).