Variance Formula

The variance is the average of the squared deviations from the mean: ^2 = 1/n (x_i - x)^2.

The Formula

Οƒ2=βˆ‘(xβˆ’ΞΌ)2n\sigma^2 = \frac{\sum(x - \mu)^2}{n}

When to use: Another spread measureβ€”variance =SD2= \text{SD}^2. Same idea, different scale.

Quick Example

Data: {2,4,6}\{2, 4, 6\}. Mean =4= 4. Deviations: βˆ’2,0,2-2, 0, 2. Squared: 4,0,44, 0, 4. Variance =(4+0+4)/3β‰ˆ2.67= (4+0+4)/3 \approx 2.67.

Notation

Οƒ2\sigma^2 for population variance, s2s^2 for sample variance

What This Formula Means

The variance is the average of the squared deviations from the mean: Οƒ2=1nβˆ‘(xiβˆ’xΛ‰)2\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2. It is the square of the standard deviation.

Another spread measureβ€”variance =SD2= \text{SD}^2. Same idea, different scale.

Formal View

Οƒ2=1nβˆ‘i=1n(xiβˆ’ΞΌ)2\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 (population); s2=1nβˆ’1βˆ‘i=1n(xiβˆ’xΛ‰)2s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 (sample)

Worked Examples

Example 1

medium
Calculate the population variance for the data set: {2,4,4,4,5,5,7,9}\{2, 4, 4, 4, 5, 5, 7, 9\}.

Answer

Οƒ2=4\sigma^2 = 4

First step

1
Find the mean: ΞΌ=2+4+4+4+5+5+7+98=408=5\mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5

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Example 2

hard
Two investments have the same mean return of 8%. Investment A returns: {6,7,8,9,10}%\{6, 7, 8, 9, 10\}\%. Investment B returns: {2,5,8,11,14}%\{2, 5, 8, 11, 14\}\%. Calculate the variance of each and interpret.

Example 3

medium
Use the computational formula Οƒ2=βˆ‘xi2nβˆ’ΞΌ2\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 for {1,2,3,4,5}\{1, 2, 3, 4, 5\}.

Common Mistakes

  • Taking the square root and calling it variance β€” that result is the standard deviation; variance is before the root.
  • Averaging the deviations without squaring β€” raw deviations sum to zero, so square them first.
  • Mixing up the divisor β€” population variance divides by nn, sample variance (s2s^2) divides by nβˆ’1n-1.

Why This Formula Matters

Variance is the algebra-friendly form of spread: variances of independent quantities add, which is why it underlies regression, ANOVA, and the central limit theorem. Standard deviation is what you report to humans; variance is what the formulas run on. Recognizing it by "Am I averaging the squared distances from the mean (and not taking the square root)?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from standard deviation and mean and range in a mixed problem set.

Frequently Asked Questions

What is the Variance formula?

The variance is the average of the squared deviations from the mean: Οƒ2=1nβˆ‘(xiβˆ’xΛ‰)2\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2. It is the square of the standard deviation.

How do you use the Variance formula?

Another spread measureβ€”variance =SD2= \text{SD}^2. Same idea, different scale.

What do the symbols mean in the Variance formula?

Οƒ2\sigma^2 for population variance, s2s^2 for sample variance

Why is the Variance formula important in Math?

Variance is the algebra-friendly form of spread: variances of independent quantities add, which is why it underlies regression, ANOVA, and the central limit theorem. Standard deviation is what you report to humans; variance is what the formulas run on. Recognizing it by "Am I averaging the squared distances from the mean (and not taking the square root)?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from standard deviation and mean and range in a mixed problem set.

What do students get wrong about Variance?

The procedure for variance is the easy part; the trap is taking the square root and calling it variance. Asking "Am I averaging the squared distances from the mean (and not taking the square root)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Variance formula?

Before studying the Variance formula, you should understand: mean, standard deviation.

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