Variance Formula

The Formula

\sigma^2 = \frac{\sum(x - \mu)^2}{n}

When to use: Another spread measure—variance = \text{SD}^2. Same idea, different scale.

Quick Example

Data: \{2, 4, 6\}. Mean = 4. Deviations: -2, 0, 2. Squared: 4, 0, 4. Variance = (4+0+4)/3 \approx 2.67.

Notation

\sigma^2 for population variance, s^2 for sample variance

What This Formula Means

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

Another spread measure—variance = \text{SD}^2. Same idea, different scale.

Formal View

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2 (population); s^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\}.

Solution

  1. 1
    Find the mean: \mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5
  2. 2
    Calculate squared deviations: (2-5)^2=9,\ (4-5)^2=1,\ (4-5)^2=1,\ (4-5)^2=1,\ (5-5)^2=0,\ (5-5)^2=0,\ (7-5)^2=4,\ (9-5)^2=16
  3. 3
    Sum squared deviations: 9+1+1+1+0+0+4+16 = 32
  4. 4
    Divide by n: \sigma^2 = \frac{32}{8} = 4

Answer

\sigma^2 = 4
Variance measures how spread out values are from the mean. By squaring the deviations we ensure positive contributions and penalize larger deviations more heavily. Standard deviation would be \sqrt{4}=2.

Example 2

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

Common Mistakes

  • Forgetting that variance is in squared units — if data is in meters, variance is in \text{m}^2, not meters
  • Confusing population variance (\div n) with sample variance (\div (n-1))
  • Taking the square root of variance and calling it variance — that is the standard deviation, not variance

Why This Formula Matters

Variance quantifies how spread out data values are from the mean, forming the foundation for standard deviation and virtually all inferential statistics. It is used in finance to measure investment risk, in quality control to monitor manufacturing consistency, and in science to assess experimental reliability.

Frequently Asked Questions

What is the Variance formula?

The variance is the average of the squared deviations from the mean: \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 measure—variance = \text{SD}^2. Same idea, different scale.

What do the symbols mean in the Variance formula?

\sigma^2 for population variance, s^2 for sample variance

Why is the Variance formula important in Math?

Variance quantifies how spread out data values are from the mean, forming the foundation for standard deviation and virtually all inferential statistics. It is used in finance to measure investment risk, in quality control to monitor manufacturing consistency, and in science to assess experimental reliability.

What do students get wrong about Variance?

Take square root of variance to get SD (back to original units).

What should I learn before the Variance formula?

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

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