Variance and Standard Deviation

Overview

Variance and standard deviation measure the spread or dispersion of data around the mean. They tell us how much individual values typically differ from the average.

Key Concepts

Measure	Symbol	Description
Variance	$\sigma^2$ (population), $s^2$ (sample)	Average squared deviation
Standard Deviation	$\sigma$ (population), $s$ (sample)	Square root of variance

Key Formulas

Population Variance

$$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$$

Sample Variance

$$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$$

Note: We divide by $(n-1)$ for samples (Bessel's correction) to get an unbiased estimate.

Population Standard Deviation

$$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$$

Sample Standard Deviation

$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$$

Computational Formulas

These are algebraically equivalent but often easier to calculate:

Variance (Computational Form)

$$s^2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n - 1}$$

Standard Deviation

$$s = \sqrt{\frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n - 1}}$$

Calculation Steps

1. Calculate the mean ($\bar{x}$)
2. Find each deviation from the mean ($x_i - \bar{x}$)
3. Square each deviation
4. Sum the squared deviations
5. Divide by $n-1$ (sample) or $N$ (population)
6. For SD, take the square root

Properties

Variance Properties

• Always non-negative ($\sigma^2 \geq 0$)
• Zero only when all values are identical
• Units are squared (e.g., meters² for length data)
• More sensitive to outliers than range

Standard Deviation Properties

• Same units as the original data
• Approximately 68% of data within 1 SD of mean (for normal distributions)
• Approximately 95% within 2 SD
• Approximately 99.7% within 3 SD

Transformations

For $Y = aX + b$:

$$\text{Var}(Y) = a^2 \cdot \text{Var}(X)$$ $$\text{SD}(Y) = \lvert a \rvert \cdot \text{SD}(X)$$

Note: Adding a constant doesn't change the spread.

Coefficient of Variation

A relative measure of variability:

$$CV = \frac{s}{\bar{x}} \times 100\%$$

Useful for comparing variability across datasets with different units or scales.

Examples

Example 1: Basic Calculation

Data: 2, 4, 4, 4, 5, 5, 7, 9

$$\bar{x} = \frac{40}{8} = 5$$

$x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
2	-3	9
4	-1	1
4	-1	1
4	-1	1
5	0	0
5	0	0
7	2	4
9	4	16
Sum	0	32

$$s^2 = \frac{32}{8-1} = \frac{32}{7} = 4.57$$ $$s = \sqrt{4.57} = 2.14$$

Example 2: Computational Formula

Same data: 2, 4, 4, 4, 5, 5, 7, 9

$$\sum x_i = 40, \quad \sum x_i^2 = 4 + 16 + 16 + 16 + 25 + 25 + 49 + 81 = 232$$ $$s^2 = \frac{232 - \frac{40^2}{8}}{7} = \frac{232 - 200}{7} = \frac{32}{7} = 4.57$$

Example 3: Comparing Datasets

Dataset	Mean	SD	CV
A	100	15	15%
B	50	10	20%

Dataset B has higher relative variability despite lower absolute SD.

Population vs Sample

Aspect	Population	Sample
Symbol	$\sigma^2$, $\sigma$	$s^2$, $s$
Divisor	$N$	$n-1$
Mean	$\mu$	$\bar{x}$
Use	Known full population	Estimating from sample