What does variance measure?

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Answer 1 · 2015-11-10T09:47:47

As the name of the topic indicates variance is a "Measure of Variability"

The variance is a measure of variability. It means that for a set of data you can say: "The higher variance, the more different data".

Examples

$A = {1, 3, 3, 3, 3, 4}$ $A={1,3,3,3,3,4}$

$¯ x = \frac{1 + 3 + 3 + 3 + 3 + 4}{6} = \frac{18}{6} = 3$ $bar(x)=(1+3+3+3+3+4)/6=18/6=3$

$σ^{2} = \frac{1}{6} \cdot ({(2 - 3)}^{2} + 4 \cdot {(3 - 3)}^{2} + {(4 - 3)}^{2})$ $sigma^2=1/6*((2-3)^2+4*(3-3)^2+(4-3)^2)$

$σ^{2} = \frac{1}{6} \cdot (1 + 1)$ $sigma^2=1/6*(1+1)$

$σ^{2} = \frac{1}{3}$ $sigma^2=1/3$

$B = {2, 4, 2, 4, 2, 4}$ $B={2,4,2,4,2,4}$

$¯ x = \frac{2 + 4 + 2 + 4 + 2 + 4}{6} = \frac{18}{6} = 3$ $bar(x)=(2+4+2+4+2+4)/6=18/6=3$

$σ^{2} = \frac{1}{6} \cdot (3 \cdot {(2 - 3)}^{2} + 3 \cdot {(4 - 3)}^{2})$ $sigma^2=1/6*(3*(2-3)^2+3*(4-3)^2)$

$σ^{2} = \frac{1}{6} \cdot (3 \cdot 1 + 3 \cdot 1)$ $sigma^2=1/6*(3*1+3*1)$

$σ^{2} = \frac{1}{6} \cdot (6)$ $sigma^2=1/6*(6)$

$σ^{2} = 1$ $sigma^2=1$

In set $A$ $A$ there are only 2 numbers other then the mean, and the difference is $1$ $1$ . The variance is small.

In set $B$ $B$ there are no elements equal to mean, and this fact makes the variance bigger.