📊 Variance & Standard Deviation

• Variance is the average squared distance from the mean: $\sigma^2=\dfrac{\sum (x-\bar{x})^2}{n}$. Standard deviation is its square root: $\sigma=\sqrt{\sigma^2}$. Always finish with the square root if SD is asked.

• Shortcut from totals: $\sigma^2=\dfrac{\sum x^2}{n}-\left(\dfrac{\sum x}{n}\right)^2$ — the mean of the squares minus the square of the mean. Rearranged: $\sum x^2=n(\sigma^2+\bar{x}^2)$.

• Scaling rules for $y=ax+b$: the mean becomes $a\bar{x}+b$, the SD becomes $|a|\sigma$, and the variance becomes $a^2\sigma^2$. Adding a constant $b$ never changes the spread — only multiplying does. So ignore any “… and then add 3” when SD or variance is asked.

• AP shortcut: for an arithmetic progression with common difference $d$ and $n$ terms, $\sigma^2=\dfrac{d^2(n^2-1)}{12}$.