Ungrouped data

$$\text{Mean } \bar{x} = \frac{\sum x}{n} \qquad \text{Range} = \text{largest} - \text{smallest}$$ Median: middle value when ordered (average of the two middle values if $n$ is even); position $= \dfrac{n + 1}{2}$.

Frequency table

$$\bar{x} = \frac{\sum fx}{\sum f}$$ Using an assumed mean $A$: $\ \bar{x} = A + \dfrac{\sum fd}{\sum f}$ where $d = x - A$.

Grouped data (use class mid-points $x$)

$$\bar{x} = \frac{\sum fx}{\sum f}$$ $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right)\times c$$ $L$ = lower boundary of the median class, $N = \sum f$, $F$ = cumulative frequency before the median class, $f$ = frequency of the median class, $c$ = class width. $$\text{Mode} = L + \left(\frac{\Delta_1}{\Delta_1 + \Delta_2}\right) c$$ where $\Delta_1 = f_m - f_{\text{before}}$ and $\Delta_2 = f_m - f_{\text{after}}$ (frequency of the modal class minus the frequency of the class before and after it), $L$ = lower boundary of the modal class, $c$ = class width.

Quartiles

$$Q_1 \text{ at } \frac{N}{4}, \quad Q_2 \text{ at } \frac{N}{2}, \quad Q_3 \text{ at } \frac{3N}{4} \text{ (from the cumulative frequency curve)}$$ $$\text{Interquartile range} = Q_3 - Q_1 \qquad \text{Quartile deviation} = \frac{Q_3 - Q_1}{2}$$

Variance and standard deviation

$$\text{Variance } \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2 \qquad \sigma = \sqrt{\text{variance}}$$ Grouped: $\ \sigma^2 = \dfrac{\sum f x^2}{\sum f} - \left(\dfrac{\sum fx}{\sum f}\right)^2$.

Data representation

• Pie chart: angle $= \dfrac{\text{frequency}}{\text{total}}\times 360^\circ$.
• Histogram: area of a bar $\propto$ frequency; frequency density $= \dfrac{\text{frequency}}{\text{class width}}$.