Percentiles, Cumulative Graphs and Categorical Displays

Percentiles and cumulative relative frequency

The $p$-th percentile of a distribution is the value such that $p\%$ of the observations are at or below it. Quartiles are simply the 25th, 50th and 75th percentiles, so the five-number summary is a set of percentiles in disguise. A cumulative relative frequency graph (ogive) plots, for each value $x$, the proportion of the data at or below $x$: it starts at $0$, ends at $1$ and never decreases. Read it in two directions — go up from a value to find its percentile, or across from a percentile to find the value.

Example: in a class of $40$ scores, $30$ are at or below $78$, so $78$ sits at the $\frac{30}{40}=75$th percentile — it is $Q_3$. A steep section of the ogive means many observations crowd there; a flat section means a gap in the data.

Categorical variables need different pictures

A bar chart shows the frequency or relative frequency of each category; the bars are separated because categories have no numeric order, and the vertical axis must start at $0$ or the picture lies. A pie chart shows part-to-whole only, so it must include every category. For a categorical variable you report the mode and the relative frequencies, never a mean — an "average eye colour" is meaningless.

Statistic versus parameter

A number computed from a sample is a statistic ($\bar x$, $\hat p$, $s$); the fixed but unknown number for the whole population is a parameter ($\mu$, $p$, $\sigma$). Keeping the symbols straight now makes every later unit readable.