Single-variable data: diagrams and histogramsEdexcel A-Level Maths: Revision notes

A histogram is used for continuous data in classes, which may have different widths. The area of each bar represents the frequency, so the vertical axis shows frequency density: $\text{frequency density}=\frac{\text{frequency}}{\text{class width}},\qquad\text{frequency}=\text{frequency density}\times\text{class width}.$ There are no gaps between bars. To estimate the frequency in part of a class, assume the data are spread evenly: for example 8 to 12 minutes uses $4.5\times2$ from the bar up to 10 and $1.2\times2$ from the bar after. If one bar's height and frequency are known, the scale for all other bars follows from the ratio of areas.

A frequency polygon is drawn by plotting the frequency (or frequency density, if the histogram is being described) at the midpoint of each class and joining the points with straight lines. It shows the shape of the distribution and is useful for comparing two data sets on the same axes. The midpoint of the class $20\le x<30$ is $25$.

The cumulative frequency is the running total of frequencies. Plot each total at the upper class boundary and join with a smooth curve (or straight lines for a polygon). For $n$ values, read off or interpolate to estimate the median at $\frac{n}{2}$, the lower quartile $Q_1$ at $\frac{n}{4}$, and the upper quartile $Q_3$ at $\frac{3n}{4}$. Linear interpolation within a class: $\text{estimate}=L+\frac{\text{position}-F_{\text{below}}}{f}\times w,$ where $L$ is the lower boundary of the class, $f$ its frequency and $w$ its width. The interquartile range is $Q_3-Q_1$. Example: 8 plants under 10 cm and 30 under 20 cm gives $Q_1\approx10+\frac{25-8}{22}\times10=17.7$.

A box and whisker plot shows the minimum, $Q_1$, median, $Q_3$ and maximum. The box covers the interquartile range. Where a rule is given, values beyond the fences are outliers and are plotted separately, with the whiskers stopping at the most extreme values that are not outliers. A common rule: outliers lie below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$. For marks with $Q_1=17$, $Q_3=26$: IQR $=9$ and the upper fence is $26+13.5=39.5$, so 45 is an outlier. To compare two box plots, compare a measure of average (median) and a measure of spread (IQR or range), in context.

If you divide each frequency by the total, you get relative frequency, and dividing by class width gives relative frequency density. A histogram drawn like this has total area 1, and the area of a bar is the relative frequency, an estimate of the probability that a randomly chosen value lies in that class. As classes get narrower and the data set larger, the histogram's outline approaches a smooth curve, the probability density function of a continuous distribution. Probability is then the area under the curve. Example: 60 of 200 eggs in a class gives area $0.3$ for that bar.