Data presentation and diagramsAQA A-Level Maths: Revision notes

A histogram shows continuous data in classes. The key idea is that area represents frequency, not height. This matters when classes have different widths. $\text{frequency density}=\frac{\text{frequency}}{\text{class width}},\qquad \text{frequency}=\text{frequency density}\times\text{class width}.$ The vertical axis is frequency density, and bars touch because the data is continuous. Example: a class 5 to 10 minutes with frequency 20 has density $\frac{20}{5}=4$. A class 10 to 20 minutes with frequency 30 has density $\frac{30}{10}=3$, so its bar is shorter even though it holds more people.

To read a histogram, convert bars into frequencies by multiplying density by width.

• The modal class is the class with the highest frequency density (the tallest bar), not necessarily the highest frequency.
• To estimate the number in part of a class, assume the data is spread evenly: multiply the density by the width of the part. Example: for the classes with densities 10, 8, 4, 1.6 on widths 2, 3, 5, 10 (frequencies 20, 24, 20, 16), the number under 4 minutes is $20+2\times8=36$, and the number over 8 minutes is $2\times4+16=24$. The shape tells you about the distribution: symmetric, or skewed with a long tail to the right (positive skew) or left (negative skew).

Divide each frequency by the total to get relative frequency, and divide by the class width to get relative frequency density. Then the area of each bar is the relative frequency, so the total area is 1. This is how a histogram connects to a probability distribution: for a randomly chosen item, the probability of falling in an interval is the area of the histogram over that interval. Example: classes 0 to 5, 5 to 10, 10 to 20, 20 to 40 with frequencies 20, 40, 30, 10 out of 100 have heights $0.04,\ 0.08,\ 0.03,\ 0.005$. Then $P(8<T<15)=2\times0.08+5\times0.03=0.31$. As data and classes get finer, the histogram approaches a smooth curve with total area 1.

• Stem-and-leaf diagram: shows every value and the shape; always include a key and order the leaves.
• Box plot: shows the median, quartiles and extremes (and any outliers). Compare two sets by median (average) and by the width of the box (spread).
• Cumulative frequency graph: plotted at upper class boundaries; read the median and quartiles from the curve, and find how many values lie below a given value.
• Bar chart / pie chart: for categorical data; compare categories by height or by angle (angle $=\frac{\text{frequency}}{\text{total}}\times360^\circ$). When interpreting any diagram, comment on average, spread and shape in the context of the question.