Histograms, stem and leaf diagrams and box plotsEdexcel International A Level Maths: Revision notes

A histogram displays continuous grouped data. Because class widths can differ, the area of each bar (not its height) is proportional to the frequency, so the vertical axis shows frequency density: $\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$ Example: 60 calls with classes 0 to 2 (12 calls), 2 to 5 (21), 5 to 10 (15), 10 to 20 (12) have frequency densities $6,\ 7,\ 3,\ 1.2$. To estimate the frequency in part of a class, assume data are evenly spread: calls between 5 and 8 minutes $\approx3\times3=9$. Bar heights are in proportion to frequency densities.

A stem and leaf diagram shows every data value. The stem is the leading digit(s) and each leaf is the last digit. Leaves are written in order and a key is always given, e.g. 4|2 means 42. Because the data are in order, the median and quartiles can be read directly. For $n$ values: the median is the $\frac{n+1}{2}$th value; $Q_1$ is at position $\frac n4$ and $Q_3$ at $\frac{3n}{4}$. If the position is a whole number take the mean of that value and the next; otherwise round up. For 15 scores the median is the 8th value (42), $Q_1$ the 4th (34) and $Q_3$ the 12th (53).

A back-to-back diagram compares two data sets on one common stem, with leaves of one set to the left (reading outwards from the stem) and the other to the right. Each side is ordered with its own key. You find the median and quartiles of each side separately. Example: 12 girls' times 38, 40, 42, 44, 45, 46, 49, 51, 53, 55, 57, 62: median $\frac{46+49}{2}=47.5$, $Q_1=\frac{42+44}{2}=43$, $Q_3=\frac{53+55}{2}=54$, IQR $=11$.

A box plot shows the five-number summary: minimum, $Q_1$, median ($Q_2$), $Q_3$ and maximum. The box spans $Q_1$ to $Q_3$ with a line at the median, and whiskers extend to the extremes. The interquartile range is $Q_3-Q_1$ and the range is maximum $-$ minimum. Outliers, where identified, are plotted separately as crosses. Skewness: if $Q_3-Q_2>Q_2-Q_1$ the distribution is positively skewed; if $Q_3-Q_2<Q_2-Q_1$ it is negatively skewed; if they are equal it is symmetrical. Quarter rule: roughly 25% of values lie in each of the four sections, so in a sample of 40, about 10 values are above $Q_3$.

To compare two distributions, make at least two comments, each in the context of the question: one about location (median) and one about spread (IQR or range). Use numbers from both distributions. Example: $Y$ has median 47 and IQR 14; $X$ has median 45 and IQR 25. So $Y$'s typical mark is slightly higher and its marks are more consistent. Finish with a conclusion if asked which performed better, and avoid vague words such as 'better' without evidence.