4.2 Presentation of dataIB Maths: Analysis and Approaches HL: Revision notes

A frequency distribution lists each value or class with its frequency (how many data items it contains). For continuous data, classes are written as inequalities without gaps or overlaps, e.g. $10 \le L < 15$, $15 \le L < 20$. Every possible value then belongs to exactly one class: 15 cm goes into $15 \le L < 20$.

Grouping loses the individual values, so anything calculated from a grouped table (median, quartiles) is an estimate.

A frequency histogram with equal class widths has touching bars whose heights are the class frequencies. It shows the shape of a distribution: where the data cluster, whether it is symmetric or skewed, and any gaps.

Here everything is described in words, e.g. 'bars of heights 4, 9, 15, 12, 7, 3 for classes of width 5 cm from 10 cm'. Read these exactly like a frequency table. (Frequency density histograms with unequal widths are not required.)

The cumulative frequency up to a value is the number of data items less than it. From a cumulative frequency graph or table you can estimate:

• median: the value at position $\frac{n}{2}$;
• lower quartile $Q_1$ at $\frac{n}{4}$ and upper quartile $Q_3$ at $\frac{3n}{4}$;
• the $p$th percentile at position $\frac{p}{100} \times n$;
• IQR $= Q_3 - Q_1$ and range $=$ maximum $-$ minimum.

Without a graph, use linear interpolation: if the position lies $r$ items into a class of width $w$ containing $f$ items starting at $a$, the estimate is $a + \frac{r}{f} \times w$.

Example: 200 apples, 90 below 140 g, 60 in $140 \le m < 160$. Median (100th) $\approx 140 + \frac{10}{60} \times 20 = 143.3$ g.

A box and whisker diagram displays the five-number summary: minimum, $Q_1$, median, $Q_3$, maximum. The box runs from $Q_1$ to $Q_3$ with a line at the median. Outliers (more than $1.5 \times \text{IQR}$ beyond a quartile) are shown as separate points, and the whisker stops at the most extreme value that is not an outlier.

On this platform the five-number summary is given in words; you give the features (quartiles, IQR, outlier boundaries), not the drawing.

When comparing two distributions, compare one measure of centre (median) and one measure of spread (IQR or range), and interpret each in context: 'the median wait at B is 40 s shorter, so a typical customer is served faster'.

Symmetry: if the median is roughly in the middle of the box and the whiskers are similar lengths, the data are roughly symmetric and may be normally distributed. If the median is much closer to $Q_1$ (longer right side), the data are positively skewed; closer to $Q_3$, negatively skewed. Skewed data are not well modelled by a normal distribution.