Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability — Topic 4.2
📋 Learning Intentions
- Construct frequency histograms and cumulative frequency graphs from grouped data tables.
- Interpret histograms by reading frequency densities and identifying distributional shape (skew, symmetry).
- Find the median, quartiles, and percentiles from a cumulative frequency graph.
- Draw box and whisker diagrams from a five-number summary and identify outliers.
- Compare two data sets using parallel box and whisker diagrams, commenting on centre and spread.
- Use the GDC to generate statistical summaries and verify graphical readings.
Real-World Introduction
When a hospital records the waiting times of 400 patients, a list of 400 numbers communicates almost nothing. Grouping those times into intervals and drawing a histogram immediately reveals whether most patients wait under 20 minutes or whether a long tail of delays exists. The shape of the histogram guides decisions about staffing and resource allocation.
Cumulative frequency graphs allow a hospital manager to answer questions such as "What percentage of patients waited less than 30 minutes?" without examining every individual record. Box and whisker diagrams then allow two hospitals — or two shifts — to be compared side by side on a single axis, making differences in median waiting time and variability immediately visible.
🌍 Why This Matters The Office for National Statistics uses cumulative frequency analysis to report income distribution across the UK population, identifying the median household income and the interquartile range as key measures of inequality. Every data visualisation in public health, economics, and environmental science relies on the same three graphical tools covered in this topic.
Core Concepts & Definitions
When data is continuous and grouped into class intervals, a bar chart is no longer appropriate because the bars must be adjacent (no gaps) and the horizontal axis must be a continuous numerical scale. The visual tool designed for this purpose is the histogram.
📖 Key Definition — Histogram A histogram is a graphical display of grouped continuous data in which each class interval is represented by a rectangle. The rectangles are adjacent (no gaps). The area of each rectangle is proportional to the frequency of that class. When all class widths are equal, the height of each bar equals the frequency (or relative frequency) of that class.
In IB Math AI, histograms almost always use equal class widths. This means the vertical axis can be labelled frequency directly, and bar height equals frequency. If class widths are unequal, the vertical