Key Idea: Topic 4.2 is about turning raw data into visual summaries. The three key displays are: frequency distributions (grouped data tables), histograms (bars show frequency density or frequency), cumulative frequency graphs (S-curves for finding medians and quartiles), and box-and-whisker plots (five-number summaries). Each display answers a slightly different question about your data.
✅ The five-number summary and quartiles
- Five-number summary
- Minimum, Q1 (lower quartile), Q2 (median), Q3 (upper quartile), Maximum. These five values completely describe the spread of a dataset.
- Q1, Q2, Q3
- Quartiles divide ordered data into four equal parts. Q2 is the median. Q1 = median of the lower half. Q3 = median of the upper half.
- IQR (interquartile range)
- IQR = Q3 − Q1. Measures spread of the middle 50% of data. Resistant to outliers (unlike range).
- Outlier (fence rule)
- A value is an outlier if it is below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. Mark outliers as individual points on a box plot.
📊 Reading cumulative frequency graphs
- Median from cumulative frequency
- Read off x at cumulative frequency = n/2. Example: 60 data points → read x at cf = 30.
- Q1 and Q3
- Q1 at cf = n/4. Q3 at cf = 3n/4. Draw horizontal lines to the curve, then drop vertically to the x-axis.
- Histogram bars
- Bars touch with no gaps (continuous data). Width × height = frequency (or use frequency density = frequency/class width for unequal classes).
Example: Data: 12, 15, 18, 20, 22, 25, 30, 35 Q2 = median = (20+22)/2 = 21 Q1 = median of {12,15,18,20} = (15+18)/2 = 16.5 Q3 = median of {22,25,30,35} = (25+30)/2 = 27.5 IQR = 27.5 − 16.5 = 11 Outlier fence: below 16.5 − 16.5 = 0 or above 27.5 + 16.5 = 44 → no outliers here.
When drawing a box plot: the box spans Q1 to Q3, the line inside is Q2, and the whiskers extend to the smallest/largest non-outlier values. For grouped data: use the midpoint of each class to estimate the mean; use the upper class boundary for cumulative frequency.
Paper 2 (GDC allowed): Enter data into lists and use 1-Var Stats to get Q1, Q3, and IQR automatically. The GDC also draws box plots. Paper 1: You may be given a completed cumulative frequency graph and asked to read off the median, Q1, or Q3 — show the horizontal and vertical lines on the graph for method marks.
IB-style question [7 marks]
The masses, in grams, of 50 apples picked from an orchard are grouped in the table below. Mass m (g): [80,100) → 6 apples; [100,120) → 14; [120,140) → 20; [140,160) → 8; [160,180) → 2. (a) Write down the modal class. (b) Estimate the mean mass of an apple. (c) Apples with mass at least 140 g are sold as 'large'. Estimate the percentage of apples that are 'large'.
Step by step:
(a) The modal class has the greatest frequency, which is 20.
(b) Estimate the mean from the class midpoints 90, 110, 130, 150, 170.
Multiply each midpoint by its frequency and add.
Divide by the total of 50 apples.
(c) 'Large' apples are in the classes [140,160) and [160,180): that is 8 + 2 = 10 apples.
Express the proportion as a percentage.
(a) [120, 140). (b) 124.4 g. (c) 20% are 'large'.