Key Idea: This topic is about reading data off displays — frequency tables, histograms, cumulative frequency curves and box plots — and pulling out the summary numbers (mode, median, quartiles, percentiles, IQR, outliers). It shows up as 'read off the graph' / 'find the median' questions on both papers.
📊 Tables & histograms
| Display | What it shows | Key reads |
|---|---|---|
| Frequency table | Each value/class with how often it occurs. | Mode = the value with the highest frequency; total = sum of frequencies. |
| Histogram | Grouped continuous data — equal-width bars that touch. | Bar height = frequency; modal class = tallest bar. |
| Bar chart | Categories — bars have gaps. | Not a histogram — don't confuse the two. |
📈 Cumulative frequency curve
| You want… | Read across from… | Then |
|---|---|---|
| Median | n ÷ 2 (cumulative axis) | across to curve, down to the value |
| Lower quartile Q₁ | n ÷ 4 | across, down |
| Upper quartile Q₃ | 3n ÷ 4 | across, down — then IQR = Q₃ − Q₁ |
| pth percentile | p% × n | across, down |
| How many below a value | the value (data axis) | read up to the curve |
| How many between a and b | both a and b | subtract the two reads |
| 'Top X% exceed' a value | (100 − X)% of n | across, down (curve counts values below) |
🟦 Box plots & outliers
| Feature | Meaning |
|---|---|
| Five-number summary | min, Q₁, median, Q₃, max |
| The box / whiskers | box spans Q₁→Q₃ (median inside); whiskers reach min & max |
| Range vs IQR | range = max − min; IQR = Q₃ − Q₁ = middle 50% |
| The 4 sections | each holds about 25% of the data |
| Comparing two sets | compare medians (centre) and IQRs/ranges (spread) |
- interquartile range — compute this first
- any value beyond a fence (below lower or above upper)
✏️ IB-style worked examples
IB-style question — read a frequency table
A survey of 50 flats records the number of bicycles owned: 0 → 12, 1 → 19, 2 → 13, 3 → 6. State the mode and find how many flats own at least 2 bicycles.
Step by step:
Mode = the value with the highest frequency.
'At least 2' means 2 or 3 — add those frequencies.
Mode = 1 bicycle; 19 flats own at least 2.
IB-style question — median & IQR from a cumulative curve
A cumulative frequency curve for 120 students' test times is drawn (totals plotted at the upper boundaries). Reading across, the curve gives 24 min at a cumulative frequency of 60, 18 min at 30 and 31 min at 90. Estimate the median and the interquartile range.
Step by step:
Median is read from n ÷ 2.
Quartiles from n ÷ 4 and 3n ÷ 4.
IQR = Q₃ − Q₁.
Median ≈ 24 min; IQR ≈ 13 min.
IB-style question — test for an outlier (1.5×IQR rule)
A data set has Q₁ = 22 and Q₃ = 38. Determine whether a value of 65 is an outlier.
Step by step:
Find the IQR.
Upper fence = Q₃ + 1.5 × IQR.
Compare 65 with the upper fence.
Upper fence = 62, and 65 > 62, so 65 is an outlier.
Important: Each running total counts everything up to the top of the class, so plot it at the upper boundary, never the midpoint. Plotting at midpoints shifts the whole curve and ruins every median/quartile read.
Tap each card to reveal the answer.
In a frequency table, 2 occurs 17 times (the most). What is the mode? 2 — the mode is the data value, not its frequency (17).
Histogram or bar chart: which has bars that touch? Histogram — continuous data, bars touch. Bar charts have gaps.
On a cumulative curve for n = 200, what level gives Q₃? 3n ÷ 4 = 150 — read across from 150, down to the value.
Five values: 11, Q₁ 19, median 25, Q₃ 34, 40. Find the IQR. IQR = 34 − 19 = 15 (Q₃ − Q₁); the range is 40 − 11 = 29.
Q₁ = 40, Q₃ = 60. Is 8 an outlier? Yes — IQR = 20, lower fence = 40 − 1.5(20) = 10, and 8 < 10.
Top 25% of 80 students exceed t — what cumulative frequency do you read across from? 60 — 75% are below, so read from 0.75 × 80 = 60.
Exam tips
- Mode = the data value, not its frequency; for grouped data name the modal CLASS.
- Histogram bars touch (continuous); bar-chart bars have gaps (categories).
- Plot cumulative totals at the UPPER class boundary, then read median at n/2, Q₁ at n/4, Q₃ at 3n/4.
- For 'top X% exceed' read across from (100 − X)% of n; for 'how many between' subtract two reads.
- Range = max − min; IQR = Q₃ − Q₁. Compute IQR before testing outliers with the 1.5×IQR fences.