Sheet № 250 · Foundation + Higher · AQA · Edexcel · OCR
Quartiles and Interquartile Range –
Quartiles and the interquartile range (IQR) are essential statistics skills tested on GCSE Maths papers at both Foundation and Higher tier. They measure the spread of the middle 50% of a data set and are more reliable than the range because they are not affected by extreme values. You need to find Q1, Q2 (the median) and Q3 from listed da
§Key definitions
Quartiles
divide an ordered data set into four equal parts:
Question:
Find Q1, Q2, Q3 and the IQR for this data: 3, 5, 7, 8, 12, 14, 17, 20, 23.
Answer:
Q1 = 6, Q2 = 12, Q3 = 18.5, IQR = 12.5.
(a)
IQR = 5.6 − 2.4 = 3.2 kg.
(b)
Upper boundary = Q3 + 1.5 × IQR = 5.6 + 1.5 × 3.2 = 5.6 + 4.8 = 10.4 kg.
§Formulas to memorise
IQR = Q3 − Q1
Q2 position = (n + 1) ÷ 2
Q1 position = (n + 1) ÷ 4
Q3 position = 3(n + 1) ÷ 4
Q1 at n ÷ 4 on the CF axis
Q2 at n ÷ 2 on the CF axis
Q3 at 3n ÷ 4 on the CF axis
An outlier is any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR
Worked example
Find Q1, Q2, Q3 and the IQR for this data: 3, 5, 7, 8, 12, 14, 17, 20, 23.
Working:
⚠ Common mistakes
- ✗Using n ÷ 2 instead of (n + 1) ÷ 2 for listed data. For listed data, use (n + 1) to find positions. For grouped data on a cumulative frequency diagram, use n ÷ 2 (without the +1).
- ✗Forgetting to order the data first. Quartiles only work on ordered data — always sort before calculating.
- ✗Confusing IQR with range. IQR = Q3 − Q1 (middle 50%). Range = max − min (all data).
- ✗Applying the outlier rule at Foundation. The 1.5 × IQR rule is Higher tier only. At Foundation, just calculate quartiles and IQR.
✦ Exam tips
- →For small data sets (listed values), use (n + 1) ÷ 4 and 3(n + 1) ÷ 4 for quartile positions.
- →For large grouped data sets (cumulative frequency), use n ÷ 4 and 3n ÷ 4 — read off the curve.
- →When the position is not a whole number, interpolate between the two adjacent values.
- →The IQR is preferred to the range because it ignores extreme values and measures the spread of the central half of the data.
- →For cumulative frequency and box plot skills, see cumulative frequency and box plots. For comparing data, see comparing data sets.