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Complete IGCSE Edexcel Higher-tier guide to drawing box plots, calculating quartiles, and comparing distributions. Worked examples for grades 7-9.
A box plot displays the five-number summary of a data set: minimum, lower quartile, median, upper quartile, and maximum. The box covers the interquartile range (IQR = UQ - LQ), containing the middle 50% of data. Box plots are drawn from cumulative frequency curves or from given data, and are used to compare two distributions by discussing medians and spreads.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The five-number summary consists of the minimum value, lower quartile (Q1 or LQ), median (Q2), upper quartile (Q3 or UQ), and maximum value. Together these five values describe the shape, centre, and spread of a data set.
Draw a number line covering the range of the data. Mark each of the five summary values with a short vertical line. Draw a box from the LQ to the UQ with a vertical line inside for the median. Extend horizontal lines (whiskers) from the box to the minimum and maximum values.
Box plots are usually drawn above a number line with a clear, accurate scale. Use a ruler for all lines and ensure the box plot aligns precisely with the scale.
From raw data: arrange values in order. The lower quartile is at the (n + 1) / 4 position, the median at (n + 1) / 2, and the upper quartile at 3(n + 1) / 4.
From a cumulative frequency curve: read across from n/4 on the y-axis for the LQ, n/2 for the median, and 3n/4 for the UQ, then read down to the x-axis for the values.
The interquartile range (IQR) is the upper quartile minus the lower quartile: IQR = UQ - LQ. It measures the spread of the central 50% of data, ignoring extreme values and outliers.
A small IQR means the middle half of the data is closely clustered. A large IQR means there is more variation in the central data. The IQR is more reliable than the range because it is not affected by outliers.
When comparing two distributions (using box plots, cumulative frequency, or summary statistics), always make two types of comparison:
| Comparison Type | What to Compare | Example Statement |
|---|---|---|
| Average | Medians (or means) | "Group A has a higher median (52) than Group B (45), showing Group A generally scored higher." |
| Spread | IQRs (or ranges) | "Group B has a larger IQR (18) than Group A (12), showing Group B's scores are more varied." |
Both statements must include the numerical values and an interpretation in context. Simply stating "the median is higher" without numbers or meaning does not earn full marks.
From a cumulative frequency curve for 120 students, the following values are read: Minimum = 15, LQ = 32, Median = 45, UQ = 58, Maximum = 78. Draw a box plot and find the IQR.
Step 1: Draw a number line from 10 to 80.
Step 2: Mark the five values: 15, 32, 45, 58, 78.
Step 3: Draw the box from 32 to 58 with median line at 45. Add whiskers to 15 and 78.
Step 4: IQR = UQ - LQ = 58 - 32 = 26.
Answer: IQR = 26. Box plot drawn with box from 32 to 58.
Two classes took the same test. Class A: median = 62, IQR = 14, range = 45. Class B: median = 55, IQR = 22, range = 52. Compare the two distributions.
Average comparison: Class A has a higher median (62) than Class B (55). This shows that Class A generally performed better on the test.
Spread comparison: Class A has a smaller IQR (14) than Class B (22). This shows that the middle 50% of scores in Class A were more consistent, while Class B had more variation in their results.
Answer: Class A performed better on average (median 62 vs 55) and their scores were more consistent (IQR 14 vs 22).
Only comparing one feature: Full marks require comparing both an average and a measure of spread. Discussing only the median (or only the IQR) earns at most half the marks.
Not including values in comparisons: Writing "Group A did better" without quoting the actual median values does not earn marks. Always include the numbers.
Reading quartiles from wrong positions: For n data points, read the LQ from n/4 and UQ from 3n/4 on the cumulative frequency axis. Using (n+1)/4 for cumulative frequency curves gives slightly wrong values.
Inaccurate box plot drawing: Box plots must be drawn to scale with precise alignment to the number line. Freehand boxes that do not match the data lose accuracy marks.
Use a ruler: All lines in a box plot must be drawn with a ruler. This includes whiskers, box edges, and the median line.
Draw construction lines on CF curves: When reading quartiles, draw horizontal and vertical construction lines on the cumulative frequency curve. These show the examiner your method.
Use the comparison template: For every comparison question, write one sentence about average (quoting medians) and one about spread (quoting IQRs), both interpreted in context.
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