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Secondary 4 G3 Mathematics Tutor Bukit Timah | Box Plots, Median and Interquartile Range

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by

eduKate Asia
Three secondary students review textbooks and notes together at a classroom table.
Three secondary students comparing notes and Mathematics data in a small-group classroom.

Did you know? Two classes can have the same median score and still show very different patterns of consistency. A box-and-whisker plot makes both the centre and the spread visible, so a parent can distinguish a typical result from a wide range of results.

Quick answer: put the values in order, determine the median, identify the lower and upper quartiles using the convention appropriate to the question, and calculate interquartile range = Q3 − Q1. For a box plot, mark the five-number summary: minimum, Q1, median, Q3 and maximum. Compare both centre and spread.

The 2027 SEC G3 Mathematics K310 syllabus lists quartiles, percentiles, box-and-whisker plots, range, interquartile range and standard deviation. In this focused Bukit Timah guide, the worked examples state the quartile convention explicitly instead of assuming every spreadsheet and textbook uses exactly the same convention.

Quick answer in three checks

  • Order data and state a consistent quartile convention.
  • Compare medians for centre and IQRs for central spread.
  • Check common axes and avoid drawing causal conclusions from box plots.

Jump to the part you need

Why the median alone is not the whole story

Imagine nine scores ranging from 2 to 15. Their median might be 8. Another nine scores may range only from 5 to 11 and also have median 8.

The median describes the middle ordered value. It does not describe how close the other scores are to one another. Range and interquartile range provide different information about that spread.

Did you know? Saying a group is ‘more consistent’ only because its median is higher confuses centre with variability. A comparison should say which measure supports the conclusion.

Start with a reliable nine-score example

Take the ordered data set A = 2, 4, 5, 6, 8, 9, 10, 11, 15. Since there are nine values, the fifth is the median, so median = 8.

Using the median-of-halves excluding the central value convention for this worked example, the lower half is 2, 4, 5, 6. Its median is (4+5)/2 = Q1 = 4.5.

The upper half is 9, 10, 11, 15. Its median is (10+11)/2 = Q3 = 10.5. Therefore IQR = 10.5 − 4.5 = 6. This is a convention stated for teaching clarity; use the question’s prescribed method when different.

The five-number summary of Group A

For that same data set, the smallest value is 2, the lower quartile is 4.5, the median is 8, the upper quartile is 10.5, and the largest value is 15.

These five ordered quantities are what the ordinary box-and-whisker plot communicates. Its rectangular box runs from Q1 to Q3; the central marker shows the median; whiskers extend toward the displayed endpoints under the convention in use.

The overall range is 15 − 2 = 13. Do not confuse it with the interquartile range 6: they describe different parts of the observed distribution.

A second class with the same median but smaller spread

Now take B = 5, 6, 7, 8, 8, 9, 9, 10, 11. Its fifth value is also median = 8.

Using the same median-of-halves convention, Q1 is (6+7)/2 = 6.5. Q3 is (9+10)/2 = 9.5. Thus IQR = 3, while its total range is 11 − 5 = 6.

Group A and Group B share a median of 8, but B’s central half is more tightly clustered because its IQR is 3 rather than 6. This conclusion concerns spread, not which group necessarily scored better on every individual question.

How to compare the two box plots meaningfully

Both boxes have a median at 8. Group A’s box stretches from 4.5 to 10.5, while Group B’s box stretches from 6.5 to 9.5. B’s box is narrower, indicating a smaller spread in its middle half.

The full whisker-to-whisker range is also narrower in B: 6 compared with 13 in A. A student’s answer might say: ‘The typical median is the same, but Group B’s results are less spread out, based on its smaller interquartile range and range.’

Do not assume a narrow box proves better teaching or identifies the cause of the pattern. It is a descriptive statement about those fictional measurements.

Quartile calculations can have more than one convention

Software packages and classroom texts may use different accepted conventions for quartiles or percentiles, especially when the number of observations is odd. Some include the central value in quartile halves while others do not; interpolation can also differ.

The worked calculations here exclude the central median from each half. That choice is clearly stated so the examples can be checked. If an examination question supplies quartile values or a method, follow those instructions.

A strong mathematical explanation identifies which data positions have been used instead of pretending there is no possible convention issue.

Do the whiskers always go to minimum and maximum?

In a simple school five-number-summary box plot, whiskers can be drawn to the observed minimum and maximum. Other statistical plotting conventions draw whiskers to the most extreme non-outlying observations and plot suspected outliers separately.

Before interpreting an unfamiliar box plot, read the axis labels and any statement about whisker endpoints or outliers. A figure’s visual conventions are part of its meaning.

Do not introduce an outlier cut-off such as 1.5 × IQR as if it were required in every K310 school question. The published syllabus focuses on interpreting these diagrams and spread measures.

Where do percentiles fit?

The median corresponds to the 50th percentile as a general description of the central position. Lower and upper quartiles are associated with approximately the 25th and 75th percentiles.

Percentiles and quartiles describe ordered positions, not percentages of the maximum possible mark. A score at the 75th percentile does not mean 75% of all available marks were earned.

Always interpret a percentile relative to the actual group of observations and the method by which the statistical boundary was calculated.

Why ‘middle half’ is a useful phrase for IQR

The interquartile range spans the interval from Q1 to Q3 and describes the spread of the central 50% of the ordered distribution in the usual quartile interpretation.

It does not claim that exactly 50% of observations lie strictly inside the open interval between Q1 and Q3, especially for small samples or repeated values. Quartile boundaries are positional summaries, not independent counted categories.

For parents, ‘the central half of the scores is more compact in Group B’ is clearer than saying ‘the graph is better’ without naming what improved.

Contrast box plots with mean and standard deviation

The mean uses every observation in a numerical average, while standard deviation summarises how the values vary around that mean. Median and IQR use ordered positions and the central spread.

Different measures respond differently to an unusually extreme value. In Group A, the high observation 15 affects its range substantially, while the central quartile summary focuses more directly on the middle positions.

The 2027 K310 syllabus includes comparing data sets with mean and standard deviation too. The right comparison measure depends on the question and distribution; never substitute IQR for standard deviation without explanation.

A short check using a changed data set

Imagine replacing Group B’s largest score 11 with 30. Its sorted values would be 5, 6, 7, 8, 8, 9, 9, 10, 30. Under our stated convention, median remains 8, Q1 remains 6.5 and Q3 remains 9.5.

The IQR stays 3, while the full range becomes 30 − 5 = 25. This shows that a dramatic high value can change the range without necessarily changing the central half.

Do not conclude the altered group is uniformly less consistent on every measure; describe precisely which measure changed.

Five common errors in interpreting a box plot

One: reading the median from the middle of the chart image rather than the marked median line. Two: confusing range with IQR. Three: forgetting the axis scale. Four: comparing box widths without checking that both plots use the same scale. Five: describing a cause that the data do not establish.

In an examination, a concise comparison should state the measure, its value or relative size, and the meaning. For example: ‘Both medians are 8, but B has smaller IQR, 3 versus 6, so its middle scores are less dispersed.’

A graph-reading mistake is often different from an arithmetic mistake. A tutor can fix the first misread axis before asking the learner to calculate more quartiles.

A practical seven-day statistics revision loop

Day 1: order numerical observations and locate medians. Day 2: calculate Q1 and Q3 under a stated convention. Day 3: draw five-number-summary plots. Day 4: compare two plots on identical axes.

Day 5: practise median, IQR and range descriptions. Day 6: investigate how one extreme observation changes the summaries. Day 7: interpret an unfamiliar question without a worked solution or a prescribed conclusion.

Keep error tags ordering, quartile convention, axis reading, range versus IQR, unjustified inference and comparison lacking numbers. Use them to schedule a targeted independent retest.

Frequently asked questions about box plots

What is the interquartile range? Q3 − Q1, the width of the central half of the ordered distribution.

Can two groups have the same median but different IQR? Yes, as the worked data sets show.

Are all box-and-whisker conventions identical? No. Read the question’s quartile and whisker conventions.

Does a higher median prove every pupil did better? No. It only compares central positions.

Can an extreme value change the range without changing the median? Yes.

Is this a G3 Mathematics topic? Yes. Quartiles, percentiles, box plots and the IQR are explicit K310 statistics content.

How small-group tutoring connects these skills

The unchanged original eduKateSG Mathematics reference describes premium tutorials of up to three students, 1.5 hours weekly, with guided corrections and independent practice at 8 Fourth Avenue, Singapore 268674, near Sixth Avenue MRT. This is the stated teaching venue; the guide does not suggest a newly opened branch or guarantee places.

A useful small-group lesson begins with each student’s unaided response, locates the earliest incorrect decision and reteaches the idea. A changed question is then attempted without hints, and a later retest checks whether the improvement lasts. The number of worksheets completed is less informative than the errors that no longer recur.

Contact eduKate Singapore about a parent–student Mathematics consultation. A recent independent school script and the student’s actual examination year are helpful starting points.

Which exam-year syllabus applies?

Singapore’s first SEC examination is in 2027, with G3 Mathematics subject code K310. The SEAB 2027 G3 list identifies the earlier Mathematics 4052 reference. Students actually sitting the 2026 GCE O-Level must follow their applicable 4052 documents and instructions, rather than assume the 2027 assessment details apply to them.

Official sources and further practice

Continue this Secondary 4 G3 Mathematics series

Editorial note: All examples in this guide are original exercises for learning. They are not leaked examinations, specimen questions or grade guarantees. Refer to the official national examination requirements for the correct year.