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How to Perform in the new G3 SEC Examinations | Learner’s Guide Vol 0107 | Mathematics: Box-Plot Comparison Workshop — Same Median, Different Spread

G3 SEC Mathematics | Learner’s Guide Vol 0107. Read the median, quartiles and spread, then make a defensible comparison.

Two classes can have the same median score and still be very different. One class may have almost everyone performing close to the middle. Another may include much lower and much higher results. If you compare only the medians, you can miss the most important distinction. A box-and-whisker plot gives you a compact way to read both centre and spread.

This G3 SEC Mathematics Learner’s Guide Vol 0107 shows you how to construct and interpret a box plot from ordered observations, locate the median and quartiles, calculate the interquartile range, compare datasets fairly and recognise what a box plot does not tell you. You will work through original examples involving test scores, journey times and measurements, then practise the kind of concise evidence-based explanation that a K310 question may require.

The central habit is to read the numerical landmarks before reacting to the picture. A taller or wider box may look more dramatic, but a comparison must name which axis quantity and interval it represents. Keep the data, the graph and the conclusion attached to one another.

Chapter index

1. Start with the five-number summary

A conventional school-level box-and-whisker plot is based on five landmarks: minimum, lower quartile, median, upper quartile and maximum. The minimum is the least value; the maximum is the greatest. The median divides the ordered observations at the centre. The lower and upper quartiles identify positions around one quarter and three quarters of the data, using the quartile convention specified or taught for the task.

Before finding these landmarks, put observations in numerical order. Consider the eight scores 2, 4, 5, 6, 6, 7, 8 and 10. The median is the mean of the fourth and fifth values: (6+6)/2=6. The lower half is 2, 4, 5, 6, whose median is (4+5)/2=4.5. The upper half is 6, 7, 8, 10, whose median is (7+8)/2=7.5. For this even-sized example using the median-of-halves convention, Q1=4.5 and Q3=7.5.

The five-number summary is therefore minimum 2, lower quartile 4.5, median 6, upper quartile 7.5 and maximum 10. The interquartile range, or IQR, is 7.5−4.5=3. The ordinary range is 10−2=8. IQR describes the spread between the quartile landmarks; range measures the entire span from minimum to maximum. Those are different summaries.

It is worth repeating the first step aloud: sort the values. If the original data arrive as 8, 2, 6, 4, 10, 7, 6, 5, the quartiles do not depend on that presentation order. A student who chooses the third number as Q1 without sorting has mistaken the position on the page for the position in the ordered dataset.

Some textbooks and digital tools use different quartile conventions, especially for odd sample sizes. In an examination, use the method specified in the question or established in your course. Our worked examples make the chosen method visible and use even numbers of observations to avoid unnecessary ambiguity.

2. What the official 2027 K310 syllabus actually covers

The official SEAB 2027 G3 Mathematics syllabus K310 lists box-and-whisker plots and cumulative-frequency diagrams under Statistics and Probability. It also includes quartiles and percentiles, range and interquartile range, standard deviation, and comparing sets of data using appropriate statistical summaries. A box plot is therefore part of the published G3 Mathematics content, not an invented exam topic.

K310 has two compulsory papers. Paper 1 lasts 2 hours 15 minutes, contains approximately 26 short-answer questions and carries 90 marks. Paper 2 also lasts 2 hours 15 minutes, contains 9 to 10 questions of varying length and carries 90 marks; its final question applies mathematics to a real-world scenario. Each paper contributes 50% of the subject assessment. The syllabus states that omission of essential working results in lost marks.

Unless a different accuracy is specified, non-exact numerical answers are generally given to three significant figures, and angles in degrees to one decimal place. Quartiles calculated exactly from small integer datasets can often be stated exactly, while numerical answers from other calculations should follow the question’s accuracy requirement. A calculator may assist with arithmetic; it cannot tell you which comparison the question is asking you to make.

You can revisit the G3 SEC learner-guide route and the Mathematics Hub for the complete course. Volume 0103 on cumulative-frequency intervals owns the cumulative-count route. This volume owns direct comparison of five-number summaries and box plots, rather than retelling the whole statistics syllabus.

A box plot can be used in either short numerical work or broader interpretation. Learn what the marks mean first. Then practise saying precisely what is comparable, what differs and which facts are unavailable.

3. Read the diagram from left to right

Imagine a horizontal box plot above a number line. The left whisker reaches the minimum, the left side of the box marks Q1, the line inside the box marks the median, the right side marks Q3 and the right whisker reaches the maximum. These locations should be read against the numerical scale, not guessed from the box’s apparent shape.

For the dataset 2, 4, 5, 6, 6, 7, 8, 10, place the marks at 2, 4.5, 6, 7.5 and 10. Draw a box from 4.5 to 7.5 and a line inside it at 6. Extend whiskers to 2 and 10 under the convention used in this example. Each part has a meaning. The distance from Q1 to Q3 is three units; the full range is eight.

Not all plotted boxes include every observation. The diagram compresses the dataset into five landmarks. Two datasets may have the same five-number summary but differ in several internal values. A box plot is useful because it simplifies; that same simplification creates a limit on what it can tell you.

The median line may not sit in the geometric centre of the box. That does not make the diagram incorrect. It may indicate unequal spacing between the lower quartile and median and between the median and upper quartile. Use the actual measured distances to describe the contrast. Do not automatically pronounce a probability distribution to be exactly symmetrical or skewed based on one small dataset.

A horizontal axis labelled “minutes” represents time values. The box’s width is measured in minutes. If another graph uses “seconds”, convert before comparison. A box that looks shorter on a different chart with different scale dimensions is not necessarily less variable in the underlying data.

Finally check whether both plots use the same numerical scale. A comparison made from two separately scaled diagrams can be misleading. Prefer plots on one common axis or work directly from the five-number values.

4. Same median, different middle spread

Consider Class A with scores 2, 4, 5, 6, 6, 7, 8, 10. Its median is 6, Q1=4.5 and Q3=7.5, so IQR=3. Now consider Class B with scores 0, 1, 2, 6, 6, 10, 11, 14. Its median is also 6. But Q1=(1+2)/2=1.5 and Q3=(10+11)/2=10.5, so IQR=9.

Both classes have the same median score of 6. The middle 50% of Class B’s results covers a larger numerical interval, so its IQR is greater. The range also differs: Class A’s range is 10−2=8, and Class B’s is 14−0=14. Those figures describe two distinct measures of spread, and both indicate a wider span for Class B in this example.

What can you conclude? The typical central value represented by the median is equal, while the scores are more dispersed according to the IQR in Class B. You cannot conclude that every Class A pupil performed better than every Class B pupil. Class B includes very high scores as well as low ones. The diagram does not support a uniform ranking of individuals.

An efficient K310 response might say, “Both groups have median 6. Group B has a larger interquartile range (9 compared with 3), indicating greater spread across the middle half of the data.” That is short, numerical and specific. It does not require elaborate language or the claim that Group B was poorly taught.

The same lesson works in travel-time comparisons. Two routes can have equal median times but different IQRs. If you value consistency around the median, the route with smaller IQR may be attractive under the observed data, but the range, sample and any special constraints may also matter. A single statistic should not be converted into a blanket guarantee.

This case continues the comparison discipline from Volume 0081 on the comparison basis: say what was measured, which groups were compared and what exactly the statistic establishes.

5. Larger median does not mean smaller or larger spread

Suppose Group C has observations 4, 5, 5, 7, 7, 8, 8, 9. Its median is 7. Lower quartile=(5+5)/2=5; upper quartile=(8+8)/2=8; IQR=3. Compare Class A from the earlier example: median 6 and IQR 3. Group C has a greater median by one unit but the same interquartile range. Its middle spread is not greater merely because its central value is higher.

Now imagine Group D with values 4, 4, 6, 6, 8, 8, 10, 10. Median=(6+8)/2=7; lower quartile=(4+6)/2=5; upper quartile=(8+10)/2=9; IQR=4. Groups C and D share median 7, yet D has the larger IQR. That comparison tells a different story from comparing Group C with Class A.

Some learners use “higher box” or “longer box” as their entire answer. A box plot cannot be interpreted that way without reading the axis and identifying which feature is higher or longer. The median’s position is a measure of centre; the width of the box along the value axis represents the IQR. The whiskers capture the minimum and maximum in the simple convention used here.

Try writing two separate sentences for each comparison. One describes central tendency: “Group C has higher median than A.” The other describes spread: “Their IQRs are equal.” It is possible for both to be true, and neither sentence automatically explains the cause of the performance difference.

Remember that a high median does not prove all members performed well. Some results may lie below Q1, and a box plot deliberately does not display every individual score. A precise statement uses the data’s summary, not an imagined list of performances.

The ability to keep centre and spread separate is one of the most important improvements a student can make in statistics.

6. Range and interquartile range answer different questions

The range measures the distance between maximum and minimum. It is affected directly by extreme values. The IQR measures the interval between Q1 and Q3, focusing on the middle part of the ordered data. Neither is universally better. Each describes a different aspect of variation.

Consider the eight values 4, 5, 5, 6, 6, 7, 7, 8. Minimum 4, maximum 8, range 4. Lower quartile=(5+5)/2=5, upper quartile=(7+7)/2=7, IQR=2. Now replace the last value, 8, with 30. The sorted series becomes 4, 5, 5, 6, 6, 7, 7, 30. Its range becomes 26. Yet Q1 remains 5, Q3 remains 7 and IQR remains 2 in this particular eight-value example.

This comparison is powerful. One extreme observation changes the range dramatically but does not affect the quartile positions here. That does not mean IQR is always entirely unaffected by changes at extremes, nor that extreme values should be ignored. It means a box plot can help separate middle spread from the outer extremes.

Suppose a school asks which delivery service is more consistent for typical deliveries. A smaller IQR may be useful for the central half of observed deliveries. If the key concern is the worst possible observed delay, maximum and range may also matter. The correct statistic depends on the question.

A learner should not say “The IQR gives the spread of all data”. It gives the difference between the upper and lower quartiles. Similarly, “The range shows whether most values are clustered” overstates its meaning. Two datasets can have the same range and very different central distributions.

A good interpretation names the measure and its limitation. “The IQR is 2 minutes, showing that the interval between the lower and upper quartiles is two minutes” is directly grounded. “Most customers will always wait no more than two minutes” would confuse spread with waiting-time level and a future guarantee.

7. Median and mean are not interchangeable

The median is the centre of the ordered data; the mean is the sum of values divided by the number of values. Both measure central tendency, but they respond differently to extreme observations. A box plot displays the median, not necessarily the mean. Unless a mean is shown or data are given, you should not read it from the centre line.

Take 4, 5, 5, 6, 6, 7, 7, 8. Its median is 6; its mean is (4+5+5+6+6+7+7+8)/8 = 48/8 = 6. Now change the final 8 to 30. The median stays 6. The mean becomes (48−8+30)/8=70/8=8.75. The extreme value raises the mean while the median remains unchanged in this example.

A learner who sees an unchanged box-plot median and declares that the mean is unchanged has made an unsupported inference. The mean can change while the five-number summary changes little in some respects. The precise data values determine the mean.

Why do we use different measures? If a dataset contains a few very large values, the mean can be pulled upwards. A median may better represent a typical ordered position. But choosing a measure is a judgement about the question. Financial amounts, scores, times and experimental values may call for different summaries.

The K310 syllabus also includes standard deviation and comparisons using mean and standard deviation. Those are distinct from median and IQR. Do not replace a request for standard deviation with an IQR calculation just because both concern spread. The command word and named statistic decide the method.

In a response, keep the nouns exact: median, mean, range, IQR and standard deviation. Calling them all “the average” makes an otherwise correct explanation harder to judge.

8. Reading two box plots on one common axis

Imagine Route A has a five-number summary of 18, 21, 24, 27 and 35 minutes. Route B has 15, 19, 24, 31 and 42 minutes. Both medians are 24 minutes. Route A’s IQR is 27−21=6 minutes; Route B’s is 31−19=12 minutes. Route B also has the larger range: 42−15=27 minutes versus 35−18=17 minutes.

A valid comparison is: “The routes have the same median. Route B has greater variability in the middle half of observed journeys and a wider overall observed range.” If a question asks for the route with more consistent journey times by the IQR criterion, A is preferable on these observed statistics.

But suppose a commuter must arrive within 20 minutes. The median alone does not establish which route meets that deadline more often. A five-number summary gives broad distribution landmarks, not the exact count of journeys under every threshold. The minimum of Route B is lower, but its upper values are also higher. The commuter may need fuller frequency data to answer a specific probability question.

The lower whisker on B begins at 15, while A begins at 18. It is accurate to say the shortest observed journey in B was shorter. It is not accurate to say B is always faster. The extreme point is only one part of the data.

A graph comparison under time pressure should name the common axis first. If both plots use minutes and one numerical scale, comparing their landmarks is straightforward. If one is in seconds, convert. If the box positions are guessed from differently scaled illustrations, the visual conclusion may be misleading.

This is where Volume 0099 on graph scales helps. The numerical values, not the size of the picture, determine the statistical comparison.

9. What a box plot cannot reveal

A box plot does not show every observation. It does not identify which student obtained a particular score unless the data are separately given. It generally does not show exact frequencies of individual scores. It does not establish why one group has a larger IQR, nor does it prove how a future population will behave.

Suppose two classes have identical five-number summaries. They might still have different individual scores inside the quartile intervals. The plots match, but the underlying datasets need not be identical. A learner who says “The classes have exactly the same results” has gone beyond the summary.

Likewise, two box plots with different medians do not prove that one teaching method caused higher scores. The classes may have differed in earlier ability, paper difficulty, sample size or other conditions. A statistical comparison is descriptive unless a relevant study design provides evidence for a causal conclusion.

An examination may ask whether there is sufficient information to determine the mean from a box plot. In general, not from a five-number summary alone. The missing internal data values can change the total sum without changing the displayed landmarks. This is an important distinction between knowing the median and knowing the mean.

Do not dismiss the box plot because it cannot answer every question. It is useful precisely because it compresses many observations into a small set of interpretable landmarks. Use it for the questions it can answer and seek raw data or a more suitable summary where necessary.

The broader lesson matches Volume 0101 on evidence sufficiency: distinguish conclusions directly supported by the representation from claims needing additional evidence.

10. An original worked case with a changed observation

A group records eight waiting times: 3, 4, 5, 6, 6, 7, 8 and 9 minutes. The median is 6. Q1=(4+5)/2=4.5, Q3=(7+8)/2=7.5 and IQR=3. The range is 9−3=6. These values summarise the original sample under our stated convention.

A ninth observation of 30 minutes is added. The dataset now has nine values, so the median and quartile conventions need attention. Rather than silently reuse the earlier positions, first note that the number of observations changed. If the question specifies a quartile method, follow it. To avoid a convention dispute in this teaching exercise, let us replace the original maximum of 9 with 30 instead of adding a value.

The revised eight times are 3, 4, 5, 6, 6, 7, 8 and 30. Median remains 6; Q1 remains 4.5; Q3 remains 7.5; IQR remains 3. The range increases from 6 to 27. The extreme delay changes the maximum and overall range while leaving the middle landmarks unchanged for this particular series.

Could the company announce that ordinary waiting-time consistency was unaffected? The data show the same IQR in this small comparison, but one severe delay occurred. A decision about service quality may need both typical variation and unacceptable tail delays. “IQR unchanged” is a defensible statistical statement; “service reliability unchanged in every respect” is broader and unsupported.

This kind of case trains two exam skills at once: recompute only what changes and avoid allowing a convenient summary to hide a relevant extreme. The correct answer may require several measures to communicate the story.

When a question introduces a ninth or tenth observation, do not recycle quartile positions mechanically. Reorder the new dataset and use the proper method. Statistical summaries depend on the data and their number, not merely on the original diagram.

11. Original practice questions

A. Order the data 7, 4, 10, 2, 8, 6, 5, 6. Find the median, lower quartile, upper quartile, IQR and range using the median-of-halves convention for eight observations.

B. Group P has five-number summary 3, 7, 10, 12, 18. Find its IQR and range.

C. Group Q has five-number summary 1, 6, 10, 15, 25. Compare Q with P in median, IQR and range.

D. Two routes have the same median of 22 minutes but IQRs of 4 minutes and 11 minutes. Which has the smaller middle-half spread? Does this guarantee a shorter trip every time?

E. A series of eight measurements is 2, 3, 4, 5, 5, 6, 7, 8. Replace 8 with 18. Calculate the original and new ranges and decide whether the median changes.

F. A plot shows minimum 10, Q1 14, median 18, Q3 22 and maximum 30. What fraction of the dataset lies in the central region of the distribution in the usual quartile interpretation? What is the IQR?

G. A pupil says “the box width is the range”. Explain the error and identify what the two whiskers and box generally represent in the convention used in this workshop.

H. A pupil says the class with a larger median necessarily has a smaller range. Give a counterexample with plausible five-number summaries.

I. A graph uses seconds for one group and minutes for the other. What must be done before comparing their IQRs?

J. Two box plots have identical five-number summaries. Does this prove every underlying observation is identical? Explain.

K. A box plot gives median 50 and IQR 8. Can you find the mean exactly? Explain why or why not.

L. A class has mean 60 but median 54. Can a box plot’s median line be read as 60? Identify which statistic that line represents.

M. A dataset has ordered values 2, 2, 3, 4, 4, 5, 6, 20. Find Q1, median, Q3 and IQR, and describe what the large final value does to the range.

N. An exam question asks you to “Compare the typical score and the consistency” of two groups. State which box-plot landmarks you would use and what each tells you.

O. Group A has a larger median but its whiskers extend beyond those of Group B. Can you conclude A always outperforms B? State a careful response.

Do these exercises in two rounds. First calculate or identify the landmarks. Then write one precise interpretation for each. The second round is where a learner discovers whether a correct number has been given too much meaning.

12. Worked answers and checks

A. Ordered data are 2, 4, 5, 6, 6, 7, 8, 10. Median=6, Q1=4.5, Q3=7.5, IQR=3 and range=8. Put the numbers in order before using positions.

B. IQR=12−7=5; range=18−3=15.

C. Both groups have median 10. Q has IQR=15−6=9, larger than P’s 5. Q has range=25−1=24, larger than P’s 15. This supports greater spread for Q by both named measures, not a causal explanation.

D. The group with IQR 4 minutes has the smaller spread between its quartiles. It does not guarantee every individual journey is faster or more consistent in all possible respects.

E. Original range=8−2=6; new range=18−2=16. Median=(5+5)/2=5 in both. One extreme value changes the range but not the median in this example.

F. The interval between Q1 and Q3 represents the central half of the ordered observations under the usual quartile interpretation. IQR=22−14=8. This is an interval of values, not a claim that the score itself is eight.

G. Box width along the value axis is Q3−Q1, the IQR. The complete span from minimum to maximum gives the range, represented by the total distance across the whiskers and box in the simple plot described here.

H. Group A could have five-number summary 0, 8, 12, 16, 30; Group B 6, 7, 10, 11, 13. A has higher median 12 than 10 but larger range 30 than 7. A higher median does not determine spread.

I. Convert values to a common unit before comparing numerical IQRs. An IQR of 120 seconds equals 2 minutes. Comparing the raw numerals without units would be meaningless.

J. No. Five landmarks do not specify every internal observation. Different samples can share those landmarks. The plot compresses information.

K. Generally no. The mean depends on the sum of all observations, which the five-number summary does not provide. Median 50 is not enough to determine the mean.

L. The centre line is the median 54, not the mean 60. A mean may need separate data or annotation.

M. Median=(4+4)/2=4. Lower half 2, 2, 3, 4 gives Q1=(2+3)/2=2.5. Upper half 4, 5, 6, 20 gives Q3=(5+6)/2=5.5. IQR=3. Range=20−2=18, strongly affected by the maximum.

N. Use medians for a comparison of central position and IQRs for the spread of the middle half. Depending on the question, also note ranges and extreme observations. Avoid equating “typical” with a guarantee for every case.

O. No. A higher median describes the centre; whiskers extending further indicate wider observed extremes in the simple convention. Some B observations may exceed A observations. The plots do not establish individual-by-individual ordering.

For every mistake, record whether you used the wrong landmark, forgot to sort, confused IQR with range, changed units, treated median as mean or wrote a claim unsupported by the five-number summary. Each label points to a different correction.

13. An efficient checking routine

First check that the landmarks are in order: minimum no greater than Q1, Q1 no greater than median, median no greater than Q3, and Q3 no greater than maximum. A calculation that violates this order contains an error or an incompatible convention that needs investigation.

Second verify the arithmetic. IQR is Q3 minus Q1; range is maximum minus minimum. Neither is a fraction of the total score. If the question asks for a comparison, calculate both groups’ values before drawing a conclusion.

Third inspect units and scales. Times must be compared in compatible units. Two graphs should use a common scale for a fair visual comparison; otherwise read the numerical values explicitly. Volume 0099 develops the graphical side.

Fourth read the question’s purpose. If it asks which group has higher typical central performance, compare medians. If it asks which has less middle variation, compare IQRs. If it asks about extreme observations, range and whisker endpoints may matter. Answer the requested job.

Fifth stop before overclaiming. A box plot does not demonstrate that one teaching programme caused improvement. A median does not identify all student results. A lower IQR does not guarantee fewer extreme delays. Keep the conclusion aligned with the statistic.

This is a short check, but it can recover marks that would otherwise be lost to a careless interpretation after correct arithmetic.

14. Seven days of deliberate practice

Day 1: sort and locate. Order ten small datasets and identify the central two observations for even-sized lists. Check that the median is always between minimum and maximum.

Day 2: lower and upper halves. Use eight-observation examples to calculate Q1 and Q3 consistently. Reconstruct the two halves before finding their medians.

Day 3: draw. Translate five-number summaries into simple box-and-whisker sketches on a labelled axis. Then read five-number summaries back from sketches. Make the representation reversible.

Day 4: compare. Work with pairs of plots that have equal median but different IQRs, and pairs with different medians but equal IQRs. Write separate sentences for centre and spread.

Day 5: extremes. Change only a maximum or minimum in a fixed eight-value dataset and recompute the summaries. Explain when the range changes while IQR remains unchanged.

Day 6: interpretation. Analyse a school, transport or measurement report with two box plots. Identify one valid conclusion and one attractive but unsupported claim.

Day 7: independent clinic. Attempt Questions A–O without notes. Mark whether each answer is numerically correct and whether its interpretation stays inside the evidence. Revisit the weakest category a few days later.

Parents can ask, “Do the two groups have the same middle score?” followed by, “How spread out are the middle results?” These two calm questions help learners separate centre from variation instead of searching for one number that supposedly tells the whole story.

15. Carry centre and spread together

A good statistics answer understands that one number rarely describes a complete group. Median shows central position; IQR shows how widely the central half of observations is spread. Range adds information about extremes. These measures work together, but they do not explain every individual result or every cause.

Continue through the Mathematics Hub, the G3 SEC learner-guide index and the earlier cumulative-frequency workshop, Volume 0103. If the task involves the visual scale rather than statistical measures, revisit Volume 0099.

The assessment details and topic scope are grounded in the 2027 SEAB G3 school-candidate syllabus index and the official Mathematics K310 syllabus. The numerical examples and practice questions are original teaching material rather than official examination questions.