A class may produce two very different answers to the same question about a graph. One learner reads the highest value as the median. Another reads the middle of the vertical scale instead of the middle of the people. Both can use the right vocabulary and still misread the data. Cumulative frequency is a way to organise counts so that the position of an observation becomes visible. The key is to keep two things distinct: the value on the horizontal axis and the number of observations at or below that value.
This guide covers the assessed 2027 SEC G3 Mathematics K310 syllabus, formerly referenced as 4052. The official 2027 K310 syllabus specifically lists cumulative frequency diagrams, box-and-whisker plots, quartiles, percentiles, interquartile range and statistical misinterpretation. Both mathematics papers last 2 hours 15 minutes and carry 90 marks each (50% each); an approved calculator may be used in both. The SEAB subject directory is the authoritative entry-code reference. Graph readings and grouped-data estimates in these examples are educational, not official worked examination answers.
This volume deepens Volume 0347 on missing frequencies and means and Volume 0351 on means and standard deviation. For an earlier transition, use PSLE Mathematics revision and error diagnosis, then the Secondary 1 Mathematics foundation route. The new job here is distributional position: what fraction of the observations lies below a value, and what conclusions survive once you acknowledge the limits of a graph.
Why a running total changes how we see a dataset
Frequency tells us how many observations belong to one value or class. Cumulative frequency tells us how many observations have accumulated up to a stated boundary. Suppose the numbers of readings in four consecutive classes are 4, 6, 8 and 2. The cumulative counts are 4, 10, 18 and 20. The last cumulative frequency is the sample size. If you accidentally add each frequency to the one immediately before rather than to the running total, the graph will misrepresent the distribution even if all individual counts were copied correctly.
The table is not yet a curve. A cumulative frequency graph places cumulative count vertically and the relevant values or upper class boundaries horizontally. The curve should not move down as values increase. A vertical plateau indicates no additional observations accumulated over the corresponding range, not that the measured quantity itself stopped changing in time. This distinction becomes especially important when the context uses words such as ‘day’, ‘distance’ or ‘temperature’. Check what is on the horizontal axis before calling a plot a time series.
A running total can answer questions such as ‘approximately how many scores were below this threshold?’ and ‘at what value is about 75% of the group at or below?’ It cannot answer every question about the raw observations. Grouping compresses detail. Two datasets can share the same grouped cumulative frequencies and still differ in the exact values within each class.
From ranks to quartiles
A median is a measure of the middle position after observations are ordered. The first quartile, Q1, marks about the lower quarter, while the third quartile, Q3, marks about three quarters. On a cumulative frequency curve with a total of N observations, a common graph-reading method is to locate N/4, N/2 and 3N/4 on the cumulative-frequency axis and read across and down. Values obtained from a graph are usually estimates because the curve and plotted points have limited resolution.
With 80 readings, the guide positions are 20, 40 and 60. With 120 readings, they are 30, 60 and 90. Do not place quartile positions at 25, 50 and 75 on the vertical axis simply because those percentages are familiar. The vertical axis gives counts. Percentage positions must be converted using the sample size unless the axis has already been labelled in percentages.
For a short ordered list of exact observations, a question may expect a median-of-halves convention or specify a particular quartile method. Different conventions for discrete datasets can produce slightly different values. Follow the method requested in the question and the school-approved approach, and avoid treating one interpolation rule as universal. The common N/4 technique discussed here is particularly useful for reading approximate positions from a cumulative frequency diagram.
Interquartile range describes a central spread
The interquartile range, abbreviated IQR, is Q3 minus Q1. It represents the width of the central half of the dataset by value. If Q1 is 12 and Q3 is 28, IQR is 16 in the units of the variable. It does not mean that exactly 16 people sit between the quartiles. Nor does it mean that every observation outside this central range is wrong.
A smaller IQR indicates that the middle half of the observations occupies a narrower range of values. This may be evidence of a more concentrated middle, but it does not automatically prove a smaller overall range or a lower standard deviation. An extreme minimum or maximum may make two groups’ total ranges very different even when their IQRs are the same. Choose the measure of spread that the question actually asks you to compare.
Ask what the measurement is before judging whether a smaller IQR is desirable. Lower variability might be valuable for reliability of a manufacturing process, but a narrow range of student scores is not automatically a better educational outcome. The median, context and objectives also matter.
Percentiles are positions, not grades
The 90th percentile is the value around which 90% of the observations are at or below, under the relevant convention. It does not mean the student scored 90 out of 100. It does not mean 90 students have the same value. Percentiles describe relative location in a distribution. In a class of 200, a 90th-percentile graph position corresponds to cumulative count 180.
When comparing percentiles across groups, compare like with like. A percentile value of 40 minutes for one set and 45 minutes for another only has a useful interpretation if both variables measure the same phenomenon in the same units, and differences in the collection method are accounted for. Percentile rank is useful for locating one observation within a distribution, but it is not a causal explanation of why the observation occupies that position.
The right way to read a threshold question is often to move from horizontal value to vertical cumulative count. The right way to read a percentile question is often to move from vertical count to horizontal value. State which direction you are reading before putting pencil to paper.
Box-and-whisker plots are compressed stories
A box plot summarises five landmarks: minimum, lower quartile Q1, median, upper quartile Q3 and maximum. The box extends from Q1 to Q3, the median is marked inside it, and the whiskers extend toward the extremes in the conventional school representation. A box plot is not a full frequency diagram. Its purpose is to make the centre and spread easier to compare at a glance.
Not all five-number summaries are symmetric. For example, minimum 4, Q1 10, median 13, Q3 24 and maximum 31 gives a box with a longer upper section, but a statement about the precise shape of the original frequency distribution needs care. A long whisker suggests a stretched tail in that direction; it does not identify individual observations or the cause of that tail.
If one group has a higher median and a wider IQR than another, say both. Do not collapse that into ‘Group A is better’. Whether higher is desirable depends on the measured variable. And do not confuse the median with the mean: a box plot conventionally shows the median, not enough information to calculate the exact mean in general.
Grouped data introduces approximation
A continuous measurement may be grouped into intervals such as 0–10, 10–20 and 20–30. The frequencies count observations in those classes. Cumulative frequencies plotted against upper class boundaries are useful for estimating median and quartiles, but the precise locations inside each class are not known from the grouped table alone.
If you interpolate along a straight segment between points, say you have made an estimate based on that representation. It is a reasonable examination technique when the graph invites reading, not a claim to have recovered the original individual measurements. The last plotted point should reach the total N. If it does not, review the frequency total or the plotting.
Read the scale: a chart may begin at a non-zero horizontal value; marked vertical intervals may jump by five or ten; horizontal values may be in minutes while the question asks for seconds. A mathematically correct percentile position will still produce a wrong answer if the axes are misread. Give a sensible number of significant figures for a graph estimate rather than pretending that a hand-drawn curve yields six decimal places.
Compare distributions without misleading the reader
A responsible comparison names the measure, the observed difference and its meaning. ‘Group A has a median time of 12 minutes compared with Group B’s 17 minutes, so a typical observed time is lower in A’ is controlled. ‘Group A is always faster’ is not justified because the distributions may overlap and the graph does not disclose every observation.
The words ‘at least’, ‘more than’, ‘below’ and ‘no more than’ need exact care. A cumulative frequency at x generally counts observations up to x under the stated class convention. To find the number above a threshold, you usually subtract the appropriate cumulative count from N. For classes with boundaries, inspect whether boundary observations are included and whether the graph only supports an approximate reading.
Keep a small note of why a graph can mislead: truncated scales, unequal class widths, poorly labelled units, different sample sizes, apparent precision beyond the graph, and causal explanations imported from outside the data. Examinations often reward identifying the mathematical limitation, not just extracting another number.
How to present exam workings
For a quartile from a cumulative frequency curve, a clear answer can be written in three lines: N = total frequency; quartile cumulative position = N/4 or 3N/4; graph reading gives Q1 or Q3 approximately the specified value and unit. For an IQR, show Q3 − Q1. If you are asked to compare two groups, explain the meaning of the medians and IQRs rather than listing four numbers without a conclusion.
Good mathematics is neither purely numerical nor unnecessarily literary. A carefully chosen sentence after a calculation shows that you know what the number represents. The SEAB syllabus explicitly assesses reasoning and communication, so missing the interpretation can lose the point of a multi-step question even if the reading itself is accurate.
Most errors occur at the first decision: wrong total, wrong percentile position, wrong axis, wrong graph direction or a comparison of unlike measures. Diagnose that first mistake before doing more practice. The aim is to learn a repeatable method for any distribution, not memorise one picture.
Twenty-five worked cases, each with an independent retest
Take these questions in order on a first pass. The early cases establish reading direction and calculation; later cases demand judgement about spread, thresholds and graphs that could mislead. The explanations are deliberately specific so that your correction identifies the real misunderstanding rather than asking you to memorise a slogan.
1. Build the cumulative totals
Question. Frequencies are 3, 7, 5 and 5 across four successive classes. Find the cumulative frequencies.
Reasoned solution. The running sums are 3, 10, 15 and 20. The last value equals the total of all four classes. Present the cumulative row beneath the original frequencies so a marker can inspect the additions. Put the quantity requested in words before calculating. The unit of the final answer should match that quantity. If the question asks ‘how many’, expect a count; if it asks for a quartile, expect a value from the measurement axis.
Common wrong turn. 3, 10, 12, 10 is not cumulative: it adds each entry to a neighbouring entry and later decreases. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. What happens to the cumulative curve if a class has frequency zero? Try it independently with units shown and compare the meaning of the result with what was asked.
2. Recover an individual frequency
Question. Cumulative counts at successive boundaries are 6 and 15. Find the frequency in the intervening class.
Reasoned solution. Subtract the earlier total from the later total: 15 − 6 = 9. A cumulative count contains everything before it; the difference isolates the new group. Check both the denominator and the boundary. A percentage uses the complete sample size unless the question expressly defines a subgroup. A threshold count depends on whether the boundary is included.
Common wrong turn. Do not report 15. That is everyone accumulated up to the later boundary. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Repeat for cumulative counts 14 and 28 and explain the subtraction. Try it independently with units shown and compare the meaning of the result with what was asked.
3. Check the sample size
Question. A cumulative frequency table finishes at 72. What is N?
Reasoned solution. N = 72 observations, assuming all classes in the dataset are included. The cumulative frequency cannot exceed N; the final value is the whole group. The graph is evidence, not decoration. Write down which axis you used and the reading direction. That makes a mistaken percentile level or scale choice visible.
Common wrong turn. A middle cumulative total is not the group size. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Use N = 72 to identify median and quartile reading positions. Try it independently with units shown and compare the meaning of the result with what was asked.
4. Locate the median level
Question. There are 80 observations. Which horizontal line should you use for the median on the cumulative count axis?
Reasoned solution. Use cumulative frequency 40 (N/2). Read across at 40, reach the curve, then down to the value axis. The result is a measured value, not the count 40 itself. Preserve the difference between exact arithmetic and an approximate graph estimate. A result can be carefully reasoned while still being approximate because its source was grouped or hand-read.
Common wrong turn. Do not report 40 units without reading the curve. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Where is the median level for N = 150? Try it independently with units shown and compare the meaning of the result with what was asked.
5. Locate the first quartile
Question. For N = 120, identify the cumulative frequency level used to estimate Q1.
Reasoned solution. The common cumulative-curve position is N/4 = 30. From y = 30, read horizontally to the curve and vertically to the x-axis to estimate the value of Q1. Once you have a number, turn it into one contextual sentence. Avoid causal language unless the design of the data collection supports a causal interpretation.
Common wrong turn. The number 30 is a count on the vertical axis, not necessarily Q1’s measured value. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Identify Q3’s level for the same dataset. Try it independently with units shown and compare the meaning of the result with what was asked.
6. Locate the third quartile
Question. For N = 120, identify the cumulative level for Q3.
Reasoned solution. The usual curve reading is 3N/4 = 90. Read the value at which about 90 observations have accumulated. State the x-axis units when reporting Q3. A reliable correction changes the earliest wrong step. If you read 75 on the vertical axis instead of 75% of the total, correcting only the final subtraction will not solve the next problem.
Common wrong turn. Writing 75 on the vertical axis ignores the total N. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain why 75% becomes 90 observations here. Try it independently with units shown and compare the meaning of the result with what was asked.
7. Calculate IQR
Question. A graph gives Q1 ≈ 18 mm and Q3 ≈ 31 mm. Find IQR.
Reasoned solution. IQR ≈ 31 − 18 = 13 mm. It is a width across the central half of the readings. Keep the approximate sign because the quartiles were read from a graph. Put the quantity requested in words before calculating. The unit of the final answer should match that quantity. If the question asks ‘how many’, expect a count; if it asks for a quartile, expect a value from the measurement axis.
Common wrong turn. The central half is 50% of observations, not 13 observations. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain what a 13 mm IQR tells you and what it does not tell you. Try it independently with units shown and compare the meaning of the result with what was asked.
8. Read a 90th percentile
Question. A graph represents 200 values. At what cumulative level do you estimate P90?
Reasoned solution. Find 0.90 × 200 = 180. Move from cumulative frequency 180 to the curve, then read the x-value. A 90th percentile is a value at a relative position. Check both the denominator and the boundary. A percentage uses the complete sample size unless the question expressly defines a subgroup. A threshold count depends on whether the boundary is included.
Common wrong turn. It is not a score of 90 marks unless the x-axis and graph happen to support that conclusion. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Compute the 65th percentile level in the same sample. Try it independently with units shown and compare the meaning of the result with what was asked.
9. Read the other direction
Question. A threshold at x = 30 corresponds to cumulative count 48 out of N = 80. What percentage is at or below it?
Reasoned solution. 48/80 × 100 = 60%. The x-value is the threshold, while the y-value is the cumulative number. This is the opposite reading direction from finding a percentile value. The graph is evidence, not decoration. Write down which axis you used and the reading direction. That makes a mistaken percentile level or scale choice visible.
Common wrong turn. Reporting 48% confuses count with percentage. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. If the threshold count were 52, what percentage would be accumulated? Try it independently with units shown and compare the meaning of the result with what was asked.
10. How many above?
Question. Of 100 recorded times, 68 are at or below 15 minutes. Approximately how many exceed 15?
Reasoned solution. 100 − 68 = 32. The complement works when the cumulative count includes the relevant boundary as stated. If a grouped graph is approximate, describe the result as approximate. Preserve the difference between exact arithmetic and an approximate graph estimate. A result can be carefully reasoned while still being approximate because its source was grouped or hand-read.
Common wrong turn. Subtract from N, not from the largest plotted time. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Express the 32 as a percentage and state the threshold in the sentence. Try it independently with units shown and compare the meaning of the result with what was asked.
11. Build a five-number summary
Question. Minimum 4, Q1 8, median 11, Q3 17, maximum 26. State the five landmarks.
Reasoned solution. List 4, 8, 11, 17, 26 in increasing order. These are the five values used for a conventional school box plot. The IQR is 9 and the overall range is 22. Once you have a number, turn it into one contextual sentence. Avoid causal language unless the design of the data collection supports a causal interpretation.
Common wrong turn. Do not insert the mean because the summary requires the median. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Describe the relative positions of the box and whiskers without inventing raw data. Try it independently with units shown and compare the meaning of the result with what was asked.
12. Interpret a box
Question. One box plot has Q1 10 and Q3 16. What does the box width mean?
Reasoned solution. It represents IQR 6 in the original measurement units. The middle 50% spans approximately 10 to 16; this does not imply that there are six observations inside the box. A reliable correction changes the earliest wrong step. If you read 75 on the vertical axis instead of 75% of the total, correcting only the final subtraction will not solve the next problem.
Common wrong turn. The width in screen centimetres means nothing without the axis scale. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Compare with another box that spans 11 to 24. Try it independently with units shown and compare the meaning of the result with what was asked.
13. Median comparison
Question. Group A’s median is 14 and Group B’s is 20, measuring minutes to finish a task.
Reasoned solution. The typical central value is lower for A by 6 minutes. That does not prove all A observations are less than all B observations. Inspect ranges and overlap before making stronger claims. Put the quantity requested in words before calculating. The unit of the final answer should match that quantity. If the question asks ‘how many’, expect a count; if it asks for a quartile, expect a value from the measurement axis.
Common wrong turn. Median difference is not a statement about every individual. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Write a cautious recommendation if lower time is desirable. Try it independently with units shown and compare the meaning of the result with what was asked.
14. Spread comparison
Question. Group A IQR 4, Group B IQR 11, both measuring the same quality variable.
Reasoned solution. A’s middle half is more concentrated by that measure. Whether this makes A preferable depends on what the variable measures and its target range. IQR alone does not prove lower standard deviation. Check both the denominator and the boundary. A percentage uses the complete sample size unless the question expressly defines a subgroup. A threshold count depends on whether the boundary is included.
Common wrong turn. Do not substitute ‘better performance’ for ‘narrower middle spread’. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. State a context where lower IQR is useful and one where it is not. Try it independently with units shown and compare the meaning of the result with what was asked.
15. A long upper whisker
Question. A box extends 8–14, median 10, upper whisker reaches 39.
Reasoned solution. The high maximum stretches the upper portion. It suggests greater distance between Q3 and the maximum than between the minimum and Q1 if those lengths are compared, but gives no cause for that pattern. The graph is evidence, not decoration. Write down which axis you used and the reading direction. That makes a mistaken percentile level or scale choice visible.
Common wrong turn. You cannot calculate the mean from these five points alone. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. What extra data would help identify whether the extreme is an unusual error? Try it independently with units shown and compare the meaning of the result with what was asked.
16. Grouped median is approximate
Question. A curve crosses cumulative level 50 between the plotted x-values 20 and 30.
Reasoned solution. The estimated median is read from the curve segment in that interval. Without the raw observations it is not safe to assert an exact value at six decimal places. Give a graph-appropriate answer and units. Preserve the difference between exact arithmetic and an approximate graph estimate. A result can be carefully reasoned while still being approximate because its source was grouped or hand-read.
Common wrong turn. Interpolation is a modelling choice, not recovered individual data. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain how a more detailed original table could reduce uncertainty. Try it independently with units shown and compare the meaning of the result with what was asked.
17. Boundary language
Question. The question says ‘more than 40’, but the graph supplies a cumulative total at 40.
Reasoned solution. Count above 40 by subtracting the number at or below 40 from N when that is the given convention. Say explicitly which boundary is included. ‘At least 40’ might require different treatment for exact integer-valued data. Once you have a number, turn it into one contextual sentence. Avoid causal language unless the design of the data collection supports a causal interpretation.
Common wrong turn. The words more than and at least cannot be swapped without thought. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Construct a small integer example with an observation exactly 40. Try it independently with units shown and compare the meaning of the result with what was asked.
18. Unequal samples
Question. One curve ends at 40 and another at 80. Which has more people?
Reasoned solution. The second dataset contains twice as many observations. Compare percentages or corresponding quartile levels when comparing distribution positions; raw cumulative heights are not automatically comparable. A reliable correction changes the earliest wrong step. If you read 75 on the vertical axis instead of 75% of the total, correcting only the final subtraction will not solve the next problem.
Common wrong turn. A larger cumulative count at a threshold may arise simply from a larger sample. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain how to compare the proportion under a threshold fairly. Try it independently with units shown and compare the meaning of the result with what was asked.
19. A reversed frequency error
Question. Cumulative totals are written as 5, 11, 9, 20. Find the issue.
Reasoned solution. The sequence decreases from 11 to 9, impossible for nonnegative class frequencies. The issue must be a copying, arithmetic, ordering or labelling error. Check the original table before plotting. Put the quantity requested in words before calculating. The unit of the final answer should match that quantity. If the question asks ‘how many’, expect a count; if it asks for a quartile, expect a value from the measurement axis.
Common wrong turn. Do not ‘fix’ the curve by smoothing away the data error. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Give one valid sequence with the same total 20. Try it independently with units shown and compare the meaning of the result with what was asked.
20. Find the frequency class
Question. Successive cumulative frequencies are 15, 29, 44. Find the two intervening class frequencies.
Reasoned solution. The differences are 14 and 15. Subtraction removes what was already counted. This works because every class frequency contributes once to a running total. Check both the denominator and the boundary. A percentage uses the complete sample size unless the question expressly defines a subgroup. A threshold count depends on whether the boundary is included.
Common wrong turn. Dividing by the number of classes is unrelated to the requested differences. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Rebuild the cumulative row and verify the last value. Try it independently with units shown and compare the meaning of the result with what was asked.
21. Compare range and IQR
Question. Group A min 0, Q1 10, Q3 20, max 100; Group B min 5, Q1 9, Q3 22, max 28.
Reasoned solution. A has range 100 and IQR 10. B has range 23 and IQR 13. A’s central half is narrower, yet its full range is far larger due to its extremes. Neither single spread measure tells the whole story. The graph is evidence, not decoration. Write down which axis you used and the reading direction. That makes a mistaken percentile level or scale choice visible.
Common wrong turn. A smaller IQR need not mean a smaller total range. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Write an evidence-safe comparison using both measures. Try it independently with units shown and compare the meaning of the result with what was asked.
22. A percentile isn’t a mark
Question. A learner is at the 80th percentile for travel time. They say they scored 80 out of 100.
Reasoned solution. The percentile describes their relative position, not a test score or percentage of correct answers. Interpret the measured variable and how higher or lower values should be understood before attaching praise or criticism. Preserve the difference between exact arithmetic and an approximate graph estimate. A result can be carefully reasoned while still being approximate because its source was grouped or hand-read.
Common wrong turn. Percentile rank cannot be translated directly into raw score. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain what information is needed to find the actual time. Try it independently with units shown and compare the meaning of the result with what was asked.
23. Graph resolution
Question. A graph has x-axis ticks every 10 seconds and a curve crossing near 37 seconds.
Reasoned solution. Report an estimate consistent with the graph’s precision, such as about 37 seconds if interpolation is reasonable. Do not claim 37.0001 seconds. Line thickness and plotting resolution limit certainty. Once you have a number, turn it into one contextual sentence. Avoid causal language unless the design of the data collection supports a causal interpretation.
Common wrong turn. Artificial decimal precision is not mathematical rigour. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Explain how a raw-data table would change the confidence of the answer. Try it independently with units shown and compare the meaning of the result with what was asked.
24. Misleading truncated axis
Question. Two box plots are displayed on different horizontal scales with different starting values.
Reasoned solution. Compare numerical axis positions and units rather than visual box width on the page. One chart may make an IQR appear three times larger through scale choice alone. A reliable correction changes the earliest wrong step. If you read 75 on the vertical axis instead of 75% of the total, correcting only the final subtraction will not solve the next problem.
Common wrong turn. Physical length on two differently scaled images is not a comparable measure. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. Describe how you would redraw both datasets fairly on one shared axis. Try it independently with units shown and compare the meaning of the result with what was asked.
25. School project decision
Question. A group compares median waiting times and IQRs for two routes with uneven sample sizes.
Reasoned solution. State the centre and spread for each route, note sample sizes, and avoid claiming a causal explanation from descriptive statistics alone. A sensible decision also asks whether times were recorded at comparable periods. Put the quantity requested in words before calculating. The unit of the final answer should match that quantity. If the question asks ‘how many’, expect a count; if it asks for a quartile, expect a value from the measurement axis.
Common wrong turn. A fast median from one peak-hour sample and a slow median from another time window may be a flawed comparison. Explain why that shortcut fails here rather than simply crossing out the final number.
Retest. List the contextual variables you would hold constant in a follow-up survey. Try it independently with units shown and compare the meaning of the result with what was asked.
A complete original mixed question
An illustrative school library records how long 80 students spend reading one article. The grouped cumulative frequency curve suggests Q1 around 14 minutes, median around 21 minutes and Q3 around 35 minutes. Another class of 80 has Q1 around 18, median around 23 and Q3 around 29. We are not given the full original observations. Compare the middle, compare the middle spread, and write one limitation of your conclusion.
First group: IQR approximately 35 − 14 = 21 minutes. Second group: IQR approximately 29 − 18 = 11 minutes. The first group has the lower median, about 21 rather than 23 minutes. The second group’s central half is considerably more concentrated by IQR. But the task does not establish which group reads better or whether the reading material had the same difficulty. Even the interpretation of a shorter time is ambiguous: it might reflect speed, skimming, a shorter passage or another factor.
If the library wants to know whether the same article produces different timings under similar conditions, it should control or record relevant variables. These may include article length, language difficulty, whether students pause for questions, timing method and time of day. Descriptive statistical differences are a sound starting observation, not an explanation of their causes.
How to revise this topic across the next fortnight
Begin with seven very small tasks: cumulative totals, recovering class frequencies, calculating N, identifying N/4, N/2 and 3N/4, reading a graph in both directions, computing IQR and interpreting a box plot. Do these without a worked example in front of you. Then move to mixed comparisons where two measures give different impressions. The hard part is often explaining a difference correctly, not subtracting two numbers.
On the next practice session, intentionally change the sample size. Many students become fluent with N = 100 because percentages look effortless. Test N = 48, 72, 120 and 200 instead. The same ideas must work regardless of the denominator. Use one different graph scale so that axis reading is tested independently of arithmetic.
In the second week, combine a cumulative frequency diagram with a written recommendation and a critical question. Have the learner answer in three parts: numerical result, contextual interpretation and limitation. Mark each part separately. A learner who reliably gets the arithmetic right but overclaims about all individuals needs a different correction from one who cannot read the scale.
Questions students and parents ask
Is IQR the same thing as standard deviation? No. Both are measures of spread, but IQR uses the width between Q1 and Q3 while standard deviation uses a different relationship to the full data and mean. See the earlier standard-deviation guide for that calculation.
Are quartiles always exact? Not from a grouped cumulative frequency curve. Graph readings often require estimates; for an exact ordered list, use the convention specified by the task or taught for that context.
Can a box plot tell me which student did best? No. It compresses a group into five values and hides most individuals. Also, higher and lower values are desirable in different contexts.
Should I call a difference significant? Avoid claiming statistical significance from a descriptive box plot alone. The G3 question may support an observed median or spread comparison, but a formal significance test is a separate matter.
What if I get different answers from classmates? Compare the sample size, class boundaries, graph scale, interpolation, precision and quartile convention before assuming someone is careless.
The one-page method to keep
Record four questions at the top of the paper: What is N? What position do I need? Which direction do I read? What does the resulting number mean in context? These prevent a high proportion of mistakes. Next to them, write IQR = Q3 − Q1. Keep that equality separate from ‘how many people are in the central half’, which is a count-based idea. Finally add one line: the graph shows distribution, not automatically cause.
For continuity, use Guide 0347 on cumulative totals from frequency tables and Guide 0351 on mean and standard deviation, then revisit the PSLE Mathematics reasoning route or Secondary 1 foundations if the underlying arithmetic is fragile. The current examination rules, subject code and content are set out in K310, verified against the 2027 SEAB G3 list on 30 October 2026.
