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Cumulative-frequency graphs with different totals cannot be compared fairly by treating the same vertical count as the same percentile. A cumulative count of 30 is halfway through a group of 60, but only a quarter of the way through a group of 120. To compare medians or percentiles, calculate the required position separately from each group’s own total.
In G3 Mathematics, start by reading what each axis measures and how many observations belong to each group. For a median estimate from a cumulative-frequency curve, use half of that group’s total on the vertical axis, move across to its curve, and then read the measurement on the horizontal axis. Use the same proportion, rather than automatically using the same count, when the question asks for a comparison of equivalent positions.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
- 10. Reconstruct counts from a grouped table before plotting
- 11. Interpolation estimates a measurement inside a group
- 12. Compare centre and spread without inventing a cause
- 13. A curve that sits higher does not always represent faster performance
- 14. Diagnose an incorrect comparison at its first wrong decision
Sections 16–20
Suppose Group A contains 40 observations and Group B contains 80. On a cumulative-frequency graph, the median-reading level is 20 for A and 40 for B. A horizontal line at 20 would locate the median of A but the lower quartile of B under the usual grouped-curve convention. It would not compare like with like.
At a fixed measurement, the graph can answer another question: how many observations fall at or below that measurement, according to the graph’s convention? Those are counts. If A has 24 such observations and B has 40, B has more observations in that category. Yet the proportions are 60% for A and 50% for B. A has the larger share.
Neither comparison is automatically wrong. The error is using one comparison to claim the other. Counts answer how many; proportions answer what share; percentile readings locate a measurement at a specified position in an ordered distribution. The question decides which is relevant.
This article focuses on that comparison problem. Use the cumulative-frequency and box-plot mastery guide for the wider topic, and return here when unequal group sizes make an apparently simple graph comparison uncertain.
The SEAB G3 Mathematics syllabus for 2027 includes cumulative-frequency diagrams, quartiles, percentiles and measures of spread within data handling and analysis. It also includes interpreting statistical representations and explaining misleading readings. Knowing the total is part of making those representations meaningful.
This is not a complete guide to collecting a representative sample or making a statistical inference about a population. The examples use invented datasets and grouped summaries to teach graph reading. Conclusions describe the groups shown unless additional evidence supports a broader claim.
The particular challenge is that a larger group naturally reaches a larger cumulative total. Its curve can appear higher at a given measurement even when a smaller proportion of that group lies below the measurement. A vertical count by itself does not account for the denominator.
The solution is to keep the group total attached to every cumulative count. You can compare counts when the question asks for counts. You can divide by the corresponding total when the question asks for proportions. You can locate matching fractional positions when comparing medians or other percentiles.
For a broader introduction to the subject levels, use what G1, G2 and G3 Mathematics mean. A particular graph skill develops through taught prerequisites and practice, rather than through assumptions about what a learner should already know from a label.
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Frequency counts observations in a category or interval. Cumulative frequency adds the frequencies up to a stated boundary. If three successive intervals contain 8, 12 and 10 observations, their running cumulative totals are 8, 20 and 30. The third cumulative total includes the observations counted in the first two.
That is why cumulative totals should not decrease as you move through increasing upper boundaries. Once an observation has been included, it stays included at a later boundary. A downward movement in a cumulative-frequency curve would need investigation: perhaps the graph is a different kind of graph, or data have been plotted incorrectly.
The final cumulative total equals the number of observations included in the distribution. Read that number for each group. Do not assume that two curves share a total merely because they share the same axes or were printed on the same page.
If a table labels the boundary “less than 20,” interpret that wording exactly. If it uses “at most 20,” the boundary includes equality. With grouped continuous measurements, exercises often use boundaries to organise intervals. Follow the convention supplied in the question instead of treating every endpoint phrase as interchangeable.
A good first annotation is “vertical: number accumulated” and “horizontal: measured quantity with unit.” This small step helps prevent reading a frequency as a score or a score as a number of people. It also prepares you to explain what a horizontal or vertical movement on the graph actually means.
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Suppose one curve finishes at cumulative frequency 60 and the other at 100. The first group has 60 observations; the second has 100. The median-reading levels on these grouped curves are therefore 30 and 50 respectively.
The vertical midpoint of the whole printed graph is not necessarily the median level for either group. If the graph extends to 120 for visual spacing, half the vertical axis would be 60. That would correspond to the entire first group, not its median. Use the data total, not the page layout.
Similarly, the highest labelled tick may sit above a curve’s final point. A graph can leave room beyond the data. The total comes from the stated sample size, table total or terminal cumulative count, checked against the question’s information.
If the total in the written question and the total you read from the curve disagree substantially, stop and inspect the scale and curve identity. You may have followed the wrong curve, missed an axis interval or mistaken a nearby label for an endpoint. Do not average two conflicting totals to make the calculation proceed.
Once the totals are secure, write a short pair: “A: N = 60, median level = 30. B: N = 100, median level = 50.” This keeps the denominators visible and reduces the chance of using the first group’s target level for both curves.
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An invented grouped distribution records task-completion times. Group A has 60 observations and Group B has 100. A supplied cumulative-frequency graph gives an estimated median time of 18 minutes when read at cumulative frequency 30 for A, and 21 minutes when read at cumulative frequency 50 for B.
The comparison is that Group A has the lower estimated median completion time: 18 minutes compared with 21 minutes. The median difference is approximately 3 minutes. The use of “estimated” reflects reading measurements from grouped curves rather than having every original observation.
It would be wrong to read both curves at cumulative frequency 30 and call both readings medians. For B, 30/100 = 30%, so that reading locates approximately the thirtieth percentile, not the median. The curves must be read at matching proportional positions for a median comparison.
The context also matters. A smaller completion time might indicate faster performance in this invented task, but a median alone does not establish accuracy, quality or the cause of a difference. If the question asks only for a statistical comparison, state the median comparison without adding unsupported explanations.
A useful written structure is: total, target cumulative level, graph reading, contextual conclusion. It shows the reasoning without excessive calculation. For example: “A has 60 observations, so its median is read at 30. B has 100, so its median is read at 50. The estimated medians are 18 and 21 minutes, so A has the lower median time.”
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On a grouped cumulative-frequency curve, the lower quartile is commonly estimated at N/4 and the upper quartile at 3N/4, where N is the group’s total. For Group A with 60 observations, the target levels are 15 and 45. For Group B with 100, they are 25 and 75.
Suppose the readings for A are 12 minutes and 24 minutes. Its estimated interquartile range is 24 − 12 = 12 minutes. Suppose B gives 16 minutes and 26 minutes. Its estimated interquartile range is 10 minutes.
This supports the statement that the middle half of B’s times is less spread out by this measure. It does not mean that every B time is closer to every other B time, or that the full range is smaller. The interquartile range describes the width between the two quartile measurements.
Be careful about the two stages. First locate the quartile positions using counts. Then read the corresponding times and subtract those times. Subtracting 45 − 15 gives a count interval of 30 observations; it does not give A’s interquartile range in minutes.
Different contexts can use different conventions for quartiles in short ungrouped lists. This guide concerns readings from grouped cumulative-frequency curves. Follow the method and convention specified in the question or taught for your course, and do not mix a raw-list position rule into a grouped-graph procedure without checking its purpose.
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At a fixed measurement, moving up to a curve and across to the vertical axis gives a cumulative count. To turn that count into a share, divide by that group’s total. The denominator is essential because the same number of observations can represent different proportions of different groups.
Suppose A contains 40 observations and 24 lie at or below a stated threshold. B contains 80 observations and 40 lie at or below the same threshold. A’s share is 24/40 = 60%. B’s share is 40/80 = 50%.
B has the larger count, 40 compared with 24. A has the larger proportion, 60% compared with 50%. If the question asks which group has more observations meeting the condition, B answers it. If it asks which group has the larger percentage meeting it, A answers it.
| Group | Total observations | Cumulative count at threshold | Percentage at threshold |
|---|---|---|---|
| B | 80 | 40 | 50% |
This example is deliberately simple so the comparison remains visible. Larger totals do not invalidate count comparisons; they change what a count comparison can mean. A count may be exactly the measure needed when planning how many seats, places or resources are required.
Read the question’s noun carefully: number, proportion, percentage, median and percentile are not synonyms. The guide to reading quantities before calculating develops that habit at a foundational level, and it remains useful in G3 statistical work.
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If a cumulative count gives the number at or below a threshold, subtract it from the group’s total to find the number above that threshold. Use the same group’s total and preserve the endpoint wording.
Using the previous example, A has 40 − 24 = 16 observations above the threshold. B has 80 − 40 = 40. Their shares above are 16/40 = 40% and 40/80 = 50%. Here B has both the larger count above and the larger percentage above.
An easy error is to subtract A’s cumulative count from B’s total because both are nearby in the working. Label each line with the group name. A short expression such as “A above = 40 − 24” prevents a mixing error that arithmetic alone would not expose.
Another error concerns equality. If the cumulative value counts observations strictly below t, subtracting from N counts observations at least t. If it counts observations at most t, subtracting counts observations strictly above t. The complementary category depends on what the first category includes.
For a smooth grouped curve, estimated counts may not resolve exactly how many observations equal one precise measurement. Do not claim a boundary distinction that the grouped information cannot support. Use the convention given in the question, and describe graph-derived counts as estimates when they come from an interpolated reading rather than an exact table boundary.
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SECTION 9 OF 20
8. Read a percentile from the count axis, then report a measurement
To estimate the eightieth percentile from a cumulative-frequency curve, locate 80% of the relevant total on the vertical count axis. Then move across to the curve and down to the measurement axis. The answer is a measurement, not the cumulative count used to find it.
For a group of 150, the target cumulative count is 0.8 × 150 = 120. If the corresponding horizontal reading is 34 minutes, the estimated eightieth percentile is 34 minutes. The number 120 is an intermediate position in the ordered distribution.
For a second group of 90, the same percentile uses 0.8 × 90 = 72. Reading both groups at 120 would not compare their eightieth percentiles; for the second group, 120 even exceeds the total.
You can reverse the direction for a different question. If the measurement threshold is given, start on the horizontal axis, read the cumulative count and divide by the total to obtain the estimated percentile share. Know which quantity the question supplies and which it asks you to find.
A compact annotation can prevent confusion: “80% → count level → curve → minutes.” This is a reading route, not a formula to memorise without meaning. The share selects a position; the curve connects that position to the measured quantity. That relationship explains why the two axes have different units.
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SECTION 10 OF 20
9. A cumulative-percentage axis already performs the normalisation
Some graphs use cumulative percentage on the vertical axis rather than cumulative frequency. In that case, the top of each complete distribution is 100%, even when the groups have different totals. Equivalent percentiles can be compared directly at the same percentage level.
For example, the median is read at 50%, the lower quartile at 25%, and the upper quartile at 75%. You do not multiply those displayed percentage levels by the total unless you specifically need the corresponding counts for another part of the question.
This is why reading the axis title is essential. A tick labelled 50 could mean 50 observations or 50%. The visual position of the tick does not determine its unit. The written axis label does.
If a graph is cumulative percentage and A contains 40 observations, a reading of 60% corresponds to 24 observations. If B contains 80 and the reading is 50%, it corresponds to 40. The normalised graph makes the shares easy to compare while the totals are still needed to recover counts.
Normalisation does not make every comparison valid. Groups must still describe comparable measurements and categories, and conclusions must respect how the data were obtained. A graph of completion times in minutes cannot be combined directly with one in seconds without conversion. Equal percentages are helpful only after the meaning of the axes and groups is secure.
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Consider an invented table of completion-time intervals. Group A has frequencies 6, 10, 14 and 10 across four consecutive intervals. Group B has frequencies 8, 16, 24 and 32 across the same intervals. The totals are 40 and 80.
A’s cumulative frequencies are 6, 16, 30 and 40. B’s are 8, 24, 48 and 80. At the third upper boundary, B’s cumulative count is 48 compared with A’s 30. But the corresponding shares are 60% for B and 75% for A.
This is another example of a taller cumulative count not automatically indicating a larger proportion below the boundary. The difference arises because B contains twice as many observations overall. The denominator changes the interpretation of the same vertical direction.
Before plotting, verify that every cumulative total adds the current interval frequency to the preceding total. Then check that the final cumulative count matches the sum of all interval frequencies. These two checks catch both running-addition errors and missed rows.
If the question gives different class intervals for the two groups, be more careful. A boundary from one table may fall inside an interval of the other. Exact count comparisons at that boundary may then be unavailable from the tables alone. A supplied curve may permit an estimate, but the estimate should not be reported as an exact original count.
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Grouped data do not preserve the exact position of every observation. A cumulative curve estimates the distribution between class boundaries. When you read a median from that curve, you generally obtain an estimate rather than recovering a uniquely known raw-data value.
Suppose a table says 16 observations are below 20 minutes and 30 are below 30 minutes, with total 40. The median target count is 20. That position lies within the 20-to-30-minute interval. The table alone does not reveal the exact individual middle observations.
If the exercise specifies straight-line interpolation between those points, the estimated time is 20 + [(20 − 16)/(30 − 16)] × 10, which is approximately 22.86 minutes. The fraction identifies how far the target count lies through the cumulative increase in that interval.
A hand-drawn smooth curve may give a slightly different estimate, depending on the supplied graph and expected reading precision. Use the requested method. Do not impose a straight-line interpolation calculation when the task is explicitly to read a provided curve.
The important reasoning remains stable: find the correct target count from the correct total, then estimate the associated measurement. Greater calculator precision cannot compensate for choosing the wrong total or the wrong axis. The exact-values and graph-estimates guide explains why the source of an answer determines how much certainty its digits can support.
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Suppose A’s estimated median is 18 minutes and its interquartile range is 12 minutes. B’s estimated median is 21 minutes and its interquartile range is 10 minutes. A has the lower median time, while B has the smaller spread across its middle half.
Both observations can belong in a comparison. It would be incomplete to say only that A is “better” without explaining what better means in the task. A lower median may be desirable for completion time, while a smaller interquartile range describes consistency by that particular measure.
Neither statistic tells you why the groups differ. Perhaps the tasks differed, participants differed, or data collection differed. Unless the question supplies a basis for causal explanation, keep the conclusion descriptive. Statistical summaries can reveal a difference without identifying its cause.
Unequal group sizes add another limitation: counts naturally differ in scale. This does not prevent comparing medians or interquartile ranges, but it means you should not treat the higher curve endpoint as evidence of better performance. The endpoint primarily tells you how many observations were included.
When writing the comparison, include the measured quantity and its unit. “A’s median completion time is approximately 3 minutes lower” is precise. “A is lower” leaves the reader guessing whether you mean a count, a time, a percentage or a spread. Clear nouns are part of statistical accuracy.
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SECTION 14 OF 20
13. A curve that sits higher does not always represent faster performance
On a cumulative-percentage graph of completion time, a higher curve at a particular time means a larger percentage has completed by that time, under the graph’s boundary convention. That can support a statement about that threshold. On a cumulative-count graph with unequal totals, the same visual comparison does not automatically support the percentage statement.
Consider A with 30 of 40 observations completed by a threshold and B with 48 of 80. B’s count curve is higher because 48 exceeds 30. Yet A’s completed share is 75%, compared with B’s 60%. The visual count advantage points in the opposite direction from the proportion comparison.
Even on percentage graphs, curves can cross. A may have a larger share below one threshold while B has a larger share below another. A single statement that one group is faster at every part of the distribution would then be too strong.
Use the measure the question specifies. For medians, compare the fiftieth-percentile measurements. For a deadline, compare the shares completed by that deadline. For resource planning, counts may be the relevant quantity. The best interpretation starts with the decision being made.
The guide to probability and statistics examination questions places these choices within a wider examination context. This article supplies the denominator check that keeps an unequal-group comparison honest.
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A student reads both medians at cumulative frequency 40 because one group has 80 observations. If the other group has 120, the first wrong decision is reusing the target count. The repair is to calculate 80/2 and 120/2 separately, not to practise moving a ruler across the graph again.
Another student correctly reads counts of 24 and 40, then claims that the second group has the larger percentage. The graph reading may be secure. The missing step is dividing each count by its own total. Give a pair of small totals that makes the distinction visible.
A third student calculates the quartile count levels correctly but reports their difference as the interquartile range. The repair is to distinguish position from measurement. Ask the student to include the unit at each stage: observations on the vertical axis, minutes on the horizontal axis.
A fourth student uses the maximum axis label as the total. The repair is to locate the curve’s terminal cumulative count or read the sample size stated in the question. Empty space above a curve is a design choice, not additional data.
The marked-paper guide explains how to turn such a specific error into a specific practice decision. “More statistics practice” is less useful than identifying whether the difficulty is total, proportion, axis direction or interpretation.
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Use the following information for Questions A to D. Group A contains 50 observations and Group B contains 120. At a given threshold, A’s cumulative count is 35 and B’s is 72. Assume the cumulative categories use the same threshold convention.
Question A: Find each group’s percentage at the threshold. State which group has the larger count and which has the larger percentage. Explain why these need not be the same group.
Question B: Find the cumulative-frequency levels needed to estimate each median. Then find the levels for each lower and upper quartile. You are finding count positions here, so do not attach a measurement unit such as minutes to these intermediate results.
Question C: Find the counts above the threshold when the cumulative category is “at or below.” Then express those counts as percentages of their own group totals. Keep the word “above” attached to the complementary calculation.
Question D: Find the target count for the ninetieth percentile in each group. Explain why reading both curves at cumulative frequency 90 would not answer this question.
Question E: A cumulative-percentage graph shows A at 65% and B at 55% at the same time threshold. Can you identify the larger share below the threshold? Can you identify the larger count without knowing the group totals? Give separate answers.
Question F: A graph extends to 150 on its vertical axis, but a group’s curve ends at 96. What median-reading level should be used if 96 is confirmed as the group’s total?
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For A, Group A’s percentage is 35/50 × 100% = 70%. Group B’s is 72/120 × 100% = 60%. B has the larger count, 72 compared with 35, while A has the larger percentage. The totals differ, so a larger numerator does not settle the comparison of shares.
For B, the median levels are 25 for A and 60 for B. The lower-quartile levels are 12.5 and 30; the upper-quartile levels are 37.5 and 90. Fractional target levels are used in grouped-curve estimation; they do not mean that an actual person has been divided into parts.
For C, A has 50 − 35 = 15 observations above, giving 30%. B has 120 − 72 = 48 above, giving 40%. These percentages complement 70% and 60% respectively because the categories partition each group under the stated endpoint convention.
For D, the ninetieth-percentile target counts are 45 and 108. A reading at 90 would exceed A’s total and would represent only 75% of B’s total. Neither would locate the requested ninetieth percentile for its group.
For E, A has the larger share, since 65% exceeds 55%. The larger count cannot be determined without the totals or equivalent count information. For F, use 96/2 = 48. The vertical axis maximum of 150 does not change the number of observations in the group.
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A useful scratch table has one row per group and four columns: total, target proportion, target count and graph reading. For a median comparison, the target proportion is one half for both groups, while the target counts may differ. That table keeps sameness and difference in their proper places.
For a threshold comparison, change the record: total, graph-derived cumulative count, fraction and percentage. This time the measurement threshold is shared, while the shares must be calculated. The structure of the record follows the question rather than forcing every graph problem into one procedure.
Ask a learner to explain one row before filling the second. “I need half of 60 because this is the median of this group” reveals more understanding than simply writing 30. Once the meaning is secure, shorten the explanation and let the student complete the record independently.
Use deliberately unequal totals during practice. Equal-size groups can conceal the count-versus-proportion distinction because the same count comparison and percentage comparison then point in the same direction. A near pair with totals 40 and 80 makes the hidden denominator visible.
The worked-example revision guide offers a route from studying a complete solution to making these decisions without prompts. The aim is not permanent dependence on a table template; it is independent recognition of which total belongs to which count.
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A final denominator test appears when a question combines groups. Using the earlier threshold example, Group A has 24 out of 40 observations at or below the threshold, and Group B has 40 out of 80. Together they have 64 such observations out of 120, giving approximately 53.3%.
The simple average of the separate percentages is (60% + 50%)/2 = 55%. That is not the combined percentage because the groups have different sizes. The larger group contributes more observations to the combined total. Add counts and totals before forming the combined share.
This calculation does not require a new cumulative-frequency rule. It requires preserving what the counts represent. Both groups must use the same measured quantity, threshold and endpoint convention before their counts can be added meaningfully. You also need to know that the groups can be combined without counting the same observation twice.
Do not try to obtain a combined median by averaging the two group medians. Medians describe ordered positions, and the combined ordering depends on how the observations from the groups fit together. The two median values and group sizes alone do not generally reconstruct that ordering.
If compatible cumulative-frequency tables are supplied at common boundaries, you can add the corresponding cumulative counts to build a combined table. The final combined total should equal the sum of the separate totals. A combined curve can then be used for the requested grouped estimates, following the question’s plotting instructions.
For example, if one common boundary has cumulative counts 16 and 24, the combined count is 40. If a later boundary has counts 30 and 48, the combined count is 78. These are sums of observations below shared boundaries, not averages of horizontal measurements. The operation follows the meaning of the quantity being combined.
This extension is particularly useful for checking whether a learner understands normalisation. Ask why the percentage calculation uses 64/120, then ask why the same style of averaging cannot simply produce a combined median. The answers distinguish additive counts from positional summaries. Both are useful statistics, but they carry different information and support different operations.
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Why can a median-reading position be a fraction?
Grouped cumulative-frequency curves estimate positions continuously between boundaries. A target such as 12.5 is a location used for reading the curve, not a claim that the dataset contains half an observation. Keep this grouped-graph procedure separate from conventions for finding medians or quartiles in a short raw list.
Must I convert every cumulative-frequency graph to percentages?
No. Counts are appropriate when the question asks how many. For a percentile comparison, you can calculate the appropriate count for each group and read directly from the frequency curves. Conversion is a tool for matching the question, not a compulsory extra step in every solution.
Can I compare medians if the groups have different sizes?
Yes, provided the measured quantity and context support the comparison. Read each median using its own total and describe the groups shown. Different sizes do not make a median comparison meaningless, but a graph alone does not automatically justify a claim about a wider population or a cause.
What if the two curves use different measurement units?
Convert the measurements to a common unit before comparing them. Also inspect whether the graphs use the same categories and boundary conventions. A consistent denominator does not repair a mismatch in what was measured.
Where should I go next?
Use the Mathematics explanation directory for the next specific decision. For a different question about statistical centre, read how removing a value changes the mean and median. The Clementi Secondary 1 Mathematics small-group guide gives the local teaching context. For this topic, the durable check is to name the total before interpreting the height: a cumulative count becomes a share only when its denominator is known.