2027 G3 SEC Mathematics K310 | Learner’s Guide Vol 0194. When a graph says the 75th percentile is 36 centimetres, does it tell us exactly which ten plants were taller than 36? Not necessarily. Percentile questions become easy to calculate once we know the count. They become difficult again when the numbers have been grouped or the threshold cuts across an interval. This chapter teaches the difference.
All measurements, journey times and selections below are invented practice data, not an official SEC examination question or a report about a real school. They are deliberately small enough for independent checking. We will insist on a clean distinction between exact counts and estimates reconstructed from grouped data.
1. The SEC mathematics task and the precise purpose of this chapter
The official 2027 SEAB G3 directory, last updated 26 October 2026, identifies Mathematics as K310. The K310 syllabus includes cumulative frequency, quartiles, percentiles, interquartile range, standard deviation and interpretation of statistical representations. There are two papers, each lasting two hours fifteen minutes and contributing half the marks; Paper 2 ends with an extended real-world problem. This chapter extends earlier practice on constructing cumulative frequencies. Its distinct task is decision-making around thresholds: how many observations lie below a cut-off, how many remain above it, and when a percentile drawn from grouped information is only an estimate. These problems reward clear working and a conclusion attached to the correct population.
2. A percentile is a position within a distribution
The 75th percentile marks a level around which roughly 75 per cent of observations lie at or below, subject to the definition and data representation in use. It is not automatically a score of 75 marks, nor a statement that a pupil answered 75 per cent of questions correctly. The percentile’s numerical value depends on the measured variable. In a journey-time distribution it may be 28 minutes; in a test distribution it may be 71 marks. Its meaning is relative position. Always complete this sentence: “Around 75 per cent of the observations are at or below approximately ___ units.” A bare percentile label without context tells the reader too little.
3. The lower tail and upper tail belong to the same whole
If 65 per cent of journeys take less than 20 minutes, then 35 per cent take 20 minutes or longer, assuming every journey is counted once and the groups are complementary. The two shares add to 100 per cent because they divide the same population. Beware the boundary. “Less than 20” excludes exactly 20, while “20 or more” includes it. “At most 20” and “more than 20” form a different pair. When a question asks for upper-tail counts, begin with the total and subtract the count in the explicitly defined lower group. Writing the boundary in words prevents a correct subtraction from answering the wrong question.
4. Raw observations can be counted exactly
Suppose eight fictional bus journey times, in minutes, are 12, 14, 14, 16, 18, 20, 22 and 25. Five journeys are under 20 minutes: 12, 14, 14, 16 and 18. Three are at least 20 minutes: 20, 22 and 25. There is no need to estimate because each observation is known. If the question asks for more than 20 minutes, only two qualify. Students sometimes treat “at least” as “greater than” and lose one observation at the boundary. Mark inclusivity using either words or the correct inequality sign before calculating anything else. This discipline becomes especially important when the source groups data into classes.
5. Grouped frequencies preserve counts but lose fine detail
A grouped table tells us how many observations fall within each interval, not the exact values within it. Imagine 40 plant-height measurements grouped as follows: 0 to under 10 cm: 4; 10 to under 20 cm: 8; 20 to under 30 cm: 12; 30 to under 40 cm: 10; 40 to under 50 cm: 6. The total is 4 + 8 + 12 + 10 + 6 = 40. We know exactly how many are below 30 cm: 4 + 8 + 12 = 24. We do not know exactly how many are above 35 cm because 35 cuts through the 30-to-under-40 class. That difference governs which answers may be exact.
6. Build cumulative frequencies deliberately
Using the 40-height example, the cumulative counts at upper class boundaries 10, 20, 30, 40 and 50 cm are 4, 12, 24, 34 and 40. Each cumulative count includes all the earlier classes, not only the current class. A common error is to copy the original frequencies 4, 8, 12, 10, 6 and label them cumulative. Another is to forget the final total. Check that cumulative counts never decrease and that the last count equals the total number of observations. If either test fails, repair the table before plotting. The shape of a cumulative curve is built from these running totals, not guessed from the raw class heights.
7. Exact upper-tail counts from a known class boundary
From that same table, the number of measurements at least 30 cm is 40 minus the 24 below 30, giving 16. This is exact because the requested threshold coincides with a recorded class boundary. The share is 16/40 = 0.4 = 40 per cent. Some students subtract 34 and report six; that would count only measurements from 40 cm upwards. A reliable written solution names the lower group first: “24 heights are below 30 cm.” Then it subtracts and returns to the requested group: “Therefore 16 are at least 30 cm.” This small explanation protects both the arithmetic and the meaning.
8. A threshold inside a class produces uncertainty
How many of the 40 measurements are at least 35 cm? The six measurements in the 40-to-under-50 class definitely qualify. Some or all of the ten in the 30-to-under-40 class may also qualify. The exact answer therefore lies between 6 and 16 inclusive, depending on the unreported measurements. A uniform-within-class estimate might divide the class approximately in half and suggest 11, but that is a model-based approximation, not an exact count. If the question provides a cumulative-frequency curve, an estimated reading at 35 cm may be appropriate. Never silently present that reading as if the original grouped table recorded each individual height.
9. The 25th percentile in our 40-value example
One quarter of 40 is 10. The tenth observation lies in the 10-to-under-20 cm class because the cumulative total rises from four at 10 cm to twelve at 20 cm. A straight-line interpolation between those plotted points estimates the cut-off as 10 + (10 − 4)/(12 − 4) × 10 = 17.5 cm. This is an estimate based on the graph’s interpolation, not the exact tenth measured height. The actual raw tenth value could differ while still satisfying the grouped frequencies. When working from a drawn curve, students normally read the height at a cumulative frequency of 10, state the units and use sensible precision for the graphical estimate.
10. The median is a middle location, not a mean
The halfway count for 40 observations is 20. In our example, that count occurs between cumulative 12 at 20 cm and cumulative 24 at 30 cm. Linear reading gives roughly 20 + (20 − 12)/(24 − 12) × 10 = 26.7 cm. Compare this with the mean: the mean would require adding all individual heights or estimating from class midpoints. The median instead uses position. A group can have a high mean because of a few extreme observations while still having a lower median. A question that asks for median should not be answered with the midpoint of the entire measurement range, unless that happens to be justified by the actual distribution.
11. The 75th-percentile boundary and the upper quarter
Three quarters of 40 is 30. The thirtieth observation lies between cumulative 24 at 30 cm and 34 at 40 cm. Interpolating along that graph segment gives about 36 cm: 30 + (30 − 24)/(34 − 24) × 10. That is an estimated upper quartile. It suggests that roughly one quarter of the observed heights are above the corresponding percentile level. Do not announce that exactly ten plants are taller than exactly 36 cm; grouped data do not provide their individual positions, and ties or interpolation conventions matter. The honest answer reports a percentile estimate and makes its precision match the information given.
12. Percentile cut-offs can be used in reverse
A question may give the threshold and ask for its percentile position rather than give a percentile and ask for its value. Suppose a cumulative graph indicates about 27 of 40 observations below a specified value. The estimated cumulative proportion is 27/40 = 67.5 per cent, so that value is around the 67.5th percentile on that graphical reading. This does not imply the measured value itself is 67.5 in physical units. Always separate the horizontal coordinate, which carries the original variable and unit, from the vertical coordinate, which carries a count or cumulative percentage. A swapped coordinate produces an answer that may look tidy but has no sensible meaning.
13. The distinction between percentile and percentile rank
“The 90th percentile is 72 marks” describes a cut-off in marks. “A pupil is at about the 90th percentile” describes the pupil’s relative position among a defined comparison group. The two statements are related but not interchangeable. A percentile rank can change when the comparison group changes, even though the pupil’s raw score stays the same. Nor does being at the 90th percentile mean answering 90 per cent of the paper correctly. Students should translate percentile statements into full sentences with a named variable, cohort and unit. If the data source does not define the cohort, any sweeping comparison with national performance is unjustified.
14. The total must match the population in the question
A school has 40 measurements from one class and 200 from the year level. A cumulative frequency of 30 in the class means 75 per cent of that class’s 40 observations; it does not mean 15 per cent of the 200-year-level group unless those 30 observations were legitimately counted within the larger population for the stated purpose. Before converting a count to a percentage, identify the source’s total. The denominator is not a cosmetic choice. A ratio with the wrong total changes the meaning of a threshold and can mislead a decision-maker about how common an outcome is.
15. Read less than, at most, above and at least as separate instructions
Take five scores: 10, 20, 20, 30, 40. Less than 20 counts one score. At most 20 counts three. Above 20 counts two. At least 20 counts four. The changes are caused by a single word in each instruction. A cumulative frequency plot may use “less than” cumulative counts at class upper boundaries. A question may ask for “20 or more,” requiring a subtraction from the total. If the source’s convention is unclear, read its interval notation and graph labels before deciding whether the threshold itself belongs to the lower or upper side. Precise boundary language is part of the mathematics.
16. A cumulative graph is not a histogram
In a histogram, the heights or areas represent frequency density or counts according to the chosen display. A cumulative-frequency graph accumulates counts. Its vertical values cannot decrease as the horizontal values increase. A steep section means many observations lie within that interval; a flatter section means fewer. The total is read at the right-hand end, not by adding cumulative heights. Draw a horizontal guide from the requested cumulative frequency to the curve and then a vertical guide to the data scale for a percentile. For a threshold count, reverse the direction: start at the horizontal value, read the accumulated count, and subtract from the total where required.
17. Axes, starting points and class boundaries
A cumulative-frequency curve generally starts at the lowest class boundary with cumulative count zero, then passes through the plotted upper-boundary totals. If the lowest class begins at 0 cm, plot 0 at 0 cm, then 4 at 10 cm, 12 at 20 cm and so forth for the example. Plotting each count at the class midpoint would shift the readings and corrupt the percentile estimates. A neat graph needs a clearly labelled horizontal unit, a vertical count scale that reaches the total, sensible intervals and a smooth joined representation where appropriate. When reading a printed graph, use the actual axis labels rather than assuming the scale uses ordinary units.
18. Interpolation is a disciplined estimate
Linear interpolation within an interval treats the segment between two cumulative points as approximately straight. That assumption produces an estimated value between the boundaries. It is reasonable for many classroom graph-reading tasks, but it does not recover lost individual measurements. If a student estimates 36 cm for the upper quartile, 35.8 or 36.2 may be reasonable depending on plotting and scale precision. A claim of 36.0000 cm falsely suggests exact knowledge. If the question calls for an answer to a specified degree of accuracy, follow that instruction; otherwise keep graph readings at precision the diagram supports.
19. Quartile and percentile conventions need context
For a short raw list, textbook methods for assigning exact quartile positions can differ, especially with small samples and ties. In this guide’s large grouped examples, we use reading positions of one quarter, one half and three quarters of the total to practise cumulative curves. When asked about raw data, use the convention taught for the question or the method specified in the examination materials. Do not force a graph-based interpolation formula into every small sorted list. The core idea remains stable: quartiles divide an ordered distribution into four portions, and an interquartile range describes the distance between the lower and upper quartiles.
20. IQR describes the middle half, not the whole spread
In the grouped height example, estimated Q1 = 17.5 cm and estimated Q3 = 36 cm give an estimated IQR of 18.5 cm. This characterises the middle half of the distribution on the chosen cumulative reading. It does not tell us the maximum or minimum height or all the individual deviations from the mean. By contrast, standard deviation summarises spread through squared deviations and averages; earlier Vol 0190 teaches that calculation. If two groups have the same median but different IQRs, their middle observations vary differently. A sensible comparison names both centre and spread instead of declaring one distribution universally “better.”
21. Decision thresholds should be linked to an objective
A canteen manager may care how many students wait longer than 15 minutes because that determines how many are at risk of missing lunch. A coach may use the 75th-percentile lap time to describe the slow end of a performance distribution. Neither metric is intrinsically good or bad. A decision threshold matters only in relation to the purpose and group. When writing a conclusion, say what the percentage or count measures and why that particular cut-off matters. Do not recommend a policy solely because a percentile sounds impressive. A strong response may also mention what the threshold does not cover, such as outcomes outside the measured period.
22. Comparing cohorts requires matched definitions
Suppose one class measures journey time from home to school and another measures time from the bus stop to school. Even if their graphs share the same units of minutes, the variables differ. A lower 75th percentile in the second group does not show that its pupils travel faster. Check starting and ending points, measurement conditions, sample and timing. If they differ, either limit the conclusion or explain how a fairer comparison could be designed. The graph is not the problem; the unexamined definition is. Mathematics interpretation frequently turns on reading the words attached to numerical measures.
23. Relative improvement and absolute percentage-point change
Imagine a service reduces the share of students waiting at least 20 minutes from 40 per cent to 30 per cent. The decrease is 10 percentage points. Relative to the initial 40 per cent, it is 10/40 = 25 per cent. These statements are both mathematically correct but answer different questions. A headline that says “waits fell by 25%” is incomplete unless it makes clear which measure fell and what base was used. This joins the interpretive discipline of Vol 0103 with later decision problems: a graph may provide accurate numbers while a sentence describing them uses an ambiguous percentage.
24. More observations can change a percentile
If a school adds new measurements to a data set, the total and cumulative ranks change. An earlier 75th-percentile cut-off need not remain unchanged. The added values may be mainly at the low end, high end or spread across the distribution. A claim that a threshold “always” places exactly one fourth of every future group above it is therefore not justified. Percentiles describe a specified distribution at a specified time. They are not fixed physical laws. A more careful forecast would say that an old percentile can be a planning benchmark, but future observations need to be checked against it.
25. A fictional school-bus threshold problem
A survey groups 60 journeys in five-minute intervals: 0–under 5 minutes, 3 journeys; 5–under 10, 9; 10–under 15, 18; 15–under 20, 20; 20–under 25, 10. The cumulative counts are 3, 12, 30, 50, 60. Exactly ten journeys lasted at least 20 minutes, or one sixth of the sample, because the threshold matches a class boundary. A claim that “half the journeys exceed 15 minutes” is not right: 60 − 30 = 30 journeys are at least 15 minutes, but “exceed” excludes journeys exactly 15 minutes and the grouped data do not separate those values. State the boundary correctly before reporting any percentage.
26. A fictional club-selection cut-off problem
An organiser wants approximately the fastest quarter of 80 recorded times. On a cumulative plot where faster means a smaller time, that selection corresponds to the lower quartile of times, not the upper quartile. If the question instead asks for the top quarter of test scores, larger values are better and the upper quartile is the relevant boundary. “Top” is about performance direction, not always about largest number. Read what is being measured. Time taken, number of errors and costs usually favour lower values, whereas scores and counts of correct responses usually favour higher ones. Identify the direction before choosing a percentile threshold.
27. From a graph reading to an exam-quality answer
Start with the total. Find the class boundary or requested percentile rank. Draw or imagine a guide line to read the cumulative count. If the question asks for values above the cut-off, subtract from the total. Write the unit for a threshold and “students,” “journeys,” or “measurements” for a count. If the answer is graphical, mark it as approximate when appropriate. Finally, interpret without overstating: “About 15 out of 60 recorded journeys lasted longer than the estimated threshold.” This sequence is quicker than guessing a number from the graph and trying to explain it afterward. It also exposes errors before they become final answers.
28. Repair practice that builds independence
When a worked solution is wrong, locate the first failure: an incorrect running total, midpoint instead of upper boundary, reversed axis reading, subtraction using the wrong total, confusion over inclusivity, or claiming a grouped estimate is exact. Correct that step, then change the story context and repeat. If the difficulty is basic percentages or reading class boundaries, revisit the PSLE learning guide. For broader Secondary 4 treatment, use the Mathematics grouped-data learning owner. The goal is not to memorise one graph shape. It is to make the same precise reasoning survive new labels, different totals and unfamiliar real-world decisions.
29. Thirty-seven original exam-style practice prompts with worked solutions
Work without the model answer first. For each item, write the total, identify the lower and upper groups, and decide whether the result is exact or estimated. Answers are educational worked solutions rather than an official SEAB marking scheme.
Exercise 1. Forty observations; cumulative 24 below 30. How many are at least 30?
Worked solution. 40 − 24 = 16. State that 16 observations are at least 30 units, using the stated boundary. The result is exact if all 40 observations are included and the cumulative count is exact.
Exercise 2. A curve has cumulative 30 at 15 minutes in a set of 60. How many lasted at least 15?
Worked solution. If the curve counts those under 15, 60 − 30 = 30. Read the source’s inclusive or exclusive convention carefully before deciding which journeys equal to 15 are counted.
Exercise 3. Why is a 75th-percentile test score not necessarily 75 marks?
Worked solution. The percentile describes position in an ordered cohort; marks describe performance on the test’s scale. The corresponding score might be 48, 62, 83 or another value.
Exercise 4. A grouped table has frequencies 3, 5, 7. Give the cumulative frequencies.
Worked solution. The running totals are 3, 8 and 15. Each cumulative value includes every earlier interval. The final value, 15, checks against the total number of observations.
Exercise 5. From the 40-height table, how many heights are below 20 cm?
Worked solution. The first two classes contribute 4 + 8 = 12. The answer is an exact class-boundary count, not an interpolated estimate.
Exercise 6. In the 40-height table, how many are at least 40 cm?
Worked solution. Six. Only the 40-to-under-50 class qualifies; the earlier 34 measurements are below 40.
Exercise 7. Can you find the exact number at least 35 cm from the grouped table?
Worked solution. No. The 35 cm threshold lies inside a class. Six observations certainly qualify and up to ten more may qualify. The exact count is not recoverable from the grouped frequencies.
Exercise 8. What assumption is made by interpolating halfway through the 30–under-40 cm class?
Worked solution. It approximates the within-class distribution as spread evenly along a straight cumulative segment. The resulting count or percentile is a reading estimate, not exact individual measurements.
Exercise 9. Find 25 per cent of 40 as a cumulative position.
Worked solution. A quarter of 40 is 10. For the grouped-height example, the tenth observation falls in the 10-to-under-20 cm class.
Exercise 10. Estimate Q1 in the height example using linear graph interpolation.
Worked solution. At 10 cm the cumulative count is four; at 20 cm it is twelve. Position 10 is six of eight counts into the interval, so Q1 ≈ 10 + 6/8 × 10 = 17.5 cm.
Exercise 11. Estimate the median location for 40 grouped heights.
Worked solution. Use cumulative position 20. The relevant interval goes from cumulative 12 to 24 across 20–30 cm, giving approximately 26.7 cm on a linear curve.
Exercise 12. Estimate Q3 for the same graph.
Worked solution. Position 30 lies in the 30–40 cm class, six of ten cumulative counts after 24. Linear graph reading gives approximately 36 cm.
Exercise 13. Estimate the interquartile range from those readings.
Worked solution. IQR ≈ 36 − 17.5 = 18.5 cm. It is a grouped-data estimate describing the middle half, not the full range.
Exercise 14. A plotted cumulative count is 45 in a sample of 75. What percentage is that?
Worked solution. 45/75 × 100% = 60%. The horizontal value at that point is around the 60th percentile under the graph’s convention.
Exercise 15. A cohort’s 90th percentile is 72 marks. Interpret it.
Worked solution. Around 90 per cent of the cohort’s observations fall at or below the corresponding cut-off under the method used. It does not mean that every pupil above it scored exactly 72 marks.
Exercise 16. A group has 40 journeys below 20 minutes out of 50. How many are at least 20?
Worked solution. 50 − 40 = 10 journeys, or 20 per cent. Use the complement of the “below 20” group.
Exercise 17. What is wrong with dividing a class’s cumulative count by the whole school’s population?
Worked solution. The numerator and denominator refer to different populations unless the question explicitly places those observations in the same set. The percentage would not describe the intended group.
Exercise 18. Does a cumulative-frequency graph ever decrease as x increases?
Worked solution. No. Counts accumulate as the threshold moves right. A decreasing plot suggests that frequencies, another type of graph or an incorrect construction has been used.
Exercise 19. Should cumulative counts be plotted at class midpoints?
Worked solution. No. For a less-than cumulative graph they are normally placed at upper class boundaries, beginning with zero at the lowest boundary. Midpoints shift the positions.
Exercise 20. A student adds all the heights of the cumulative graph to get n. Explain the mistake.
Worked solution. The same earlier observations appear in many cumulative totals. The correct total is the final cumulative value, not the sum of running totals.
Exercise 21. What does a steeper interval on a cumulative graph mean?
Worked solution. More observations fall within that horizontal interval, relative to another interval of comparable width. It describes frequency concentrated there, not the average value of the observations.
Exercise 22. What is the difference between 40% to 30% and a 25% relative decrease?
Worked solution. The absolute difference is 10 percentage points. Relative to the starting 40%, the decrease is 10/40 = 25%.
Exercise 23. Which quartile identifies the fastest quarter when the variable is time taken?
Worked solution. The lower quartile of time, because smaller times are faster. “Best” need not mean “largest numerical value.”
Exercise 24. Which quartile describes the highest-scoring quarter?
Worked solution. The upper quartile of scores, because higher marks represent stronger performance in that variable.
Exercise 25. The sample total doubles after new observations are added. Does Q3 stay unchanged?
Worked solution. Not necessarily. Percentiles describe ranks in the new distribution. Added observations can alter the cut-off and its interpretation.
Exercise 26. A cumulative plot rises from 12 at x = 20 to 24 at x = 30. Estimate x at cumulative 18.
Worked solution. 18 is halfway between 12 and 24, so the straight-line graphical estimate is halfway between 20 and 30: 25 units.
Exercise 27. For that same segment, estimate cumulative frequency at x = 25.
Worked solution. Using the straight-line approximation, the value is about 18. The actual individual count at that internal threshold could differ because grouped data do not reveal exact observations.
Exercise 28. A table lists 0–under 5: 3, 5–under 10: 9, 10–under 15: 18, 15–under 20: 20, 20–under 25: 10. Total?
Worked solution. The total is 3 + 9 + 18 + 20 + 10 = 60. The cumulative sequence is 3, 12, 30, 50, 60.
Exercise 29. How many of those journeys lasted at least 20 minutes?
Worked solution. 60 − 50 = 10. The boundary aligns with a class edge, so the grouped data support an exact count.
Exercise 30. Exactly how many lasted more than 15 minutes in that table?
Worked solution. It cannot be known precisely because “more than 15” excludes exactly 15, which is within the 15-to-under-20 class. We can determine at least 15, not necessarily strictly more than 15.
Exercise 31. The graph’s estimated 75th-percentile journey is 18.4 minutes. How should you state the answer?
Worked solution. Say “approximately 18.4 minutes,” or use a precision compatible with the graph. Do not imply the graph revealed an exact timestamp for the 75th ordered journey.
Exercise 32. Two classes have the same median but different upper quartiles. What follows?
Worked solution. Their central percentile is similar, but their upper-side distributions differ in at least the estimated quartile boundary. That alone does not establish which group has better learning.
Exercise 33. Why might IQR and standard deviation give different impressions?
Worked solution. IQR focuses on the middle half; standard deviation incorporates squared deviations from the mean and is sensitive to extreme observations. They describe spread in different ways.
Exercise 34. Why is “top 10%” incomplete without a named group?
Worked solution. A percentile ranking depends on the comparison cohort. Top 10% of one class does not automatically mean top 10% of the whole year or all candidates.
Exercise 35. What is the strongest exact conclusion from the 40-height class table about 30 cm?
Worked solution. 24 are below 30 cm and 16 are at least 30 cm, so 40% are at least 30 cm. The conclusion names the threshold, count and denominator.
Exercise 36. You know a percentile from a different year’s cohort. Does it predict an identical cut-off this year?
Worked solution. No. Changes in the distribution can change percentiles. It may be a comparison benchmark, but an unchanged future cut-off is not guaranteed.
Exercise 37. A curve says 35 out of 50 are at or below a threshold. What fraction lie above it?
Worked solution. 15 out of 50, which is 30%, provided “at or below” and “above” partition the same 50 observations. Preserve that exact boundary language.
30. Extended integrated case: choosing a service standard
In an invented school pilot, 80 journey times are recorded in minutes. The grouped frequencies are: 0–under 5, 5; 5–under 10, 15; 10–under 15, 28; 15–under 20, 20; and 20–under 25, 12. First check the total: 5 + 15 + 28 + 20 + 12 = 80. The cumulative frequencies at 5, 10, 15, 20 and 25 minutes are 5, 20, 48, 68 and 80. Exactly 12 journeys were at least 20 minutes. Exactly 32 journeys were at least 15 minutes. The two counts differ because their thresholds cut the table at different class boundaries.
A parent committee suggests that at least 75 per cent of journeys should take under 20 minutes. The grouped data show 68/80 = 85 per cent under 20, so this particular sample meets that numerical condition. This does not establish that the journey times will meet the target in every future week: the sample period, route conditions and pupil mix matter. An alternative target is that the 75th-percentile time should remain below 20 minutes. Cumulative position 0.75 × 80 = 60 lies in the 15–under-20 class, where the cumulative count rises from 48 to 68. The linear graph estimate is 15 + (60−48)/(68−48) × 5 = 18 minutes. It is an estimate derived from the grouped curve.
Finally, the committee receives a claim that “only 15 per cent of journeys ever take 20 minutes or longer.” In this observed sample 12/80 = 15 per cent, but the word “ever” reaches beyond the recorded period. The correct public wording is “In the 80 recorded journeys, 15 per cent lasted at least 20 minutes.” The calculation stays the same while the unsupported promise disappears. That is the kind of real-world interpretation the K310 syllabus rewards.
31. Independent transfer and links
Recreate the cumulative totals from a fresh grouped table before consulting your notes. Then answer three different requests using the same curve: estimate the median, count observations above a class boundary, and explain why an interior threshold requires estimation. If you confuse quartiles with spread, return to Vol 0107 on box plots and Vol 0190 on standard deviation. Keep the Secondary 4 statistics reference for longer-term review. The next Biology volume uses a similar standard of precision: measuring water transport does not automatically tell you everything about movement of manufactured food through a plant.
