Two classes can have the same middle reading and very different distributions
Suppose two sets of fictional breathing-rate observations have a median of eighteen per minute. One set is tightly grouped around eighteen; the other ranges widely from low to high. Calling the groups “the same” because their medians match ignores the distribution. A stronger Mathematics answer considers the centre, spread, sample size, any missing values and the meaning of the graph. Each statistic answers a different question. Your job is to choose the one relevant to the problem and explain what it can and cannot show.
This volume follows Vol 0568, which owns unequal-window rates and pooling, then Vol 0570’s evidence hearing and Vol 0571’s alveolar explanation. Here every input is already a rate measured on a consistent per-minute basis. We will not recalculate raw observation windows; instead, we study sorted values, mean, median, range, quartiles, interquartile range, box plots and standard deviation. The fictional data provide a setting for G3 Mathematics, not medical benchmarks or information about any real pupils.
Quick route: verify each observation is comparable; arrange the rates in ascending order; choose mean or median for the required comparison; calculate lower and upper quartiles according to the stated convention; find the interquartile range; interpret a box plot without confusing it with a frequency graph; compare variation; and state a conclusion that stays within the sample. A precise statement such as “this sample has a larger IQR” is preferable to the unsupported claim that a class is healthier.
Verified 2027 SEC Mathematics K310 syllabus and examination format
The SEAB 2027 G3 syllabus catalogue, updated on 26 October 2026, lists G3 Mathematics as K310, with former O-Level code 4052 given for reference. The official K310 syllabus includes statistical diagrams, mean, mode and median, quartiles, percentiles, range, interquartile range, standard deviation, and comparison of datasets. It also expects explanations of how statistical diagrams may mislead. Paper 1 has about 26 short-answer questions and Paper 2 has nine to ten questions, including an extended real-world application problem. Each paper is 90 marks, 50% of the assessment, and 2 hours 15 minutes. Unless the question specifies a different accuracy, non-exact numerical answers are given to three significant figures.
In earlier learning, the PSLE guide on how averages can conceal different totals makes the first useful distinction. The site’s deliberate Mathematics practice guide explains how to repair a specific weak link instead of doing more of every problem. Here that weak link is often the temptation to report one mean without studying the shape of a dataset.
Start with the variable, not the formula
Write what a number represents. “Twenty-four” alone is not a rate; “twenty-four observed breathing cycles per minute during the specified window” is. A dataset of counts recorded over different time durations cannot fairly be compared until every value has been converted correctly. Vol 0568 handles that conversion. This chapter begins after it has been done. For distribution analysis, identify the measured variable, its unit, the number of valid observations and any exclusions. A correctly calculated statistic applied to incomparable values does not become meaningful just because the arithmetic is neat.
Check that the data are independent enough for the question
A class might contribute multiple readings per person. If the problem asks for a distribution of readings, list each record as directed. If it asks for a distribution of students, repeated records for one person should not automatically count as multiple individuals. The distinction changes the sample size, frequencies and interpretation. Exam questions usually supply the needed context. Do not silently treat repeated measurements as independent pupils. State whether the unit of observation is a session, a person or a reading before calculating the summary.
Ordering is the first statistical operation
To find the median or quartiles, arrange values from smallest to largest. For eight observations, the median lies between the fourth and fifth positions. Sorting is easy to skip because it feels elementary, but one misplaced value changes the answers for several measures at once. Keep the original list and the ordered list separately. Circle the positions you use for the median, lower quartile and upper quartile. Then check the count of observations; missing one item changes the central positions.
Mean is a balance point, not a typical guaranteed observation
The arithmetic mean equals the sum of values divided by their number. It can be a fraction even when every observation is a whole number. For example, eight observations may have a mean of 20.125 although no recorded rate equals 20.125. The mean reflects every value and is sensitive to unusually high or low values. It does not say that every observation was near the centre. When reporting it, state the population or sample under discussion and distinguish it from the median.
Median describes the middle position
With eight sorted observations, the median is the mean of the fourth and fifth values. With nine sorted observations, it is the fifth value. A median does not necessarily tell you how spread out the other readings are. Two samples can have the same median yet very different upper and lower tails. The median is often useful when a distribution contains extreme observations, but it should not be called automatically “more accurate” than the mean; each is an accurate measure with a different interpretation.
Mode and frequency answer a different question
The mode is the most frequently occurring value, when such a value is identifiable. A set can have more than one mode, and some datasets have no uniquely most frequent observation. The mode does not have to equal the mean or median. In a frequency table, the number attached to an observation is its frequency, not the observation’s size. For raw numerical observations, find repeated values deliberately rather than selecting the largest number as the mode.
Range shows extremes, not the middle spread
The range is maximum minus minimum. It responds directly to the two outermost values and ignores the detailed distribution of all the observations in between. A set can have a large range because of one unusually large observation while most values remain close together. Use range if the problem asks for total width between extremes, but compare it with the interquartile range when asked about the central portion of the data. Always keep the original units: a range of twelve breaths per minute is not a percentage.
Quartiles divide the ordered dataset by position
The lower quartile Q1 and upper quartile Q3 mark positions toward the lower and upper ends of the ordered data. For eight values, a common school convention is to take the median of the lower four and the median of the upper four. A question may provide its own quartile convention or cumulative-frequency approach; follow the specified method. Do not switch methods halfway through a comparison. The key is to describe both position and meaning, not merely memorise a calculator key.
Interquartile range concerns the middle half
IQR = Q3 − Q1. It describes the spread between the first and third quartiles, covering the middle half of the ordered observations in the conventional interpretation. An IQR of fifteen is wider than an IQR of six; it indicates greater spread in the central portion of that sample. It does not by itself say which set has the higher average. In a strong comparison, name the centre measure separately from the spread measure, with the correct values and units.
Construct a box plot from a five-number summary
A basic box plot requires minimum, Q1, median, Q3 and maximum. The box runs from Q1 to Q3 and contains the median line. The whiskers connect the box to extremes under the convention used in the question. Some statistical software distinguishes outliers and adjusts whiskers; do not import a software convention into a school problem without instructions. Label the axis and use a consistent numerical scale when comparing plots. A box plot does not display every frequency directly.
Do not read box width as sample size
A wide box indicates a large interquartile spread on a correctly scaled axis, not that the dataset necessarily includes more individuals. A narrow box can represent many observations close together. If two box plots are drawn on the same numerical axis, compare median positions and quartile ranges. If they are drawn on separate scales, the apparent physical lengths on the page may mislead. The graph’s meaning comes from the axis, not the amount of ink used.
Calculate standard deviation with the stated formula
The official K310 syllabus includes standard deviation for grouped and ungrouped data. For a complete dataset, one useful formula is the square root of mean of squares minus square of the mean. A larger standard deviation generally describes greater dispersion around the mean. It is not measured in squared units after taking the square root; for rates, the standard deviation has the same unit as the rates. Carefully distinguish the quantity inside the square root from the final standard deviation.
Do not confuse variance with standard deviation
Variance is the mean squared deviation under the population-style calculation; it has squared units. Standard deviation is the square root of that variance and returns to the original units. If a calculation gives fourteen in squared units, the standard deviation is square root of fourteen, about 3.74 in the original units. Reporting fourteen as the standard deviation is a common error. If the calculator has sample and population standard-deviation options, choose the one corresponding to the syllabus formula and the problem’s definition.
Outliers need a rule, not a feeling
An unusually large or small observation may be called an outlier under an explicit rule, such as a threshold based on the IQR. But a student should not label any inconvenient result invalid simply because it looks far from the rest. A genuine extreme observation can be important. Conversely, a transcription error should be corrected from verified records. If a question asks for an outlier rule, state the rule and calculate its cutoffs. If no rule is supplied, describe a value as unusually high or low without pretending to have performed a formal test.
A changed maximum can alter range without altering median
Suppose the largest value in a sorted eight-item sample increases while the other seven are unchanged. The range increases; the median may remain the same because the fourth and fifth positions have not changed. The IQR may or may not change, depending on the quartile positions and the extent of the alteration. The mean changes because it includes all values. This example shows why separate statistics are needed: centre, middle spread and extremes are related but not identical pieces of information.
Cumulative frequency is not the same as ordinary frequency
An ordinary frequency states how many observations fall in a category or value. Cumulative frequency counts how many have occurred up to a boundary. It must not decrease as the boundary increases. A cumulative-frequency diagram can be used to estimate medians and quartiles from positions, while a simple bar chart communicates counts by categories. The examiner may ask you to choose an appropriate representation and explain why an alternative is less helpful. Match the graph to the question.
Read histogram bars carefully
For equal class intervals, as named in the K310 syllabus, a histogram can show how observations are distributed across continuous numerical intervals. The height reflects frequency under equal widths. Do not assume every observation is exactly at the midpoint, or read a grouped chart as though raw data are known. Mean estimates from grouped data use class midpoints and frequencies. They are estimates unless a question defines values at those midpoints. Explain whether your calculation uses raw values or grouped approximations.
A trend in a chart does not establish a cause
If two distributions differ, that difference may be an observation worth describing. It does not tell us why it occurred unless the question supplies appropriate evidence. In this chapter, breathing rates are a mathematical teaching context and do not establish health status, tobacco exposure, fitness or respiratory efficiency. The previous Biology volume owns alveolar mechanisms, while this one owns statistical comparison. Scientific plausibility is not a licence to convert a statistical summary into a diagnosis.
Always compare like with like
The same measure, unit and quartile convention must be used for both groups. Comparing one group’s median with another’s mean is usually not a fair answer to a question about centre. Comparing one group’s range with another’s IQR describes different parts of the distributions. If one set has ten valid readings and the other has eight, mention the different sample sizes where relevant. Data can still be compared, but a categorical claim such as “more reliable” demands more than a larger count alone.
Name both centre and variability
A useful comparison template is: “Group A has a lower median than Group B, while Group A has a smaller IQR; therefore the middle of A’s distribution is lower and its central observations are more tightly grouped.” Replace the values appropriately. If the medians match, state that plainly and then discuss the differing spread. Avoid “better” or “worse” unless the task defines a meaningful direction. The statistical vocabulary “higher”, “lower”, “more spread out” and “less variable” is often all you need.
Keep calculation and interpretation separate
On a calculation line, show a sum, a division or a quartile difference. On an interpretation line, explain the result in context: “The middle half of group B’s recorded rates spans fifteen breaths per minute.” That is more informative than writing “IQR = 15”. A question that asks to calculate may need only working and answer. A question that asks to compare requires a sentence explaining what the numbers show. Learn to recognise which job the command verb assigns.
Round at the end and retain enough precision
Use exact values through intermediate steps. If a mean is 20.125 and a standard deviation calculation uses its square, rounding the mean early to twenty can significantly alter the result. The 2027 Mathematics syllabus generally specifies three significant figures for non-exact numerical answers unless another accuracy is requested. Box-plot positions derived from whole-number observations may produce exact halves. Distinguish an exact 12.5 from an estimated 12.5 rounded from other information.
Check whether the description matches the graphic
A caption can misstate the chart even if each plotted number is correct. A box plot showing a wide middle interval cannot support “all values are clustered tightly around the median.” A truncated axis may exaggerate apparent separation between group medians. Some visualisations draw values using different vertical scales, making comparisons unreliable. Use the labelled axes and five-number summaries to challenge the story a graphic tells. The final step of statistical literacy is not calculation, but interpretation.
A final exam paragraph should be bounded
For a class exercise, say “In these fictional recorded observations, the medians are equal while one sample has greater spread.” Do not claim that the group with the higher mean has stronger lungs or that the other is unhealthy. The mathematical conclusion is useful without a health narrative. A good response is specific about its measure and scope, supplies the values, and stops before unsupported causal claims. This kind of restraint is exactly what transfers from primary school arithmetic into advanced statistical reasoning.
Fully worked comparison of two fictional eight-record samples
Dataset A: 12, 14, 16, 18, 18, 20, 22, 24 breaths per minute. Dataset B: 10, 12, 13, 18, 18, 26, 29, 35 breaths per minute. These are already sorted, fictional practice values with eight complete records in each group. A has total 144 and mean 18. B has total 161 and mean 20.125. Both have median 18, the average of the fourth and fifth values. Even before calculating spread, the sets reveal why a common median does not imply identical means or distributions.
Dataset A five-number summary: minimum 12; Q1 = (14 + 16)/2 = 15; median 18; Q3 = (20 + 22)/2 = 21; maximum 24. Therefore IQR = 21 − 15 = 6 and range = 24 − 12 = 12 breaths per minute. The mean is 18. The sum of squared values is 2704; the population-style variance by the syllabus formula is 2704/8 − 18² = 14, and the standard deviation is square root of 14, approximately 3.74 breaths per minute.
Dataset B five-number summary: minimum 10; Q1 = (12 + 13)/2 = 12.5; median 18; Q3 = (26 + 29)/2 = 27.5; maximum 35. Therefore IQR = 27.5 − 12.5 = 15 and range = 35 − 10 = 25 breaths per minute. The mean is 20.125. The sum of squared values is 3803; the population-style variance is 3803/8 − 20.125² = 70.359375, and the standard deviation is approximately 8.39 breaths per minute.
The comparison can be written in one clear paragraph: both samples have median 18, but B has a larger IQR (15 rather than 6), range (25 rather than 12) and population standard deviation (about 8.39 rather than 3.74). Thus B’s observed values are more spread out. Its mean is higher, but that is not evidence of better or worse health. Nothing in these fictional observations identifies causes, medical conditions or exposure. The correct Mathematics conclusion concerns the recorded distribution.
Thirty original examination decisions and answer checks
Each exercise targets a distinct calculation or interpretation. Work from the data supplied in the question, not from the number of the preceding exercise. The answer explanation identifies the method and the common trap; where the result is exact it is left exact.
Worked problem 1: The sorted data are 2, 4, 6, 8, 10. Find the median.
Answer. The median is 6, the third value. Method. For five observations the middle position is (5+1)/2 = 3. Trap to avoid. Do not divide the sum by five when the question asks for median. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 2: The sorted data are 3, 5, 8, 12, 15, 21. Find the median.
Answer. The median is (8+12)/2 = 10. Method. An even number of observations has two central positions. Trap to avoid. Do not select only the third value. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 3: The values are 4, 6, 8, 10. Find the mean.
Answer. The mean is 28/4 = 7. Method. The numerator is the sum of all four observations. Trap to avoid. Do not use the midrange unless the data happen to support it. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 4: The values are 3, 3, 4, 5, 8. Find the mode.
Answer. The mode is 3. Method. Three occurs twice; the other values occur once. Trap to avoid. The largest value, eight, is not the mode. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 5: The sorted values are 5, 7, 9, 11, 13, 15, 17, 19. Find Q1 using median-of-halves.
Answer. Q1 = (7+9)/2 = 8. Method. The lower four values are 5,7,9,11. Trap to avoid. Do not accidentally include values from the upper half. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 6: For the same eight values, find Q3.
Answer. Q3 = (15+17)/2 = 16. Method. The upper four are 13,15,17,19. Trap to avoid. Keep the same quartile convention. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 7: For that set, calculate IQR.
Answer. IQR = 16−8 = 8. Method. It measures the interval between Q1 and Q3. Trap to avoid. Do not subtract maximum and minimum. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 8: A dataset has minimum 11 and maximum 29. Find range.
Answer. Range = 29−11 = 18. Method. The range measures total span of extreme values. Trap to avoid. Keep the original units in a contextual answer. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 9: Two samples share median 20; A has IQR 4, B has IQR 13. Compare central spread.
Answer. B’s middle half is more spread out than A’s because 13 exceeds 4. Method. The medians are equal but the IQRs differ. Trap to avoid. Do not say B has the higher median. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 10: One group has mean 18 and median 13. Can we say every value is 18?
Answer. No; a mean is an aggregate and need not equal any individual observation. Method. The difference between mean and median suggests that distribution details matter. Trap to avoid. Do not infer a guaranteed number of high observations from these two statistics alone. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 11: The ordered values are 2, 4, 7, 8, 11, 13, 15, 18. Give the five-number summary.
Answer. Minimum 2, Q1 5.5, median 9.5, Q3 14, maximum 18. Method. Q1 is (4+7)/2, median (8+11)/2, and Q3 (13+15)/2. Trap to avoid. Check all eight positions before calculating. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 12: A plotted box extends from 12 to 22. What is the IQR?
Answer. The IQR is 10 units. Method. The box spans Q1 to Q3. Trap to avoid. Do not use whisker endpoints for IQR. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 13: A median line is at 17 inside a box from 12 to 24. What is the median?
Answer. It is 17. Method. The line inside a standard box marks the median. Trap to avoid. Do not average the box edges. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 14: Two box plots use different axis scales. Can we compare their physical box widths directly?
Answer. No; compare numeric quartile coordinates on comparable scales. Method. Physical length on the page depends on the axis scale. Trap to avoid. Do not decide from appearance alone. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 15: Eight observations have total 160. Find mean.
Answer. Mean = 160/8 = 20. Method. Divide the sum by the number of included records. Trap to avoid. Do not confuse number of records with their duration. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 16: A ninth valid observation of 29 is added to eight values totalling 160. Find new mean.
Answer. New mean = (160+29)/9 = 21. Method. The total and denominator both change. Trap to avoid. Do not divide 189 by eight. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 17: The largest value rises from 30 to 40 while all others stay fixed. What necessarily changes?
Answer. The range increases by 10, provided the minimum is unchanged. Method. The upper extreme changes, so maximum-minus-minimum changes. Trap to avoid. The median need not change. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 18: Can the IQR be negative?
Answer. No, because Q3 is at least Q1 in an ordered dataset. Method. The difference Q3−Q1 cannot be less than zero. Trap to avoid. A negative result signals subtraction in the wrong order. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 19: If every value increases by 5, what happens to the mean?
Answer. The mean increases by 5. Method. Adding a constant to every observation adds that constant to their mean. Trap to avoid. The range and IQR do not change from this shift. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 20: If every value increases by 5, what happens to IQR?
Answer. It remains unchanged. Method. Both quartiles increase by 5, so their difference stays the same. Trap to avoid. Do not increase IQR just because the raw values are larger. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 21: If every value is doubled, what happens to IQR?
Answer. The IQR doubles. Method. Q1 and Q3 both double, so their difference doubles. Trap to avoid. The same scaling affects range and standard deviation. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 22: A variance is 16 (units squared). Find standard deviation.
Answer. Standard deviation = 4 in the original units. Method. Take the positive square root of the variance. Trap to avoid. Do not present 16 as the standard deviation. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 23: A dataset is 2, 2, 2, 2. Find standard deviation.
Answer. It is zero. Method. Every value equals the mean, so every deviation is zero. Trap to avoid. There is no spread about the mean. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 24: The values are 2, 4, 6. Calculate population-style standard deviation.
Answer. Mean 4; variance ((−2)²+0²+2²)/3 = 8/3; standard deviation about 1.63. Method. The calculation divides by three for this stated population-style formula. Trap to avoid. Do not divide by two without a sample-formula instruction. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 25: A bar chart shows ordinary frequencies 3, 5, 2. Find total observations.
Answer. There are 10 observations. Method. Add each category’s count: 3+5+2. Trap to avoid. Do not use the largest category frequency as the total. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 26: Cumulative frequencies at increasing boundaries are 4, 9, 15, 22. How many fall in the last interval?
Answer. Seven observations. Method. Subtract 15 from 22 to isolate the last interval. Trap to avoid. The cumulative total 22 is not the interval frequency. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 27: A histogram uses equal class widths. What does a taller bar show?
Answer. A greater frequency in that class under equal widths. Method. Equal widths make heights directly comparable as frequencies. Trap to avoid. Do not infer the exact individual values inside the class. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 28: A box plot has the median near Q1. Does this prove all upper values are outliers?
Answer. No; it shows asymmetry of central spacing, not an outlier classification. Method. Outlier rules require additional calculations or criteria. Trap to avoid. Do not invent outlier status from median position alone. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 29: A report says B has IQR 15 and A has IQR 6, so B is healthier. Evaluate.
Answer. The statistical comparison is valid, but the health conclusion is unsupported. Method. IQR describes spread of observations and not medical status. Trap to avoid. Separate a numerical result from an unjustified interpretation. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
Worked problem 30: What is a sufficient final statistical comparison for the two worked samples?
Answer. Both medians are 18, but B has the larger IQR and standard deviation; its sample is more variable. Method. The answer names centre and two distinct measures of spread. Trap to avoid. It stops at the scope of fictional observations rather than inventing causes. Transfer test. Before using the same method on a fresh dataset, identify the observation unit, the positions of the values and the exact comparison requested. Show working sufficient to make each step verifiable.
A mini assessment: from raw figures to a careful conclusion
Original task: A teacher provides two sets of eight fictional per-minute observations, A and B, as above. Part one asks for mean and median; part two for quartiles, IQR and range; part three for a box-plot comparison; part four asks whether the findings reveal a respiratory illness. Full-credit working should show all eight sorted values, the appropriate central positions, five-number summaries and a direct comparison based on the reported statistics.
Part one reasoning: A mean equals 18 while B mean equals 20.125; both medians equal 18. A good answer notes that the mean and median can disagree, because one is based on the total and the other on the central positions. A poor answer interprets the difference as evidence of an environmental cause. No explanatory variable was measured.
Part two and three reasoning: A’s five-number summary is 12, 15, 18, 21, 24; B’s is 10, 12.5, 18, 27.5, 35. Place these points correctly on a single common scale. The second box is wider, corresponding to the larger IQR. Both median lines sit at eighteen. The left and right whisker extents should match the minima and maxima under the stated simple five-number convention. The conclusion is that B’s recorded values are more dispersed.
Part four reasoning: There is insufficient evidence to infer any pupil’s illness or exposure. The numbers are fictional, and their only established meaning is a collection of observed per-minute counts for a Mathematics task. This does not weaken the statistical work. It shows that statistical literacy includes knowing which conclusion is not contained in the dataset.
Revision method: calculate, explain, vary
Practise first with ordered lists of eight values, then with nine, then with frequency tables and finally with diagrams. After each solution, say which statistic answers which question. Mean answers about the average across all values; median answers about central rank; IQR answers about central spread; range describes extremes; standard deviation describes dispersion about the mean. Next vary one record and predict which statistics will be affected. This develops reasoning that transfers beyond the particular fictional data.
Use Vol 0568 when observations have unequal recording windows, because correct rates come before statistics. Use Vol 0571 for the biological processes that a numerical rate does not measure. Use Vol 0570 when an audience challenges a caption or asks for a conclusion stronger than the records justify. This volume is the dedicated owner for statistical comparisons across per-minute distributions.
Official references and limitations
The syllabus map and paper format come from SEAB’s 2027 G3 subject index and the K310 Mathematics syllabus. All practice observations are invented; they are not clinical reference values, medical recommendations or SEAB specimen examination questions. The worked statistics help explain methods, not individual health outcomes.
