Secondary Mathematics · Worked Repair Guide 27
Data questions become dangerous when calculation is treated as the entire job. A sample can be biased before any graph is drawn. A scatter plot can show association without explaining why it exists. A line of best fit can be useful for prediction without becoming a law of nature. The mathematics therefore includes both calculation and limits.
This guide develops one central habit: separate what the data show from what the data do not establish. Sampling determines what evidence was collected. A scatter plot summarises pairs. Correlation describes association. A fitted line supports approximate prediction. Causation requires stronger reasoning than a pattern in points.
All surveys, marks, times and measurements below are invented for teaching. They do not describe actual students, schools or populations. Use Averages, Spread and Data Interpretation and Quartiles, Cumulative Frequency, Box Plots and Standard Deviation for the neighbouring descriptive-statistics routes.
1. A population is the whole group of interest
If a study asks about all 3000 students in an invented school network, the population contains those 3000 students.
A sample is the subset actually observed or surveyed.
If 120 students are sampled, the sampling fraction is 120/3000=0.04=4%.
A large sample fraction does not automatically guarantee an unbiased sample. Selection method matters.
2. A census observes every member of the defined population
A census attempts to collect information from every member of the population.
It can avoid sampling variation caused by selecting only part of the group, but it can still suffer from measurement error, missing responses or badly worded questions.
“Census” therefore does not mean “perfect data”. It describes coverage, not automatic accuracy.
Sampling is often chosen because a census is expensive, slow, destructive or impractical.
3. Simple random sampling gives units an explicit chance of selection
In a simple random sample of size n from a population of size N, every possible sample of that size is selected according to the random mechanism.
A practical method could number all population members and use a suitable random-number process to select distinct labels.
Choosing “people who happen to be nearby” is not simple random sampling even if the researcher did not deliberately favour anyone.
Randomness is a selection procedure, not a synonym for casual choice.
4. Systematic sampling uses a regular interval after a starting point
Suppose an ordered list contains 2000 units and a sample of 80 is required. The sampling interval is 2000/80=25.
A common method chooses a random start among the first 25 positions, then selects every 25th unit.
This can be efficient, but periodic structure in the list may create bias if the interval aligns with a repeating pattern.
Systematic does not mean automatically representative; inspect how the list is ordered.
5. Stratified sampling preserves chosen group proportions
Suppose a population has 600 members in group A and 400 in group B, total 1000. A proportional stratified sample of 120 would allocate:
A:120×600/1000=72. B:120×400/1000=48.
Selection should then occur within each stratum using an appropriate random method.
Stratification is useful when the chosen categories are important to representation, but it does not correct poor measurement inside each group.
6. Convenience samples can be badly biased
An invented survey about travel habits conducted only among people waiting at one bus stop is easy to administer but may overrepresent regular bus users.
The problem is not small sample size alone. The selection mechanism is related to the variable being studied.
Increasing the same convenience sample from 50 to 500 can make the estimate more precise about that biased group without making it representative of the intended population.
Precision and representativeness are different ideas.
7. Question wording can create response bias
Compare “Do you support the sensible new study schedule?” with “Do you support the new study schedule?”
The word “sensible” signals an expected answer and may influence responses.
Questions can also be ambiguous, double-barrelled or dependent on inaccurate memory.
Statistical quality begins before calculation, at the design of what is measured and how it is asked.
8. Non-response can alter the effective sample
Suppose 500 selected people are contacted but only 160 respond.
The response rate is 160/500=32%.
If respondents systematically differ from non-respondents on the study variable, results can be biased.
Random initial selection does not guarantee that the final responding group remains representative.
9. Scatter plots display paired numerical observations
A scatter plot places one numerical variable on each axis and plots each paired observation as a point.
For example, an invented data set may pair study time x with quiz score y for each learner.
The plot can reveal direction, form, strength and unusual points.
It does not retain personal stories or automatically reveal which variable causes the other.
10. Positive and negative correlation describe direction
Positive correlation means larger x values tend to be associated with larger y values.
Negative correlation means larger x values tend to be associated with smaller y values.
No clear correlation means the plot does not display a strong systematic direction of association.
“Positive” does not mean desirable and “negative” does not mean bad. The words describe direction.
11. Strength concerns how tightly points follow a pattern
A strong positive relationship has points clustered closely around an increasing pattern. A weak positive relationship has much more scatter.
A relationship can be strong but nonlinear. Points lying closely around a curved shape may have a clear association even if a straight line is a poor summary.
Therefore “strong correlation” should not be inferred solely from whether a line can be drawn.
Inspect form before fitting a linear model.
12. An outlier may be important rather than disposable
A point far from the main pattern may represent a recording error, a rare valid case, a different subgroup or a change in process.
Do not delete it merely because it weakens the fitted relationship.
In an investigation, check the observation and report any justified exclusion transparently.
In a school exercise, work with the data as supplied unless instructed otherwise.
13. Correlation does not by itself establish causation
If two variables are associated, several explanations remain possible: x may influence y, y may influence x, another variable may affect both, or the observed pattern may reflect selection or chance.
For example, a positive association between two school-related measures does not prove that changing one will cause the other to improve.
Experimental design, randomisation, controls and subject-matter reasoning can strengthen causal conclusions, but a scatter plot alone cannot do so.
A responsible interpretation describes the association first.
14. A line of best fit is an approximate summary
A hand-drawn line of best fit should follow the central tendency of a roughly linear cloud of points, with points reasonably balanced above and below.
The line does not have to pass through every point. If it did, the data would need to be exactly linear.
When a question supplies or asks for a fitted equation, treat it as a model for the observed range rather than an exact rule for every case.
Different reasonable hand-drawn lines can produce slightly different predictions.
15. Use a fitted line to estimate one variable from the other
Suppose an invented line of best fit is y=2x+5.
At x=8, predicted y=21.
If predicted y=31, solve 31=2x+5 to obtain x=13.
The equation gives a model estimate, not proof that an actual observation at x=8 must equal 21.
16. Residuals measure vertical prediction error
A residual can be defined as observed y minus predicted y.
If the line predicts 21 and the observed value is 24, residual=24−21=+3.
A positive residual means the point lies above the fitted line; a negative residual means below.
A good linear fit tends to leave residuals without an obvious systematic pattern. Curvature in residuals suggests that a straight line may be missing structure.
17. Interpolation stays within the observed x-range
If observed x-values run from 2 to 10, estimating at x=7 is interpolation.
It still carries uncertainty, but it uses the model inside the range where data were observed.
Estimating at x=15 is extrapolation because it moves beyond the observed range.
Extrapolation is generally riskier because the relationship may change outside the data range.
18. Sampling conclusions should match the sampling design
A random sample from one school does not automatically represent every school in a country.
A well-designed sample can support inference to the population from which it was selected, subject to sampling uncertainty and measurement quality.
Claims should therefore name the population being described.
A common overreach is to widen the conclusion beyond the sampling frame simply because the sample size is large.
19. Capstone: design, fit and interpret without overclaiming
An invented population has 70% group A and 30% group B. A proportional stratified sample of 50 would include 35 from A and 15 from B.
Suppose a roughly linear scatter plot from the sample is summarised by y=1.5x+4 within observed x-values from 2 to 12.
At x=8, the fitted estimate is y=16. At x=20, the formula gives 34, but this is extrapolation well outside the observed x-range.
If the point at x=8 actually has y=18, its residual is +2.
A careful conclusion is: “Within this sample, the two variables showed a positive roughly linear association. The fitted line estimated y≈16 at x=8.” It does not say that x causes y or that the same relationship must hold in every other population.
20. Independent practice
- Define the population in a study.
- A sample of 120 is taken from a population of 3000. Find the sampling fraction as a percentage.
- A list has 2000 members and a systematic sample of 80 is required. Find the sampling interval.
- A population has 600 in group A and 400 in group B. Find proportional allocations for a sample of 120.
- Explain one problem with surveying only people who are easiest to reach.
- Explain why “Do you support the sensible new rule?” may be biased.
- Points rise from lower left to upper right. State the correlation direction.
- Points show no systematic upward or downward trend. State the likely description.
- Points lie tightly around a downward-sloping line. Describe direction and strength.
- Explain why correlation alone does not establish causation.
- For fitted line y=2x+5, estimate y at x=8.
- For the same line, solve for x when y=31.
- If observed y=24 when predicted y=21, find residual observed−predicted.
- Observed x-values run from 2 to 10. Classify prediction at x=7 as interpolation or extrapolation.
- Classify prediction at x=15.
- Explain how an outlier can affect a fitted line.
- Explain why a sample mean need not equal the population mean exactly.
- Explain one way non-response can bias a survey.
- What design feature normally supports stronger causal inference than an observational scatter plot alone?
- A population is 70% A and 30% B. Find proportional allocations in a sample of 50.
21. Worked answers
1. The entire defined group about which the study seeks information.
2. 4%. 120/3000×100.
3. 25. 2000/80.
4. A:72, B:48.
5. Convenience selection can overrepresent people who are easy to reach and underrepresent others, making the sample unrepresentative.
6. The word “sensible” leads respondents toward approval and can influence the measured response.
7. Positive correlation.
8. No clear correlation.
9. Strong negative correlation.
10. Other variables, reverse direction, selection effects or chance can produce association; a scatter pattern alone does not isolate a causal mechanism.
11. 21.
12. 13. 31=2x+5.
13. +3.
14. Interpolation. x=7 lies within 2 to10.
15. Extrapolation.
16. An unusual point can pull a fitted line and alter its gradient or intercept; investigate rather than deleting automatically.
17. A sample contains only part of the population and is subject to sampling variation.
18. If non-respondents systematically differ from respondents on the study variable, the responding data may no longer represent the selected sample well.
19. A suitably designed randomised experiment with controls, where feasible and ethical, generally supports stronger causal reasoning.
20. A:35, B:15.
22. Diagnose data errors by separating design, calculation and interpretation
Common failures include calling a convenience sample random, reading positive correlation as “good”, treating any decreasing graph as a strong negative linear relationship, extrapolating far beyond observed data without caution, or writing causal claims from observational association.
A useful repair note says “who was eligible and how were they selected?”, “correlation describes association”, “fit form before line”, or “outside observed range → extrapolation”.
Then ask the learner to rewrite the conclusion in one bounded sentence naming the sample, variables and evidence actually shown.
23. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Averages, Spread and Data Interpretation for basic summaries and Quartiles, Cumulative Frequency, Box Plots and Standard Deviation for advanced spread.
Within Batch 07, continue to Functions, Domain, Range and Graph Behaviour for fitted-line function language, Trigonometry, Pythagoras, Bearings and Elevation, or Surface Area, Volume and Composite Solids.
24. Sources and scope
The populations, samples, survey questions, fitted lines and practice tasks are original teaching material. Sampling terminology and correlation interpretation are used in their standard statistical sense.
For the current Singapore Secondary curriculum doorway, see MOE: Curriculum for secondary schools. Match sampling methods, residuals and causal-design discussion to the learner’s actual subject level and school programme.