Real-World Mathematics: Probability, Statistics, Data and Decisions

Question → data-generating process → sample → summary → uncertainty → comparison → decision → review.

1. Data begins with a question, not a spreadsheet

A table of numbers has no automatic meaning. Before analysis, we need to know what each row represents, how each value was measured, when the data was collected, which population it came from and what question the data is supposed to answer.

Good statistical reasoning therefore starts before arithmetic. A badly framed question can produce precise-looking summaries of the wrong phenomenon.

2. Population and sample are different objects

A population is the full group we want to understand. A sample is the subset we actually observe. Sampling is necessary because measuring everyone may be expensive, slow or impossible.

The mathematical problem is not merely sample size. The sample must also represent the population well enough for the intended conclusion.

Worked example: convenience bias

Suppose a school wants to estimate average travel time for all students but surveys only students who arrive before 7:15 a.m. The sample may over-represent students with longer or more cautious journeys. A large biased sample can still give a biased answer.

3. Mean, median and mode answer different questions

The mean uses every value and is sensitive to extreme observations. The median locates the middle and is often more resistant to outliers. The mode identifies the most frequent value or category.

Worked example: skewed waiting times

Waiting times in minutes are 2, 3, 3, 4, 4, 5 and 28. Mean = 49/7 = 7 minutes. Median = 4 minutes. The mean is pulled upward by one unusually long wait. Neither summary is “wrong”; they answer different aspects of the distribution.

If the operational question is typical experience, median may be more informative. If the question is total staff time lost, the mean may matter because the long delay contributes real minutes.

4. A centre without spread is incomplete

Two data sets can have the same mean and very different consistency. Spread measures describe how dispersed the observations are.

• Range: maximum − minimum.
• Interquartile range: spread of the middle 50%.
• Variance and standard deviation: more advanced measures using distances from the mean.

In quality control, reliability and service planning, variation can matter as much as the average.

Worked example: same average, different reliability

Machine A produces lengths 9.9, 10.0, 10.1, 10.0 and 10.0 cm. Machine B produces 9.0, 10.0, 11.0, 10.0 and 10.0 cm. Both have mean 10.0 cm, but Machine A is far more consistent.

5. Probability is a model of uncertainty

A probability lies between 0 and 1, or between 0% and 100%. It describes the modelled chance of an event. It does not promise what will happen in one individual case.

A 70% chance of rain does not mean it must rain for 70% of the day. A 5% defect probability does not mean exactly five defects in every 100 items. Probability describes uncertainty across repeated or comparable situations under stated conditions.

6. Experimental probability and theoretical probability serve different roles

Theoretical probability comes from a mathematical model of possible outcomes. Experimental probability comes from observed frequency.

For a fair six-sided die, theoretical probability of rolling a 6 is 1/6. If a die produces 22 sixes in 100 rolls, experimental probability is 0.22. The difference may be ordinary random variation, or it may raise a question about the fairness of the die. More evidence helps distinguish these possibilities.

7. Independence means one event does not change the probability of another

If two events are independent, knowing one occurred does not change the probability of the other. Repeated coin tosses in the ideal model are independent. Drawing cards without replacement is not independent because the deck composition changes after each draw.

This distinction determines whether probabilities multiply in a simple way.

Worked example

Probability of two heads in two independent fair coin tosses = 1/2 × 1/2 = 1/4.

For two red cards drawn without replacement from a standard 52-card deck with 26 red cards, probability = 26/52 × 25/51, because the second probability changes after the first draw.

8. Conditional probability asks what changes after new information

Conditional probability is the probability of A given that B is known. The conditioning information changes the reference group.

Worked example: quality inspection

Suppose 200 items are produced. Forty come from Line X. Of those 40, six are defective. Conditional defect rate given Line X = 6/40 = 15%.

If there are 10 defects across all 200 items, overall defect rate = 10/200 = 5%. These are different probabilities because the denominator changes when we condition on Line X.

9. Base rates matter when interpreting tests and alerts

A test can be accurate in a technical sense and still produce many false alarms when the underlying condition is rare. This is a base-rate problem.

Imagine 10,000 items where only 1% are truly defective: 100 defects and 9,900 good items. Suppose a test catches 90% of defects and falsely flags 5% of good items. It flags about 90 true defects and 495 good items. Among 585 flags, only about 15.4% are true defects.

The lesson is not that the test is useless. It is that interpreting a positive result requires both test behaviour and the underlying base rate.

10. Expected value combines outcomes with probabilities

Expected value is a probability-weighted average of possible outcomes. It is useful for repeated decisions and long-run comparisons, but it is not a promise about one trial.

Worked example

A simple game pays $10 with probability 0.2 and $0 otherwise. Expected payout = 0.2×10 + 0.8×0 = $2.

Playing once will not produce $2. The result is either $10 or $0. The expected value summarises the long-run average under repeated identical trials.

11. Risk is not captured by expected value alone

Two options can have the same expected value and different variability. A guaranteed $50 has expected value $50. A 50% chance of $100 and 50% chance of $0 also has expected value $50. The distribution of outcomes is different.

Real decisions may care about downside, reliability, maximum loss, thresholds and tolerance for variation. Expected value is one summary, not the entire decision.

12. Relative risk and absolute risk tell different stories

If an event rate rises from 1% to 2%, the absolute increase is 1 percentage point, while the relative increase is 100%. Both statements are mathematically correct, but they create very different impressions if reported alone.

Responsible quantitative communication often reports both the base rate and the change.

13. Correlation does not by itself establish causation

Correlation describes association. Two variables may move together because one causes the other, because a third variable influences both, because of selection effects, because of common trends or simply because of chance.

A scatter plot can reveal association, clusters and outliers. It cannot alone identify the causal mechanism.

Pattern is evidence to investigate, not permission to invent a cause.

14. Averages can hide subgroups

An overall average can improve while one subgroup worsens, especially when group sizes change. Good analysis therefore checks whether important categories behave differently from the aggregate.

This is one reason tables, distributions and subgroup comparisons often matter more than a single headline number.

15. Sampling variability means different samples give different answers

If we repeatedly draw random samples from the same population, their means and proportions will not be identical. This is sampling variability.

Larger well-designed samples generally reduce random sampling noise, but they do not automatically remove systematic bias. Ten thousand responses to a biased question can be less useful than a smaller representative sample with a well-designed instrument.

16. Confidence is not certainty

More advanced statistics uses intervals and uncertainty estimates to describe what values are compatible with the data and method. These tools do not make uncertainty disappear. They make uncertainty visible enough to reason about.

A narrow interval may indicate high precision under the model; it does not rescue poor measurement, biased sampling or an irrelevant question.

17. Outliers can be errors, rare events or the most important observations

An outlier is unusually far from the rest of the data. It may arise from data entry, measurement failure, a genuinely rare event or a different underlying process.

Deleting an outlier simply because it is inconvenient is not good statistics. Investigate why it occurred and report the treatment transparently.

18. Quality control is probability meeting tolerance

Manufacturing systems measure items, compare them with specification limits and track patterns over time. A single out-of-tolerance item may trigger inspection; a shift in the distribution may reveal process drift even before many items fail.

This connects the companion guide on measurement, units, scale and estimation directly to statistics. Measurement creates data; statistical structure helps decide whether the process is stable.

19. Forecasting is conditional on patterns continuing

A forecast uses past and present information to estimate future outcomes. The model may use averages, trends, seasonality or more advanced statistical structure.

The further a forecast extends, the more opportunities there are for conditions to change. Forecasts should therefore communicate horizon, assumptions and uncertainty rather than presenting one number as inevitable.

20. Classification creates thresholds and trade-offs

Many systems classify observations: pass/fail, fraud/not fraud, urgent/not urgent, positive/negative. A threshold controls which cases fall into which group.

Lowering a threshold may catch more true cases but also create more false alarms. Raising it may reduce false alarms but miss more true cases. There is rarely a threshold that eliminates both kinds of error.

The correct threshold depends on the costs and consequences of mistakes, not only on mathematical accuracy.

21. Dashboards compress evidence—and can hide it

A dashboard might show averages, percentages, trends and alerts. Compression is useful for decisions, but every compressed metric discards detail.

Before trusting a dashboard, ask:

• What is the denominator?
• What period is being compared?
• How much data sits underneath the metric?
• Are subgroups combined?
• Has the measurement definition changed?
• Is variation hidden behind an average?

22. Data cleaning is part of Mathematics, not clerical work

Missing values, duplicated records, inconsistent units and impossible entries can distort analysis. A clean formula applied to dirty data remains a dirty conclusion.

Simple validation rules are powerful: ages should fall in a plausible range; percentages should not exceed logical bounds unless the quantity allows it; dates should be ordered; units should be consistent; identifiers should not be duplicated without explanation.

23. A practical data-to-decision loop

1. State the question.
2. Define the population and quantities.
3. Understand how the data was generated.
4. Check quality, units and missingness.
5. Summarise centre and spread.
6. Model uncertainty where relevant.
7. Compare alternatives or groups.
8. Test whether the conclusion is sensitive to assumptions.
9. Make the decision and record what evidence would cause revision.

24. Worked real-world mini cases

Case A: service waiting time

A service centre reports average wait of 6 minutes. If most customers wait 2–4 minutes but a few wait 30 minutes, the average alone hides the tail. Reporting median, a high percentile or the distribution may better describe reliability.

Case B: defect monitoring

A factory historically sees 1% defects. A new batch has 4 defects in 100 items. That single batch does not prove the underlying rate is now 4%, but it is evidence worth comparing with normal variation and process history.

Case C: school survey

A voluntary online survey receives mostly responses from highly engaged students. The sample size may be large, but non-response and self-selection can make the sample unrepresentative. Statistical reasoning includes asking who is missing.

Case D: route reliability

Route A averages 30 minutes with standard travel usually between 28 and 32. Route B averages 28 minutes but often ranges from 20 to 45. If arrival reliability matters, the lower mean does not automatically make Route B the better choice.

25. Common failure modes

• Average-only thinking: centre is reported without spread.
• Biased sample: a convenient group is treated as the full population.
• Base-rate neglect: test results are interpreted without underlying prevalence.
• Probability as promise: a probability is treated as a guaranteed frequency in a small number of cases.
• Correlation-to-causation leap: association is presented as a mechanism.
• Absolute/relative confusion: percentage points and relative change are mixed.
• Outlier deletion: unusual values are removed without investigation.
• Dashboard blindness: headline metrics are trusted without checking definitions and denominators.
• Forecast certainty: a prediction is presented without horizon or uncertainty.

26. Practice set

1. Find the mean of 4, 7, 8, 9, 12.
2. Find the median of 3, 3, 5, 8, 20.
3. Find the range of 11, 14, 17, 19, 25.
4. Data set A and B both have mean 10. A values are tightly clustered; B values are widely spread. Which is more consistent?
5. A bag has 3 red and 7 blue counters. Find P(red).
6. Two independent fair coins are tossed. Find P(two heads).
7. A card is drawn from 5 red and 5 blue cards without replacement, then a second is drawn. Find P(two red).
8. In a group of 80 people, 20 are in Team A and 6 of Team A are late. Find P(late | Team A).
9. An event has probability 0.08. What percentage is this?
10. A game pays $12 with probability 0.25 and $0 otherwise. Find expected payout.
11. Option A guarantees $30. Option B gives $60 with probability 0.5 and $0 otherwise. Compare expected values.
12. A rate rises from 2% to 3%. State the absolute increase in percentage points and relative increase.
13. Why can a large convenience sample still be biased?
14. Explain why correlation alone does not prove causation.
15. A machine gives readings 10.01, 10.00, 9.99, 10.00. What does this suggest about precision?
16. A survey of all school students is answered mainly by graduating students. Name the likely problem.
17. A rare condition affects 1 in 1000 people. Why does base rate matter when interpreting a positive screening result?
18. Route A mean = 25 min with little variation; Route B mean = 23 min with large variation. What extra information is needed before deciding which is better?

27. Answers and reasoning

1. 8.
2. 5.
3. 14.
4. Data set A.
5. 3/10 = 0.3.
6. 1/4.
7. 5/10 × 4/9 = 2/9.
8. 6/20 = 0.30 = 30%.
9. 8%.
10. 0.25×12 = $3.
11. Both have expected value $30, but their risk/variation differs.
12. 1 percentage point; relative increase = 1/2 = 50%.
13. Bias comes from who is included or excluded, not only from sample size.
14. Other variables, selection effects or chance can create association without a direct causal mechanism.
15. The readings are tightly grouped, suggesting high repeatability/precision, though accuracy requires comparison with a trusted reference.
16. Non-response or selection bias; the responding group may not represent the whole school.
17. When the condition is rare, even a modest false-positive rate can create many more false positives than true positives.
18. Decision objective, acceptable delay risk, distribution of travel times and consequences of being late.

28. What statistical thinking teaches beyond data

Probability teaches humility about uncertainty. Statistics teaches that summaries are representations, not the raw world. Sampling teaches that who is observed matters. Conditional probability teaches that new information changes the reference group. Expected value teaches long-run comparison. Spread teaches that consistency matters. Causal caution teaches that patterns need mechanisms and stronger evidence.

A good data decision does not pretend uncertainty is gone. It shows enough of the uncertainty that the decision can be judged.

Continue the Real-World Mathematics series

• Measurement, Units, Scale and Estimation
• Ratios, Rates, Percentages and Proportions
• Graphs, Functions, Optimisation and Systems
• Return to the BTT Mathematics Hub