The point of collecting data is to say something about a whole population. This page is about doing that honestly — drawing conclusions that the data actually support.
In a sample of $250$ households, $95$ have a pet. The town has $14\,000$ households. Estimate how many have a pet.
A sample of $40$ apples from an orchard has a mean mass of $118$ g. The orchard has $12\,000$ apples. Estimate the total mass of the crop, in kilograms.
- Quote the actual figures from the data.
- Name the statistic you are using (median, mean, IQR…).
- Say what it means in the context of the question.
- Use cautious language: "suggests", "estimates", "on average".
- Mention any limitation of the data.
| Weak conclusion | Strong conclusion |
|---|---|
| "Group A is better." | "Group A has a higher median score ($68$ vs $61$), so on average they performed better." |
| "The data go up." | "Sales rose from £$18\,000$ to £$35\,000$ over the two years, an upward trend." |
| "Most people agree." | "$62\%$ of the $400$ people sampled agreed, suggesting a majority — though the sample was taken in one town only." |
A survey of $80$ students found a mean of $2.4$ hours of homework per night. A student writes: "This proves all students do lots of homework." Rewrite the conclusion properly.
Better: "The sample of $80$ students had a mean of $2.4$ hours of homework per night, which suggests that students in this school typically spend a little over two hours on homework. However, the mean alone does not show how much this varies between students."
| Trick | How to spot it |
|---|---|
| Choosing the flattering average | The mean, median and mode can differ a lot — check which one is quoted |
| No sample size given | "$80\%$ of people agree" — out of how many? Four? |
| Biased sample | Who was asked, and where? |
| Percentage without a base | "Sales up $200\%$" — from $1$ to $3$? |
| Cherry-picked time period | Starting the graph at an unusually low point |
| Confusing correlation with cause | Two things rising together need not be linked |
A company's salaries are: £$18$k, £$19$k, £$19$k, £$20$k, £$22$k, £$25$k, £$180$k. The manager says "the average salary here is £$43$k". Comment.
The claim is misleading. The single £$180$k salary is an outlier that drags the mean far above what anyone else earns. The median of £$20$k is a much fairer description of a typical salary here.
A headline reads: "Cases of a rare illness double — a $100\%$ increase!" The number rose from $3$ to $6$ in a population of $2$ million. Comment on the headline.
The headline is technically true but highly misleading. Quoting a percentage change from a very small base makes a negligible change sound alarming. The actual numbers should be given alongside.
Rainfall (mm) was recorded for $30$ days in two towns.
| Mean | Median | Range | IQR | |
|---|---|---|---|---|
| Ashby | $4.2$ | $3.0$ | $22$ | $3.5$ |
| Barton | $4.0$ | $3.8$ | $9$ | $3.2$ |
Compare the rainfall in the two towns.
Conclusion: the two towns have similar typical rainfall, but Ashby experiences occasional very heavy downpours that Barton does not.
| Fault | Example | Fix |
|---|---|---|
| Leading | "Don't you agree the café is too expensive?" | "How would you rate the café's prices?" |
| Vague | "Do you exercise often?" | "How many hours did you exercise last week?" |
| Overlapping options | $0$–$5$, $5$–$10$, $10$–$15$ | $0$–$4$, $5$–$9$, $10$–$14$ |
| Gaps in options | $0$–$5$, $7$–$10$ | Make the boxes continuous |
| No option fits | Boxes only up to "$10$ or fewer" | Add "more than $10$" |
| Personal or intrusive | "How much do you earn?" | Use broad bands, or omit |
| Two questions in one | "Do you like maths and science?" | Split into two questions |
A questionnaire asks: "How much TV do you watch? □ $0$–$1$ hour □ $1$–$3$ hours □ $3$–$5$ hours". Give two criticisms and write an improved version.
Improved: "How many hours of TV did you watch yesterday?"
□ less than $1$ □ $1$ to less than $3$ □ $3$ to less than $5$ □ $5$ or more
Scaling up
Sample proportion $\times$ population size.
State the assumption
That the sample is representative.
Good conclusion
Figures $+$ named statistic $+$ context.
Cautious language
"Suggests", not "proves".
Which average?
An outlier means the median is usually fairer than the mean.
Percentages
Always ask "out of how many?"
Comparing
Average $+$ spread; compare proportions, not raw counts.
Questionnaires
Non-overlapping, exhaustive, not leading, with a time frame.
Limitations
Mention sample size and how the sample was chosen.
In a sample of $120$ people, $42$ cycle to work. The town has $30\,000$ workers. Estimate how many cycle, and state an assumption.
▶ Show solution
Proportion $= \dfrac{42}{120} = 0.35$
Estimate $= 0.35 \times 30\,000 = 10\,500$ workers
Assumption: the sample is representative of all workers in the town.
A sample of $50$ bags of flour has a mean mass of $1.02$ kg. Estimate the total mass of $8000$ bags.
▶ Show solution
Estimated total $= 1.02 \times 8000 = 8160$ kg
Give two criticisms of the question: "Do you agree that our excellent new library is a great improvement? Yes / No".
▶ Show solution
1. It is leading. The words "excellent" and "great improvement" push the respondent towards saying yes.
2. The options are too limited. There is no way to express a neutral view or to say you have not used the library.
A shop's daily takings are £$210$, £$225$, £$218$, £$232$ and £$1450$. Which average best represents a typical day, and why?
▶ Show solution
Mean $= \dfrac{2335}{5} = £467$; median $= £225$ (the middle value once ordered).
The £$1450$ is an outlier that pulls the mean far above four of the five days.
The median (£$225$) is the better average, because it is not distorted by the one exceptional day.
Rewrite these response boxes so they are suitable: "$0$–$10$, $10$–$20$, $20$–$30$".
▶ Show solution
The boxes overlap at $10$ and $20$, and there is nothing for values above $30$.
Improved: "$0$ to less than $10$", "$10$ to less than $20$", "$20$ to less than $30$", "$30$ or more".
A newspaper reports "Crime up $50\%$ in our village!" The number of reported crimes rose from $4$ to $6$. Comment on the report.
▶ Show solution
The percentage is correct: $4 \to 6$ is a $50\%$ increase.
But the actual change is only two crimes. With such small numbers, an increase of two could easily be normal year-to-year variation rather than a real trend.
The report is misleading because it uses a percentage from a very small base to make a tiny change sound dramatic. The raw figures should be given.
Two shops record customer waiting times. Shop A: median $4$ min, IQR $2$ min. Shop B: median $3$ min, IQR $7$ min. Write a comparison and say which shop you would prefer.
▶ Show solution
Average: Shop B has the lower median ($3$ min vs $4$ min), so a typical wait is shorter at Shop B.
Spread: Shop A has a much smaller IQR ($2$ min vs $7$ min), so waiting times at Shop A are far more predictable.
Preference: Shop A, if you value knowing roughly how long you will wait — at Shop B you might be served very quickly or face a very long wait.
An advert claims "$9$ out of $10$ dentists recommend our toothpaste". What three questions should you ask before believing it?
▶ Show solution
1. How many dentists were asked? If only $10$ were surveyed, "$9$ out of $10$" means nine people.
2. How were they chosen? If the company selected them, or paid them, the sample is biased.
3. Recommend it over what? Recommending it as better than nothing is very different from recommending it over rival brands.
A school surveys $60$ of its $900$ students about a proposed change to the timetable. $39$ are in favour.
(a) Estimate how many of all $900$ students are in favour. (b) The survey was carried out in the sixth-form common room. Explain why the estimate may not be reliable. (c) Suggest an improvement.
▶ Show solution
(a) Proportion $= \dfrac{39}{60} = 0.65$
Estimate $= 0.65 \times 900 = 585$ students
(b) The sample was taken only from sixth-formers, who are not representative of the whole school. Younger students may be affected differently by the timetable change and could hold quite different views, so the sample is biased.
(c) Take a stratified random sample across all year groups, in proportion to their sizes, so that every year is fairly represented.
A charity surveyed $200$ donors and found the mean donation was £$38$ and the median was £$15$.
(a) What does the difference between the mean and median suggest about the data? (b) The charity has $8500$ donors. Estimate the total donated, and say which average you used and why. (c) The charity's advert says "Our typical donor gives £$38$". Comment on this claim. (d) Suggest one further statistic that would make the picture clearer.
▶ Show solution
(a) The mean is much larger than the median, which means the data are positively skewed — a small number of very large donations are pulling the mean up, while most donations are much smaller.
(b) To estimate a total, use the mean, because mean $\times$ number of items gives the total.
Estimate $= 38 \times 8500 = £323\,000$
(Using the median would give $15 \times 8500 = £127\,500$, which would underestimate the total because it ignores the large donations.)
(c) The claim is misleading. "Typical" suggests what most donors give, and the median of £$15$ shows that half of all donors give £$15$ or less. Very few donors are likely to give close to £$38$.
(d) A measure of spread such as the interquartile range, or the quartiles themselves, would show how varied the donations are. A histogram of the donation sizes would show the skew directly.