Worked example · Easy
Classify variables
Decide whether favourite drink and number of siblings are categorical or numerical.
Favourite drink is categorical. Number of siblings is discrete numerical data because it is counted.
Chapter 45 · Completely rewritten
Categorical, Numerical and Grouped Data
Clear explanations, original worked examples and a full graded exercise set, mapped to the textbook.
What you will learn
Collect and classify data responsibly; interpret tables; choose useful statistical measures; calculate statistics for ungrouped and frequency data; estimate averages from grouped data.
Begin with the easy examples, then attempt the medium and exam-style examples before using the hidden solutions.
1
Subtopic 1
Collecting and grouping data
Categorical data describes groups, such as transport method or favourite sport. Numerical data is recorded as numbers; discrete values are counted, while continuous measurements may take any value in a measurement interval.
Raw data can be recorded in a tally chart and summarised in a frequency table. Check that class intervals do not overlap and account for every observation.
| Number of books | Tally | Frequency |
|---|---|---|
| 0 | || | 2 |
| 1 | |||| | 4 |
| 2 | ||| | 3 |
| Total | 9 |
Essential method
\(\text{Total observations}=\sum f\). A frequency tells how often a value or category occurs. For grouped data, use consistent class boundaries and sensible class widths.
Exam tip: Five-bar tally groups make counting easier; check the frequencies add to the sample size.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Medium
Frequency table
A survey records shirt sizes S, M, M, L, S, M, L, M. Find the frequency of M.
Count M four times, so \(\boxed{f_M=4}\); the total frequency is eight.
Worked example · Exam-style
Grouped travel times
Times in minutes are 12, 19, 23, 27, 31, 36. Use groups 10–19, 20–29 and 30–39.
Counts are \(\boxed{2,2,2}\) respectively. Each time is in exactly one group and \(2+2+2=6\).
2
Subtopic 2
Surveys
A population is the complete group of interest; a sample is a subset. A good survey identifies the population, uses a suitable sample size and avoids selection bias. Random selection gives every eligible member an equal chance of selection; stratifying can ensure important groups are fairly represented.
Results from a sample are estimates, not certain descriptions of the population.
Essential method
For proportional stratified sampling, \(n_{\rm group}=\frac{\text{group population}}{\text{total population}}\times n_{\rm sample}\), followed by appropriate whole-number allocation.
Exam tip: A large biased sample can still be misleading; size alone does not guarantee reliability.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Biased food survey
A school asks only students who buy lunch about the quality of the canteen food. Identify a limitation.
Students who bring lunch or avoid the canteen are excluded. The selection can be biased.
Worked example · Medium
Random sample
A teacher numbers 200 eligible pupils and uses a random-number generator to choose 30 distinct numbers. Describe the sampling method.
This is a simple random sample if every pupil has the same chance of selection.
Worked example · Exam-style
Proportional sampling
A school has 360 juniors and 240 seniors. Select a proportional sample of 50.
Total 600. Juniors \(=(360/600)50=\boxed{30}\); seniors \(=(240/600)50=\boxed{20}\).
3
Subtopic 3
Designing a questionnaire
Questions should be brief, neutral, unambiguous and easy to answer. For closed questions, response categories must be mutually exclusive (no overlap) and exhaustive (cover all plausible answers). Specify units and time periods.
Open questions permit free responses but are more difficult to analyse. Pilot a questionnaire to find unclear wording before using it.
Essential method
For class intervals such as ages 10–19, 20–29, 30–39, each eligible integer belongs in exactly one interval. For measured values, clear half-open intervals can be used, e.g. \(10\le t<20\).
Exam tip: Avoid leading wording (“Don’t you agree...?”), double questions, and overlapping categories.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Spot the leading question
“Surely the new cafeteria is much better, isn’t it?” Explain the problem.
The wording suggests a desirable answer. Replace it with a neutral question such as “How do you rate the cafeteria?”
Worked example · Medium
Fix overlapping ages
A survey offers 10–20, 20–30 and 30–40 years. Explain the problem.
A respondent aged 20 or 30 fits two choices. Use 10–19, 20–29 and 30–39 for integer ages.
Worked example · Exam-style
Improve a travel survey
Rewrite “How often and how far do you cycle to school?” as answerable questions.
Ask two separate questions: “On how many days last week did you cycle to school?” and “Approximately how many kilometres is each one-way trip?”
4
Subtopic 4
Two-way tables
A two-way table organises counts by two classifications, such as year group and transport method. Row totals, column totals and the grand total must agree. A missing cell can be found by subtracting the known counts from its row or column total.
| Group | Bus | Walk | Total |
|---|---|---|---|
| Junior | 12 | 18 | 30 |
| Senior | 10 | 10 | 20 |
| Total | 22 | 28 | 50 |
Essential method
\(\text{row total}=\sum\text{entries in that row}\), \(\text{grand total}=\sum\text{row totals}=\sum\text{column totals}\).
Exam tip: The “both conditions” cell is not the same as a row or column total.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Read a joint frequency
In a table, 9 girls and 6 boys cycle to school. How many cyclists are there?
Add the cycling column: \(9+6=\boxed{15}\).
Worked example · Medium
Complete a row
A row contains 12 walking, 8 cycling and an unknown bus count, with row total 30.
Bus count \(=30-12-8=\boxed{10}\).
Worked example · Exam-style
Conditional comparison
Of 30 juniors, 12 take the bus; of 20 seniors, 10 take the bus. Compare bus-use percentages.
Juniors \(=12/30=\boxed{40\%}\); seniors \(=10/20=\boxed{50\%}\). Bus travel is more common among seniors as a percentage.
5
Subtopic 5
Averages and range
The mode is the most frequent value; the median is the middle value in numerical order. With an even number of observations, average the two central numbers. Data may be bimodal or have no mode.
The median is a measure of centre, while range is a measure of spread; they answer different questions.
Essential method
Ordered data with \(n\) values have middle position \((n+1)/2\) when \(n\) is odd. When \(n\) is even, average positions \(n/2\) and \(n/2+1\).
Exam tip: Sort the observations before taking a median, and distinguish repeated values from repeated positions.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Find the mode
Find the mode of 2, 4, 4, 5, 6, 4, 7.
4 occurs three times, more than any other value. \(\boxed{\text{Mode}=4}\).
Worked example · Medium
Even-number median
Find the median of 6, 1, 9, 3, 7, 5.
Order: 1,3,5,6,7,9. Middle pair 5 and 6, so median \(=(5+6)/2=\boxed{5.5}\).
Worked example · Exam-style
Two modes
For 3,3,5,5,6,8, find the median and mode(s).
Middle pair 5,5 gives median \(\boxed5\). Frequencies for 3 and 5 both equal 2, so modes are \(\boxed{3,5}\).
6
Subtopic 6
Mean and range
The mean uses every numerical observation: add the data values then divide by how many there are. The range is the difference between the largest and smallest values and measures overall spread.
Essential method
\(\bar x=\dfrac{\sum x}{n}\), and \(\text{range}=\max(x)-\min(x)\).
Exam tip: A single extreme value can greatly change the mean and range.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Simple mean
Find the mean of 4, 6, 8.
Sum \(=18\); count \(=3\); mean \(18/3=\boxed6\).
Worked example · Medium
Mean and range
Find the mean and range of 3, 7, 7, 11.
Mean \(=(3+7+7+11)/4=28/4=\boxed7\). Range \(=11-3=\boxed8\).
Worked example · Exam-style
Missing observation
Five observations have mean 12. Four are 8, 11, 13 and 14. Find the fifth.
Total \(=5\times12=60\). Known total \(46\); missing \(=60-46=\boxed{14}\).
7
Subtopic 7
Which average to use when comparing data
The mean uses every value but is sensitive to extremes. The median gives a typical middle value and is often better for skewed salaries or house prices. The mode identifies the most common value, useful for stock sizes or categorical data.
To compare groups, explain both typical size (an average) and consistency (for example, range), in the real-world context.
Essential method
Smaller range often indicates less overall spread. When comparing data sets, choose comparable measurements and sample sizes and justify your preferred statistic.
Exam tip: “Higher mean” alone is not enough: say what the higher mean means in the story.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Mean changed by outlier
A group earns $20k, $22k, $24k, $26k and $100k. Compare mean and median.
Mean \(=192/5=\boxed{38.4\text{k}}\). Median \(=\boxed{24\text{k}}\). The unusual $100k inflates the mean.
Worked example · Medium
Consistency
Team A scores 8,9,10,11; team B scores 4,8,12,14. Compare ranges.
A range \(=11-8=\boxed3\); B range \(=14-4=\boxed{10}\). Team A is more consistent by range.
Worked example · Exam-style
Choosing shop stock
A shop sold sizes 6,7,7,7,8,8,9. Which statistic helps order the most popular size?
The mode \(=\boxed7\). It gives the most frequent size and is most directly useful for stocking decisions.
8
Subtopic 8
Working with larger data sets
In a frequency table, each value \(x\) is repeated \(f\) times. Multiply \(x\) by its frequency, sum these products and divide by total frequency to find the mean. Use cumulative frequency to locate the middle observation(s). The modal value is the one with greatest frequency.
| Value \(x\) | Frequency \(f\) | \(fx\) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 3 | 6 |
| 3 | 1 | 3 |
| Total | 6 | 11 |
Essential method
\(\bar x=\frac{\sum fx}{\sum f}\). For the median, identify the \(\tfrac{n+1}{2}\)th value (odd \(n\)) or the two central values (even \(n\)).
Exam tip: Do not divide by the number of table rows; divide by the sum of frequencies.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Frequency mean
Values 1,2,3 have respective frequencies 2,3,1. Find the mean.
\(\sum fx=1(2)+2(3)+3(1)=11\), \(\sum f=6\). Mean \(=\boxed{11/6\approx1.83}\).
Worked example · Medium
Mode and median
Values 0,1,2,3 have frequencies 1,4,3,2. Find mode and median.
Mode 1 has greatest frequency 4. Total 10; fifth value is 1 and sixth is 2, so median \(=(1+2)/2=\boxed{1.5}\).
Worked example · Exam-style
Complete a frequency table
Marks 1,2,3 have frequencies 4,\(k\),2 and total 15. Find \(k\) and the mean.
\(k=15-4-2=9\). Weighted sum \(=4+18+6=28\). Mean \(=\boxed{28/15\approx1.87}\).
9
Subtopic 9
Working with grouped and continuous data
Discrete data is counted; continuous data is measured. Continuous classes need boundaries without gaps, such as \(10\le h<20\), \(20\le h<30\). When observations are grouped, their original values are unknown, so the mean is an estimate using class midpoints.
The modal class has the highest frequency (for equal-width classes), and the median class contains the middle observation by cumulative frequency. Grouping loses precision.
| Class | Midpoint \(m\) | \(f\) | \(fm\) |
|---|---|---|---|
| 0–10 | 5 | 3 | 15 |
| 10–20 | 15 | 5 | 75 |
| Total | 8 | 90 |
Essential method
\(\text{midpoint}=\frac{\text{lower boundary}+\text{upper boundary}}2\); \(\bar x\approx\frac{\sum f(\text{midpoint})}{\sum f}\).
Exam tip: The grouped mean is estimated, not exact. For unequal class widths, compare frequency density rather than raw frequency to identify the most concentrated interval.
Worked examples · Easy → Medium → Exam-style
Every solution explains the mathematical method and gives the result.
Worked example · Easy
Class midpoint
Find the midpoint of the class \(20\le x<30\).
\((20+30)/2=\boxed{25}\).
Worked example · Medium
Estimated mean
Classes 0–10 and 10–20 have frequencies 3 and 5. Estimate the mean.
Midpoints 5 and 15; \(\sum fm=3(5)+5(15)=90\), total 8; mean \(\approx\boxed{11.25}\).
Worked example · Exam-style
Grouped distribution
Heights (cm) 140–150, 150–160, 160–170 have frequencies 2,5,3. Find modal class and estimated mean.
Modal class \(\boxed{150\!–\!160}\) (frequency 5). Midpoints 145,155,165: \((2(145)+5(155)+3(165))/10=1560/10=\boxed{156\text{ cm}}\).
Chapter 45 — Student exercises
Questions are arranged by textbook subtopic. Within each section they start with confidence-building tasks and finish with extended examination-style reasoning. Try each question before opening its solution.
Collecting and grouping data
1Question 1Foundation
Classify shoe colour and shoe size (measured as numbered categories).
Show Answer
Step-by-step solution: Colour is categorical; numbered sizes are numerical discrete labels for an ordered set of categories.
2Question 2Foundation
The values are 1,2,2,3,2,1,4. What is the frequency of 2?
Show Answer
Step-by-step solution: Count three occurrences of 2. \(\boxed{3}\).
3Question 3Intermediate
Ten responses have frequencies 2, 5 and 3. Show the total sample size.
Show Answer
Step-by-step solution: \(2+5+3=\boxed{10}\).
4Question 4Exam-style
The measurements are 11,15,21,24,29,30,36. Group them into 10–19, 20–29 and 30–39.
Show Answer
Step-by-step solution: Frequencies are \(\boxed{2,3,2}\); totals \(2+3+2=7\).
Surveys
5Question 5Foundation
What is the population in a survey of all 800 pupils at a school?
Show Answer
Step-by-step solution: The population is \(\boxed{800\text{ pupils}}\); a sample would contain fewer selected pupils.
6Question 6Foundation
Why is asking only members of a gym about exercise habits biased for a whole-town survey?
Show Answer
Step-by-step solution: Gym members may be more active than residents generally; many residents have no chance to respond.
7Question 7Intermediate
A college has 150 first-year and 100 second-year students. Choose a proportional sample of 25.
Show Answer
Step-by-step solution: First-years \((150/250)25=\boxed{15}\); second-years \((100/250)25=\boxed{10}\).
8Question 8Exam-style
A survey has 1000 respondents, all recruited on a gaming forum. Explain why it may not represent all adults.
Show Answer
Step-by-step solution: The sample is large but the recruitment excludes many nongamers. Thus selection bias remains; sample size does not remove it.
Designing a questionnaire
9Question 9Foundation
Why should a question specify “per week” when asking about hours worked?
Show Answer
Step-by-step solution: Without a time period, responses may mean hours per day, week or month and are not comparable.
10Question 10Foundation
Is “Are you happy with the wonderful new library?” a neutral question?
Show Answer
Step-by-step solution: No. “Wonderful” is leading. Ask “How satisfied are you with the new library?”
11Question 11Intermediate
Create nonoverlapping age classes for ages 0–29 in three equal-width groups.
Show Answer
Step-by-step solution: Use \(\boxed{0\!–\!9,10\!–\!19,20\!–\!29}\) years.
12Question 12Exam-style
A survey asks for exact salary but offers only $0–$999 and $1000–$1999. Give two improvements.
Show Answer
Step-by-step solution: Add further salary categories or “$2000 or more,” and clarify whether the salary is weekly, monthly or annual.
Two-way tables
13Question 13Foundation
A row contains values 4, 7 and 9. Find its total.
Show Answer
Step-by-step solution: \(4+7+9=\boxed{20}\).
14Question 14Foundation
A column total is 25. Two cells contain 8 and 10. Find the missing cell.
Show Answer
Step-by-step solution: \(25-8-10=\boxed7\).
15Question 15Intermediate
A two-way table has row totals 16, 18 and 11. Find the grand total.
Show Answer
Step-by-step solution: \(16+18+11=\boxed{45}\).
16Question 16Exam-style
A club has 28 adults and 12 children. There are 18 female adults and 7 female children. Find the number of males.
Show Answer
Step-by-step solution: Female total \(=18+7=25\). Grand total \(=28+12=40\). Males \(=40-25=\boxed{15}\).
Averages and range
17Question 17Foundation
Find the mode of 1,2,2,2,4.
Show Answer
Step-by-step solution: 2 occurs three times; \(\boxed2\).
18Question 18Foundation
Find the median of 8,1,3,6,4.
Show Answer
Step-by-step solution: Ordered: 1,3,4,6,8. Middle value \(=\boxed4\).
19Question 19Intermediate
Find the median of 2,9,5,8.
Show Answer
Step-by-step solution: Ordered 2,5,8,9; middle average \((5+8)/2=\boxed{6.5}\).
20Question 20Exam-style
Data are 4,4,6,6,8,10. State the modes and median.
Show Answer
Step-by-step solution: Modes \(\boxed{4\text{ and }6}\). Ordered middle pair 6,6 gives median \(\boxed6\).
Mean and range
21Question 21Foundation
Find the mean of 2,4,6,8.
Show Answer
Step-by-step solution: Sum \(=20\), count 4; \(\boxed5\).
22Question 22Foundation
Find the range of 6,11,2,15,9.
Show Answer
Step-by-step solution: Largest 15, smallest 2; \(15-2=\boxed{13}\).
23Question 23Intermediate
Five numbers sum to 90. Find their mean.
Show Answer
Step-by-step solution: \(90/5=\boxed{18}\).
24Question 24Exam-style
Four scores have a mean of 7. Three are 3,6 and 8. Find the last score.
Show Answer
Step-by-step solution: Required total \(=4(7)=28\). Known total \(=17\). Missing \(=28-17=\boxed{11}\).
Which average to use when comparing data
25Question 25Foundation
Which average best describes the most common shoe size: mean, median or mode?
Show Answer
Step-by-step solution: \(\boxed{\text{Mode}}\), because it identifies the size bought most frequently.
26Question 26Foundation
Which measure is less distorted by one exceptionally high salary?
Show Answer
Step-by-step solution: \(\boxed{\text{Median}}\), because it depends on ordered positions rather than the size of the extreme value.
27Question 27Intermediate
Class A has scores 4,5,6; Class B 1,5,9. Compare consistency by range.
Show Answer
Step-by-step solution: A range \(=2\); B range \(=8\). A is more consistent.
28Question 28Exam-style
Shop A daily sales are 10,11,12,13; shop B sales 7,10,13,16. Compare mean and range.
Show Answer
Step-by-step solution: Both means \(=11.5\). A range \(=3\), B range \(=9\). Typical sales are equal, but A is more consistent.
Working with larger data sets
29Question 29Foundation
Values 1 and 3 have frequencies 2 and 4. Find total frequency.
Show Answer
Step-by-step solution: \(2+4=\boxed6\).
30Question 30Foundation
Values 0,1,2 have frequencies 2,3,1. Find modal value.
Show Answer
Step-by-step solution: Frequency 3 is highest, corresponding to \(\boxed1\).
31Question 31Intermediate
Values 2,4,6 have frequencies 1,2,1. Find mean.
Show Answer
Step-by-step solution: Weighted sum \(2+8+6=16\), total 4; mean \(=\boxed4\).
32Question 32Exam-style
Scores 0,1,2,3 have frequencies 2,5,4,1. Find the median.
Show Answer
Step-by-step solution: Total \(n=12\); positions 6 and 7 are both score 1 (positions 3–7). Median \(=\boxed1\).
Working with grouped and continuous data
33Question 33Foundation
Find the midpoint of the interval 40–60.
Show Answer
Step-by-step solution: \((40+60)/2=\boxed{50}\).
34Question 34Foundation
The non-overlapping classes 0≤x<5, 5≤x<10 and 10≤x<15 have frequencies 2, 6 and 3. Find the modal class.
Show Answer
Step-by-step solution: The largest frequency is 6, so \(\boxed{5\!–\!10}\).
35Question 35Intermediate
Classes 0–10 and 10–20 have frequencies 4 and 6. Estimate the mean.
Show Answer
Step-by-step solution: Midpoints 5,15; weighted sum \(4(5)+6(15)=110\). Total 10; mean \(\approx\boxed{11}\).
36Question 36Exam-style
Classes 20–30, 30–40, 40–50 have frequencies 3,8,5. Identify the median class and estimate the mean.
Show Answer
Step-by-step solution: Total 16; middle positions 8 and 9 lie in 30–40 (cumulative 3,11,16). Midpoints 25,35,45: sum \(75+280+225=580\). Estimated mean \(580/16=\boxed{36.25}\).
37Question 37Exam-style challenge
Two schools survey the times taken by 20 students at each school to travel to school. The time classes (in minutes) are \(0\leq t<10\), \(10\leq t<20\) and \(20\leq t<30\). Their frequencies are School A: \(2,6,12\) and School B: \(4,12,4\). Estimate the mean travel time for each school and the combined mean for all 40 students. Which school has the higher estimated mean?
Show Answer
Step 1 — Midpoints: \(5,15,25\) minutes.
Step 2 — Estimated mean for A: \(\frac{2(5)+6(15)+12(25)}{20}=\frac{400}{20}=\boxed{20\text{ min}}\).
Step 3 — Estimated mean for B: \(\frac{4(5)+12(15)+4(25)}{20}=\frac{300}{20}=\boxed{15\text{ min}}\).
Step 4 — Combined estimated mean: \(\frac{400+300}{40}=\boxed{17.5\text{ min}}\).
Interpretation: School A has the higher estimated mean by \(\boxed{5\text{ min}}\). These are estimates because class midpoints replace the individual journey times.
High-grade revision checklist
- Can I solve unfamiliar questions using Collecting and grouping data and explain my method?
- Can I solve unfamiliar questions using Surveys and explain my method?
- Can I solve unfamiliar questions using Designing a questionnaire and explain my method?
- Can I solve unfamiliar questions using Two-way tables and explain my method?
- Can I solve unfamiliar questions using Averages and range and explain my method?
- Can I solve unfamiliar questions using Mean and range and explain my method?
- Can I solve unfamiliar questions using Which average to use when comparing data and explain my method?
- Can I solve unfamiliar questions using Working with larger data sets and explain my method?
- Can I solve unfamiliar questions using Working with grouped and continuous data and explain my method?
Before the exam, practise without looking at the answers, check units and rounding, and write geometric or statistical reasons where they are needed.