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Problem-solving Pack
MathematicsYear 8 · Statistics
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 48
Reaction-distance dataset (CdN Numberphile, May 2026). A class of 20 Year 8 students measured their reaction distance (cm):
15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28.
(a) Find the mean.
(b) Find the median.
(c) Find the mode.
(d) Find the range.
(e) Identify any value that may be an outlier and justify.
15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28.
(a) Find the mean.
(b) Find the median.
(c) Find the mode.
(d) Find the range.
(e) Identify any value that may be an outlier and justify.
Working space
2Problem 2 of 48
Effect of outlier on mean vs median. Continuing the reaction-distance dataset: remove the largest value (28) and recompute the mean and median.
(a) Find the new mean.
(b) Find the new median.
(c) Compare the shifts. Which measure is more robust?
(a) Find the new mean.
(b) Find the new median.
(c) Compare the shifts. Which measure is more robust?
Working space
3Problem 3 of 48
Mean with an unknown value. A class of 24 students has a mean test mark of 62. A new student joins and the class mean is now 63.
(a) Find the new student's mark.
(b) The teacher discovers one mark was misread: 78 should have been 87. Find the corrected mean.
(a) Find the new student's mark.
(b) The teacher discovers one mark was misread: 78 should have been 87. Find the corrected mean.
Working space
4Problem 4 of 48
Comparing two classes (means + ranges). Class A reaction distances had mean 15 cm and range 18 cm. Class B had mean 17 cm and range 10 cm.
(a) Which class had a faster (lower) average reaction?
(b) Which class was more consistent (smaller range)?
(c) Write a one-sentence comparison.
(a) Which class had a faster (lower) average reaction?
(b) Which class was more consistent (smaller range)?
(c) Write a one-sentence comparison.
Working space
5Problem 5 of 48
Frequency-table mean. A frequency table records the number of pets per student in a class:
| Pets | 0 | 1 | 2 | 3 | 4 |
|------|---|---|---|---|---|
| Frequency | 5 | 8 | 4 | 2 | 1 |
(a) Find the total number of students.
(b) Find the mean number of pets per student.
(c) Find the median and modal number of pets.
| Pets | 0 | 1 | 2 | 3 | 4 |
|------|---|---|---|---|---|
| Frequency | 5 | 8 | 4 | 2 | 1 |
(a) Find the total number of students.
(b) Find the mean number of pets per student.
(c) Find the median and modal number of pets.
Working space
6Problem 6 of 48
Compute and compare measures. Five students' weekly hours of homework: 8, 12, 6, 14, 10.
(a) Find mean, median, mode, range.
(b) An additional value 30 is added. Find the new mean, median and range.
(c) Which measure changes the most?
(a) Find mean, median, mode, range.
(b) An additional value 30 is added. Find the new mean, median and range.
(c) Which measure changes the most?
Working space
7Problem 7 of 48
Find a missing value with given mean. A set of 5 numbers is 4, 9, , 12, 16. The mean is 10.
(a) Find .
(b) The number 16 is removed. Find the new mean.
(a) Find .
(b) The number 16 is removed. Find the new mean.
Working space
8Problem 8 of 48
Median with even count. A set of 8 numbers, sorted, is: 5, 7, 9, 11, 12, 13, 16, 20.
(a) Find the mean.
(b) Find the median.
(c) Find the range.
(d) Identify any outlier and justify.
(a) Find the mean.
(b) Find the median.
(c) Find the range.
(d) Identify any outlier and justify.
Working space
9Problem 9 of 48
Combining two classes. Class A has 12 students with mean 14. Class B has 18 students with mean 16.
(a) Find the combined sum.
(b) Find the combined mean.
(c) Is the combined median equal to the average of the two class medians? Explain.
(a) Find the combined sum.
(b) Find the combined mean.
(c) Is the combined median equal to the average of the two class medians? Explain.
Working space
10Problem 10 of 48
Mean and median together. Five integers have sum 60 and median 12.
(a) Suggest one set of values.
(b) Find a set where the mean equals the median.
(c) Find a set where the mean is much larger than the median, and identify why.
(a) Suggest one set of values.
(b) Find a set where the mean equals the median.
(c) Find a set where the mean is much larger than the median, and identify why.
Working space
11Problem 11 of 48
Modelling with tallies. A teacher tallies the number of pets per student over 30 students:
0: //// //// /
1: //// ////
2: ////
3: /
(a) Construct a frequency table.
(b) Find the mean, median, and mode.
(c) State the percentage of students with at least 2 pets.
0: //// //// /
1: //// ////
2: ////
3: /
(a) Construct a frequency table.
(b) Find the mean, median, and mode.
(c) State the percentage of students with at least 2 pets.
Working space
12Problem 12 of 48
Cleaning data. A pupil records the heights (cm) of 10 students: 150, 152, 148, 155, 1500, 153, 149, 151, 154, 150. One value is clearly a typo.
(a) Identify the typo.
(b) Compute the (a) mean with the typo and (b) without.
(c) Compute the median in both cases.
(d) Which measure was the typo more disruptive to?
(a) Identify the typo.
(b) Compute the (a) mean with the typo and (b) without.
(c) Compute the median in both cases.
(d) Which measure was the typo more disruptive to?
Working space
13Problem 13 of 48
Stratified sampling. A school has 200 pupils in Year 7, 180 in Year 8, and 220 in Year 9. A survey of 90 pupils is to be taken using stratified random sampling.
(a) How many from each year?
(b) Compare with simple random sampling of 90 from the whole school. Which is more likely to over- or under-represent a year?
(c) Suppose 20 new Year 7 pupils are added and the sample size stays 90. How does the Year-7 sample size change?
(a) How many from each year?
(b) Compare with simple random sampling of 90 from the whole school. Which is more likely to over- or under-represent a year?
(c) Suppose 20 new Year 7 pupils are added and the sample size stays 90. How does the Year-7 sample size change?
Working space
14Problem 14 of 48
Sampling bias scenarios. For each scenario, state whether the sample is likely biased and explain.
(a) Surveying mass transport use only by interviewing people on a tram.
(b) Estimating average height from a basketball team.
(c) Phone polling people about TV viewing habits at 9 am on a weekday.
(a) Surveying mass transport use only by interviewing people on a tram.
(b) Estimating average height from a basketball team.
(c) Phone polling people about TV viewing habits at 9 am on a weekday.
Working space
15Problem 15 of 48
Predicting from a sample. A sample of 25 Year 8 pupils contains 3 left-handers. The Y8 cohort has 400 pupils.
(a) Estimate the number of left-handed Y8 pupils.
(b) Identify one assumption being made.
(c) The school's overall stat is "about 10% of pupils are left-handed". Does your estimate agree? Discuss.
(a) Estimate the number of left-handed Y8 pupils.
(b) Identify one assumption being made.
(c) The school's overall stat is "about 10% of pupils are left-handed". Does your estimate agree? Discuss.
Working space
16Problem 16 of 48
Designing a survey. You want to know how many Y8 pupils enjoy maths. Design a survey:
(a) Write three good survey questions.
(b) Specify a sample size and sampling method.
(c) State two ways to reduce bias.
(a) Write three good survey questions.
(b) Specify a sample size and sampling method.
(c) State two ways to reduce bias.
Working space
17Problem 17 of 48
Sample size and confidence. Two polls predict election support:
- Poll A: 60% support, sample size 100, MoE ±10%.
- Poll B: 60% support, sample size 1000, MoE ±3%.
(a) Which poll is more reliable? Why?
(b) For Poll A, the true support could be anywhere from ? to ?
(c) For Poll B, ditto.
(d) The candidate needs > 50% to win. Which poll is more conclusive?
- Poll A: 60% support, sample size 100, MoE ±10%.
- Poll B: 60% support, sample size 1000, MoE ±3%.
(a) Which poll is more reliable? Why?
(b) For Poll A, the true support could be anywhere from ? to ?
(c) For Poll B, ditto.
(d) The candidate needs > 50% to win. Which poll is more conclusive?
Working space
18Problem 18 of 48
Stratified vs simple random. A school has the following enrolment:
| Year | 7 | 8 | 9 | 10 | 11 |
|------|---|---|---|----|----|
| Count | 150 | 140 | 130 | 120 | 100 |
A sample of 64 is required, stratified by year.
(a) Compute the sample size for each year (round to nearest integer).
(b) The school decides to oversample Y11 to ensure ≥ 15 are in the sample. Revise.
| Year | 7 | 8 | 9 | 10 | 11 |
|------|---|---|---|----|----|
| Count | 150 | 140 | 130 | 120 | 100 |
A sample of 64 is required, stratified by year.
(a) Compute the sample size for each year (round to nearest integer).
(b) The school decides to oversample Y11 to ensure ≥ 15 are in the sample. Revise.
Working space
19Problem 19 of 48
Bias detection. Identify the type of bias in each scenario and suggest a fix.
(a) Asking "Don't you agree that the maths curriculum is great?"
(b) Calling phone numbers only during work hours.
(c) Surveying only students at the after-school club about extracurriculars.
(a) Asking "Don't you agree that the maths curriculum is great?"
(b) Calling phone numbers only during work hours.
(c) Surveying only students at the after-school club about extracurriculars.
Working space
20Problem 20 of 48
Cleaning data. A spreadsheet of student heights (cm) includes: 152, 148, 1500, NA, 153, 155, "?", 149, 151, 1.6.
(a) Identify problematic entries.
(b) Suggest a cleaning plan.
(c) Compute the mean before and after cleaning.
(a) Identify problematic entries.
(b) Suggest a cleaning plan.
(c) Compute the mean before and after cleaning.
Working space
21Problem 21 of 48
Sample-size planning. A school of 800 students wants to estimate the percentage who walk to school, to within ±5%.
(a) Estimate the required sample size (rule of thumb: where is the desired margin in decimal form).
(b) Compare with sampling 30 students. What is the approximate margin of error?
(a) Estimate the required sample size (rule of thumb: where is the desired margin in decimal form).
(b) Compare with sampling 30 students. What is the approximate margin of error?
Working space
22Problem 22 of 48
Non-response bias. A survey is sent to 500 students, but only 200 respond.
(a) Compute the response rate.
(b) Why is non-response a problem for estimates?
(c) Suggest two ways to reduce non-response.
(a) Compute the response rate.
(b) Why is non-response a problem for estimates?
(c) Suggest two ways to reduce non-response.
Working space
23Problem 23 of 48
Polling election support. A poll surveys 1000 voters: 540 support A, 460 support B. The election needs >50% to win.
(a) Find the sample percentages.
(b) State the margin of error (rule of thumb).
(c) Can the poll predict the winner with confidence?
(a) Find the sample percentages.
(b) State the margin of error (rule of thumb).
(c) Can the poll predict the winner with confidence?
Working space
24Problem 24 of 48
Question design audit. A school survey asks: "How often do you study at home?"
(a) Suggest 5 mutually exclusive, exhaustive response options.
(b) Critique the question — what assumptions does it make?
(c) Rewrite the question to remove the assumption.
(a) Suggest 5 mutually exclusive, exhaustive response options.
(b) Critique the question — what assumptions does it make?
(c) Rewrite the question to remove the assumption.
Working space
25Problem 25 of 48
Designing a pie chart. A survey of 360 students on favourite music:
| Pop | Rock | Hip-Hop | Classical | Other |
|-----|------|---------|-----------|-------|
| 144 | 108 | 72 | 18 | 18 |
(a) Find the angle for each slice.
(b) Find the percentage for each.
(c) State the modal category.
| Pop | Rock | Hip-Hop | Classical | Other |
|-----|------|---------|-----------|-------|
| 144 | 108 | 72 | 18 | 18 |
(a) Find the angle for each slice.
(b) Find the percentage for each.
(c) State the modal category.
Working space
26Problem 26 of 48
Histogram with unequal widths. Data is grouped into:
- 0–4: 12 students
- 4–8: 20
- 8–16: 24
- 16–20: 4
(a) Find the frequency density for each class.
(b) Which class has the tallest bar?
(c) Class 8–16 has more students but a shorter bar — explain.
- 0–4: 12 students
- 4–8: 20
- 8–16: 24
- 16–20: 4
(a) Find the frequency density for each class.
(b) Which class has the tallest bar?
(c) Class 8–16 has more students but a shorter bar — explain.
Working space
27Problem 27 of 48
Bar vs pie chart comparison. Data on 200 students' favourite subjects:
| Maths | English | Science | History | Art |
|-------|---------|---------|---------|-----|
| 60 | 50 | 40 | 30 | 20 |
(a) Construct a bar chart description (bar heights).
(b) Construct a pie chart description (slice angles).
(c) Which is easier to read for comparing subjects?
(d) Which is easier to read for showing total proportions?
| Maths | English | Science | History | Art |
|-------|---------|---------|---------|-----|
| 60 | 50 | 40 | 30 | 20 |
(a) Construct a bar chart description (bar heights).
(b) Construct a pie chart description (slice angles).
(c) Which is easier to read for comparing subjects?
(d) Which is easier to read for showing total proportions?
Working space
28Problem 28 of 48
Line chart: rainfall over a year. Monthly rainfall (mm) in a town: 45, 38, 52, 60, 72, 80, 85, 90, 75, 65, 55, 48.
(a) Sketch a description of the line chart.
(b) Identify the wettest month.
(c) Find the mean monthly rainfall.
(d) Find the annual total.
(a) Sketch a description of the line chart.
(b) Identify the wettest month.
(c) Find the mean monthly rainfall.
(d) Find the annual total.
Working space
29Problem 29 of 48
Choosing the right chart. For each scenario, choose: (a) bar chart, (b) pie chart, (c) line chart, (d) histogram, (e) scatter plot.
(i) Daily temperature over a week.
(ii) Favourite ice-cream flavour of 50 students.
(iii) Heights of 100 students grouped into intervals.
(iv) Comparing sales in 3 different shops.
(v) Relationship between hours studied and exam mark.
(i) Daily temperature over a week.
(ii) Favourite ice-cream flavour of 50 students.
(iii) Heights of 100 students grouped into intervals.
(iv) Comparing sales in 3 different shops.
(v) Relationship between hours studied and exam mark.
Working space
30Problem 30 of 48
Misleading axes. A newspaper shows a bar chart of crime rates per 1000: 2024 = 28; 2025 = 27. The y-axis starts at 26.
(a) What does the chart visually suggest?
(b) Compute the actual percentage change.
(c) Draw the chart with a y-axis starting at 0 and describe the visual difference.
(d) When is a non-zero baseline justified?
(a) What does the chart visually suggest?
(b) Compute the actual percentage change.
(c) Draw the chart with a y-axis starting at 0 and describe the visual difference.
(d) When is a non-zero baseline justified?
Working space
31Problem 31 of 48
Grouped bar chart. A school records year-end average marks for two consecutive years across 3 subjects:
| Subject | 2024 | 2025 |
|---------|------|------|
| Maths | 65 | 70 |
| English | 72 | 70 |
| Science | 60 | 68 |
(a) Construct a grouped bar chart description.
(b) Find the percentage change for each subject.
(c) Find the school's overall average for each year.
| Subject | 2024 | 2025 |
|---------|------|------|
| Maths | 65 | 70 |
| English | 72 | 70 |
| Science | 60 | 68 |
(a) Construct a grouped bar chart description.
(b) Find the percentage change for each subject.
(c) Find the school's overall average for each year.
Working space
32Problem 32 of 48
Frequency table to histogram. Reaction distances (cm) for 20 students:
| Class | 8–11 | 12–15 | 16–19 | 20–23 | 24–27 | 28–31 |
|-------|------|-------|-------|-------|-------|-------|
| Frequency | 3 | 7 | 5 | 4 | 0 | 1 |
(a) State the class width (assume equal widths).
(b) State the modal class.
(c) Describe the histogram shape.
(d) Identify any outlier classes.
| Class | 8–11 | 12–15 | 16–19 | 20–23 | 24–27 | 28–31 |
|-------|------|-------|-------|-------|-------|-------|
| Frequency | 3 | 7 | 5 | 4 | 0 | 1 |
(a) State the class width (assume equal widths).
(b) State the modal class.
(c) Describe the histogram shape.
(d) Identify any outlier classes.
Working space
33Problem 33 of 48
Pie chart calculation. A pie chart shows the budget of a school: Teachers 60%, Buildings 20%, Supplies 10%, Other 10%. Total annual budget is £600 000.
(a) Find the £ allocated to each category.
(b) Find the angle of each slice.
(c) Compute the percentage of the budget not used for teachers.
(a) Find the £ allocated to each category.
(b) Find the angle of each slice.
(c) Compute the percentage of the budget not used for teachers.
Working space
34Problem 34 of 48
Constructing a histogram. Heights of 50 plants (cm) are grouped:
| Class | Freq |
|-------|------|
| 5–10 | 4 |
| 10–15 | 8 |
| 15–20 | 18 |
| 20–25 | 12 |
| 25–30 | 8 |
(a) State the class width.
(b) Compute the frequency densities.
(c) State the modal class.
(d) Estimate the mean height using midpoints.
| Class | Freq |
|-------|------|
| 5–10 | 4 |
| 10–15 | 8 |
| 15–20 | 18 |
| 20–25 | 12 |
| 25–30 | 8 |
(a) State the class width.
(b) Compute the frequency densities.
(c) State the modal class.
(d) Estimate the mean height using midpoints.
Working space
35Problem 35 of 48
Comparing line charts. Two stocks' monthly prices (chf):
| Month | A | B |
|-------|---|---|
| Jan | 100 | 100 |
| Feb | 110 | 95 |
| Mar | 105 | 90 |
| Apr | 115 | 85 |
| May | 120 | 80 |
(a) Identify the overall trend for each.
(b) Find the percentage change for each.
(c) Which would you invest in?
| Month | A | B |
|-------|---|---|
| Jan | 100 | 100 |
| Feb | 110 | 95 |
| Mar | 105 | 90 |
| Apr | 115 | 85 |
| May | 120 | 80 |
(a) Identify the overall trend for each.
(b) Find the percentage change for each.
(c) Which would you invest in?
Working space
36Problem 36 of 48
Visual literacy. Identify the issue in each chart description.
(a) A pie chart with 15 slices, including 8 slices each of "Other".
(b) A bar chart with no y-axis label.
(c) A line chart connecting non-time-series data (favourite colour vs frequency).
(d) A histogram with overlapping class intervals.
(a) A pie chart with 15 slices, including 8 slices each of "Other".
(b) A bar chart with no y-axis label.
(c) A line chart connecting non-time-series data (favourite colour vs frequency).
(d) A histogram with overlapping class intervals.
Working space
37Problem 37 of 48
Reaction-distance dataset (CdN Numberphile). A class of 20 Year 8 students measured reaction distance (cm):
10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28.
(a) Find the five-number summary: min, Q1, median, Q3, max.
(b) Find the IQR.
(c) Sketch a boxplot.
(d) Using the 1.5×IQR rule, identify outliers.
10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28.
(a) Find the five-number summary: min, Q1, median, Q3, max.
(b) Find the IQR.
(c) Sketch a boxplot.
(d) Using the 1.5×IQR rule, identify outliers.
Working space
38Problem 38 of 48
Outlier sensitivity. The dataset has mean 16.25 and median 15.5.
(a) Remove the largest value (28). Find the new mean and median.
(b) Compare the changes.
(c) Which measure is more robust to outliers? Explain.
(a) Remove the largest value (28). Find the new mean and median.
(b) Compare the changes.
(c) Which measure is more robust to outliers? Explain.
Working space
39Problem 39 of 48
Comparing two boxplots. Class A and Class B reaction distances:
- Class A: min 8, Q1 12, median 15, Q3 18, max 25.
- Class B: min 10, Q1 13, median 16, Q3 22, max 30.
(a) Which class has a higher median?
(b) Which class is more consistent (smaller IQR)?
(c) Describe the skew of each.
(d) Make a one-sentence comparison.
- Class A: min 8, Q1 12, median 15, Q3 18, max 25.
- Class B: min 10, Q1 13, median 16, Q3 22, max 30.
(a) Which class has a higher median?
(b) Which class is more consistent (smaller IQR)?
(c) Describe the skew of each.
(d) Make a one-sentence comparison.
Working space
40Problem 40 of 48
Combining two classes. Class A: 20 students, mean 16.25. Class B: 30 students, mean 14.50.
(a) Find the combined mean.
(b) Why is this NOT the average of the two means?
(a) Find the combined mean.
(b) Why is this NOT the average of the two means?
Working space
41Problem 41 of 48
Skew investigation. A distribution of salaries in a small company has mean £45 000 and median £32 000.
(a) Identify the skew direction.
(b) What does this tell you about the salary distribution?
(c) The CEO's salary is £200 000. Is this an outlier? Justify.
(d) Which measure of central tendency is more representative of a typical employee?
(a) Identify the skew direction.
(b) What does this tell you about the salary distribution?
(c) The CEO's salary is £200 000. Is this an outlier? Justify.
(d) Which measure of central tendency is more representative of a typical employee?
Working space
42Problem 42 of 48
Sampling-based inference. A sample of 30 students has mean 75 and SD 10. The school of 600 students is the population.
(a) Estimate the population mean.
(b) Comment on the reliability with .
(c) Estimate the percentage of students within ±10 marks of 75 (one SD).
(d) Estimate the percentage between 65 and 85.
(a) Estimate the population mean.
(b) Comment on the reliability with .
(c) Estimate the percentage of students within ±10 marks of 75 (one SD).
(d) Estimate the percentage between 65 and 85.
Working space
43Problem 43 of 48
Inference from a small sample. A poll of 50 people finds 30 prefer brand A. Estimate the percentage preference in the population, and state your confidence.
Working space
44Problem 44 of 48
Comparing two data sets. Class A and Class B test scores:
A: 60, 70, 75, 80, 85.
B: 50, 60, 75, 80, 100.
(a) Find mean, median, range for each.
(b) Which has higher mean? Higher median?
(c) Which has more spread?
(d) Comment on skew.
A: 60, 70, 75, 80, 85.
B: 50, 60, 75, 80, 100.
(a) Find mean, median, range for each.
(b) Which has higher mean? Higher median?
(c) Which has more spread?
(d) Comment on skew.
Working space
45Problem 45 of 48
Inference validity. A school claims its students have an average grade of 75. To verify:
(a) Suggest two ways the school could test this claim.
(b) State two issues that might cause the published average to be unreliable.
(c) If sampling 50 students, what would be an acceptable margin of error?
(a) Suggest two ways the school could test this claim.
(b) State two issues that might cause the published average to be unreliable.
(c) If sampling 50 students, what would be an acceptable margin of error?
Working space
46Problem 46 of 48
MAD calculation. Compute the Mean Absolute Deviation (MAD) of the data set 5, 8, 10, 12, 15.
(a) Find the mean.
(b) Find absolute deviations.
(c) Find the MAD.
(d) Interpret what MAD tells us.
(a) Find the mean.
(b) Find absolute deviations.
(c) Find the MAD.
(d) Interpret what MAD tells us.
Working space
47Problem 47 of 48
Box-and-whisker investigation. 30 students' weekly reading hours have 5-number summary: 1, 3, 6, 10, 25.
(a) Find IQR.
(b) Identify any outliers using the 1.5×IQR rule.
(c) Sketch the boxplot and describe the skew.
(d) What might the value 25 represent?
(a) Find IQR.
(b) Identify any outliers using the 1.5×IQR rule.
(c) Sketch the boxplot and describe the skew.
(d) What might the value 25 represent?
Working space
48Problem 48 of 48
Validating an inference. A teacher claims: "students who attend tutorials get higher marks". To support this:
(a) Suggest a data-collection method.
(b) State what statistical measures would help test the claim.
(c) List two confounding factors that could mislead the conclusion.
(d) Describe how a control group would help.
(a) Suggest a data-collection method.
(b) State what statistical measures would help test the claim.
(c) List two confounding factors that could mislead the conclusion.
(d) Describe how a control group would help.
Working space