Pack B — Fluency

Mathematics

Year 8 · Statistics

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–40

1.
Find the mean of: 12, 10, 14, 18, 16. Show the sum and the division.
2.
Find the median of: 22, 19, 15, 18, 21.
3.
Find the mode of: 14, 11, 14, 17, 11, 14, 19.
4.
Find the range of: 8, 13, 6, 17, 11, 20.
5.
A tally chart shows: ★★★★ ★★★ for favourite colour Red (7 students). What is the frequency for Red?
6.
In a class of 30, the frequency table shows: Football 12, Basketball 8, Tennis 5, Other 5. What fraction play Football?
7.
For the data 5, 8, 10, 7, 8, find: (a) mean, (b) median, (c) mode.
8.
A data set has range 12 and minimum value 5. Find the maximum.
9.
For the data 2, 4, 4, 5, 7, 30, identify a potential outlier and justify.
10.
A frequency table: x-value 1, 2, 3, 4; frequency 4, 3, 2, 1. Find the total count.
11.
Distinguish: a (a) survey, (b) census, (c) population, (d) sample. Give one-sentence definitions.
12.
A school has 600 students. A class of 30 is surveyed. Identify the population and the sample.
13.
A researcher gives each student a number and uses a random-number generator to pick 30. What sampling method is this?
14.
A sample of 40 out of 500 pupils is taken. What percentage is the sample of the population?
15.
In a sample of 40 students, 5 were left-handed. Estimate the number of left-handed students in a school of 600.
16.
A survey question reads "Do you agree that the canteen serves poor food?" Why is this biased?
17.
A survey of 100 pet owners finds 65% have dogs. Estimate how many dog owners there are in a community of 800 pet owners.
18.
A teacher surveys only students who attend the library to ask about reading habits. State one source of bias.
19.
In a sample of 50 cars, 12 are red. Estimate the percentage of red cars in the population.
20.
A teacher takes the first 30 students arriving at school as a sample. Identify a potential bias.
21.
Which type of chart is best for showing how a whole is divided into parts (e.g. how the school budget is split)?
22.
Which chart is most appropriate for comparing the heights of 5 different buildings?
23.
Which chart is most appropriate for showing how a temperature changes over a day?
24.
Which chart is best for continuous numerical data (e.g. heights of 100 pupils grouped into intervals)?
25.
In a pie chart of 200 students, 80 chose Football. What angle does Football occupy?
26.
In a survey of 200 students, 56 preferred Pasta. What angle on a pie chart represents Pasta?
27.
In a survey of 200 students, 56 preferred Pasta. What percentage preferred Pasta?
28.
On a bar chart of favourite fruits, the Apple bar is 12 units tall, representing 24 students. What does 1 unit represent?
29.
A line chart shows temperature: 9 am = 12°C, 12 pm = 18°C, 3 pm = 22°C, 6 pm = 15°C. What was the peak temperature?
30.
Find the midpoint of the class interval 16–19 cm.
31.
Define "spread" in statistics in one sentence.
32.
For the data 4, 6, 8, 9, 12, find the (a) mean, (b) range.
33.
Describe the symmetry of a data set whose mean = median.
34.
Describe the skew of a distribution where mean > median.
35.
Describe the skew of a distribution where mean < median.
36.
Identify the outlier in the data 5, 6, 7, 7, 8, 9, 50.
37.
A dataset has mean 10. Describe what each tail represents.
38.
A uniform distribution has all values occurring at the same frequency. Sketch what this looks like.
39.
For the data 3, 4, 4, 5, 5, 5, 6, 7, identify the symmetry.
40.
For the data 2, 3, 5, 6, 9, find the mean absolute deviation (MAD).

SilverQuestions 41–80

41.
Find the median of: 20, 17, 14, 11, 19, 16.
42.
The mean of 8, 11, 9, 14 and $x$ is 11. Find $x$ .
43.
For the data set 4, 7, 9, 12, 18, find the mean and median.
44.
Find the mean of the frequency table: value 1, 2, 3, 4; freq 5, 3, 2, 2.
45.
A data set has mean 20. If a new value of 50 is added, what is the new mean? (The set had 9 values originally.)
46.
Find the mean, median and mode of: 7, 8, 5, 7, 9, 7, 11, 4.
47.
A class's test marks are 6, 8, 7, 9, 5, 10, 7, 8, 6, 4. (a) Find the mean. (b) Find the median.
48.
A class records the number of pets per student: 0, 0, 0, 1, 1, 1, 1, 2, 2, 3. Construct a frequency table and find the mode.
49.
The range of a set of 6 numbers is 20 and the smallest is 5. The largest is then replaced by 40. Find the new range.
50.
Identify whether each measure (mean, median, mode) is affected by an outlier of 100 added to the data {5, 8, 10, 12}.
51.
A school has Year 7: 200, Year 8: 180, Year 9: 220. A stratified sample of 60 is to be taken. How many from each year?
52.
A poll of 200 voters finds 110 support candidate A. Estimate the percentage of supporters in the wider population, and comment on how confident this estimate is.
53.
A researcher wants to learn about Year 8 lunch preferences. Suggest a sampling plan that avoids bias.
54.
Identify whether each scenario is a census, a sample or biased: (a) UK government population census every 10 years. (b) Surveying every 10th student exiting the school. (c) Asking the football team about favourite sport.
55.
A class of 30 students is surveyed about study time. The mean is 7.4 hours/week. Is this representative of all Y8 students? Suggest two reasons it might not be.
56.
Improve this question: "How often do you eat the disgusting cafeteria food?" Provide a fairer version.
57.
A sample of 40 students has 18 left-handed. The expected proportion is 10%. Comment on whether the sample is consistent with this.
58.
A teacher conducts a survey on a school trip. List two reasons why this sample is unlikely to represent the whole school.
59.
Identify the most representative sampling method for a school survey: (a) volunteers in the library; (b) random class draw; (c) every 10th student exiting; (d) stratified across year-groups.
60.
A data set has the values 152, 148, 1500, 155 (heights in cm). Clean it and explain.
61.
A frequency table of grouped data: 0–5 → 4; 5–10 → 7; 10–15 → 6; 15–20 → 3. Find the total students.
62.
In a pie chart of 30 students, the angles for "Liked Maths" (yes/no) are 168° (yes) and 192° (no). How many said yes?
63.
A class interval has lower bound 12 and upper bound 15, frequency 7. Find the (a) class width and (b) frequency density.
64.
A bar chart of favourite colour shows: Red 12, Blue 18, Green 8. Convert to percentages of the total.
65.
On a histogram with class widths 4, the frequencies are 12, 20, 24, 4 for classes 0-4, 4-8, 8-12, 12-16. Find the frequency densities.
66.
A line graph shows monthly rainfall (mm): Jan 50, Feb 40, Mar 60, Apr 90. Find the (a) total, (b) mean rainfall, (c) month with most rain.
67.
In a pie chart of student transport (n = 240), angles are: Walk 90°, Bus 120°, Car 90°, Cycle 60°. How many students walk?
68.
For a histogram, if a class has frequency 30 and density 5, what is its class width?
69.
A bar chart shows daily ice-cream sales: Mon 25, Tue 30, Wed 22, Thu 35, Fri 50. (a) Find the week's total. (b) Find the modal day.
70.
A histogram shows ages 0-20: density 2; 20-40: density 4; 40-60: density 3; 60-80: density 1. Find total frequency (assuming widths of 20).
71.
For the data 4, 7, 8, 10, 15, find the (a) mean, (b) median, (c) MAD.
72.
Two classes' reaction times (ms) have means 200 and 210, ranges 50 and 30. Which class is faster on average and which more consistent?
73.
A dataset has the mean = 50 and median = 45. State the skew direction.
74.
For the data 5, 8, 9, 10, 12, 15, 30, find the median and explain whether 30 is likely an outlier.
75.
A test has marks 12, 14, 16, 18, 20 with mean 16. Compute the (a) MAD, (b) variance.
76.
Compare the spread of two data sets. Set A: 8, 9, 10, 11, 12 (range 4); Set B: 6, 8, 10, 12, 14 (range 8). Which has greater spread?
77.
For a uniform distribution of values 1, 2, 3, 4, 5, 6 (each appearing once), find the (a) mean, (b) MAD.
78.
Class A vs Class B mean test marks: A = 75, B = 80; standard deviations: A = 10, B = 4. Which class is more consistent?
79.
A teacher says "the class average is 72, so most students scored around 72." Critique this statement.
80.
Two cricketers have batting averages: A 45, B 42. A's standard deviation is 25, B's is 8. Whose performance is more reliable?

GoldQuestions 81–120

81.
A class of 20 students has reaction-distance data (cm): 15, 11, 19, 14, 12, 17, 21, 14, 17, 11, 23, 15, 20, 17, 13, 10, 20, 16, 12, 28. Find the (a) mean, (b) median, (c) range.
82.
For the reaction-distance dataset, identify the value most likely to be an outlier and justify.
83.
Find the missing frequency: $1 \to 3$ , $2 \to 5$ , $3 \to x$ , $4 \to 4$ , $5 \to 2$ . Total = 20.
84.
The mean of 5 numbers is 12. The mean of 4 of them is 10. Find the 5th number.
85.
A class of 24 has mean test score 62. Adding a 25th student raises the mean to 63. Find the new student's score.
86.
The mean of 8 numbers is 15. One number, 9, is replaced by 25. Find the new mean.
87.
For a frequency table: x = 0, 1, 2, 3, 4; f = 2, 5, 8, 3, 2. Find the median.
88.
A test taker scores $a, b, c, d, e$ where the mean is 75 and median is 78. Two extra results are added: 60 and 90. Find the new mean.
89.
Compare the means of two classes: Class A has 25 students with mean 72; Class B has 15 students with mean 80. Find the combined mean.
90.
A teacher records weekly mark improvements (in %): 3, 5, $-2$ , 4, 6, $-1$ , 0, 8. Find the (a) mean, (b) range, (c) the number of weeks where improvement was negative.
91.
A school survey on phone use has 800 students. A stratified random sample of 80 is taken: Year 7 (200), Year 8 (180), Year 9 (220), Year 10 (100), Year 11 (100). How many from each year?
92.
A sample of 30 students reports an average of 6.5 h homework per week with range 4. Estimate the school-wide average (school of 600 students) and discuss the confidence in this estimate.
93.
For a survey about lunch preferences in a school of 1000 students, you want results accurate to ±3%. Approximately how large a sample is needed?
94.
A school surveys 120 students about whether they bring their own lunch. 78 say yes. (a) What percentage in the sample? (b) Estimate the count in a year-group of 600. (c) Comment on accuracy.
95.
A survey question is "Do you prefer cats over dogs?" State two ways this question is poorly designed and improve it.
96.
A teacher considers two sampling plans for a school of 600. (i) 60 random students from the whole school; (ii) the first 60 students arriving at school. State which is better and why.
97.
A doctor wants to estimate the average daily steps for the local town (~50 000 people). Suggest two methods, with pros and cons.
98.
A school survey contains the question: "How many hours of TV do you watch a week? (a) 0–5, (b) 6–10, (c) 10–15, (d) more than 15." Identify the problem with the response options.
99.
Sample with replacement vs without replacement. A class of 30 students has names placed in a hat. Five draws are made (with replacement). On average, how many distinct names are drawn?
100.
In an opinion poll, respondents are asked: "How satisfied are you with the school cafeteria? Very satisfied / Satisfied / Neutral / Dissatisfied / Very dissatisfied." Identify whether this is biased.
101.
A frequency table has classes (8-11), (12-15), (16-19), (20-23), (24-27), (28-31) with frequencies 3, 7, 5, 4, 0, 1. (a) State the modal class. (b) How many students total?
102.
A bar chart compares 2 datasets: Class A (red bars) and Class B (blue bars), categories Maths, English, Science. The mean scores are A: 75, 68, 80; B: 70, 72, 78. Describe one feature visible from the chart.
103.
A pie chart of 360 students shows transport: Walk 25%, Bus 35%, Car 30%, Cycle 10%. Find (a) the angle for each slice, (b) the number of students in each category.
104.
Construct a pie chart for 30 students whose favourite subjects are: Maths 12, English 9, Science 6, Other 3. State each slice angle.
105.
On a histogram with unequal class widths, frequencies 12, 20, 24, 4 correspond to widths 4, 4, 8, 4. Find the densities and identify which class has the tallest bar.
106.
A line chart shows weekly profits (chf) at a café: Jan 1200, Feb 1500, Mar 1300, Apr 1700, May 1900. Find the (a) mean, (b) range, (c) month with biggest change from prior month.
107.
A stacked bar chart shows boys vs girls in Y7, Y8, Y9: Y7 (boys 100, girls 100), Y8 (90, 90), Y9 (110, 110). Total in each year?
108.
For the same data (Y7: 200, Y8: 180, Y9: 220), choose between (a) grouped bar, (b) stacked bar, (c) line chart. Which is best for comparing within-year boys vs girls?
109.
A pie chart slice represents 25% of 800 people. Find (a) the number of people, (b) the angle of the slice.
110.
Choose the most misleading chart for the data: 50 yes, 50 no (a) double bar; (b) pie chart with very different colours; (c) bar chart with y-axis starting at 40.
111.
For the reaction-distance dataset (sorted): 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28: find (a) Q1, (b) Q3, (c) IQR.
112.
Using Q1 = 12.5, Q3 = 19.5, find the lower and upper outlier fences ( $Q_1 - 1.5 \cdot \text{IQR}$ and $Q_3 + 1.5 \cdot \text{IQR}$ ). Which values are outliers from a max 28 and min 10?
113.
Two boxplots: Class A min 8, Q1 12, median 15, Q3 18, max 25. Class B min 10, Q1 13, median 16, Q3 22, max 30. Which has (a) higher median, (b) larger IQR, (c) more skew?
114.
Suggest one inference about reaction times that can be drawn from a sample with mean 16 and median 16, vs one with mean 18 and median 14.
115.
A class of 30 has mean 75, range 40. A new student with score 100 joins. Find the new mean.
116.
Mean salary in company A is £50k with SD £5k; in company B mean £45k with SD £20k. Compare typical salaries.
117.
A class's 5-number summary is 10, 15, 22, 28, 50. (a) Compute IQR. (b) Identify any outliers using 1.5×IQR rule.
118.
In a survey of 1000 households, mean income $50k, median$ 40k. (a) State the skew. (b) Explain what this tells you about wealth distribution.
119.
A teacher's annual class mean tests have decreased from 75 to 65 over 5 years. Does this prove the teacher's teaching is getting worse? List two factors to consider.
120.
Two athletes record race times (s): A: 12.0, 12.2, 12.1, 12.3, 12.0. B: 11.8, 12.8, 11.9, 12.7, 12.0. Compare via mean and range.

PlatinumQuestions 121–160

121.
A data set has mean 16.25 and median 15.5. After removing the largest value (28), find the new mean and median.
122.
Class A: 25 students, mean 70, median 72. Class B: 15 students, mean 80. Find combined mean and discuss whether you can deduce the combined median.
123.
In a set of 6 numbers, the mean is 12, the median is 11.5, the mode is 9, and the range is 14. The numbers are integers and 9 appears twice. Find a possible data set.
124.
A frequency table for grouped data has midpoints 9.5, 13.5, 17.5, 21.5 with frequencies 3, 7, 6, 4. Find the estimated mean.
125.
A set of 7 positive integers has mode 4, median 5, mean 6, and range 9. Find a possible set.
126.
A test produces marks with mean 60, median 62 and mode 70 (a positively skewed distribution). For each of the following changes, state whether mean / median / mode increase, decrease or stay the same: (a) add 100 to every score; (b) remove the score 70 (the only one); (c) multiply every score by 2.
127.
Five numbers have mean 12 and range 6. The smallest is 9. What is the largest? Construct one valid set.
128.
A frequency table has values $x = 1, 2, 3, 4, 5$ with frequencies $f = 2, 5, a, 3, 2$ where the mean is exactly 3. Find $a$ .
129.
The mean of the integers 1 to $n$ is 8. Find $n$ .
130.
For a list of 6 numbers, mean = median = mode = 7. Construct one valid set, and show it has range 0 or 2.
131.
A school of 800 students will be surveyed about transport. Year 7: 200, Year 8: 200, Year 9: 200, Year 10: 100, Year 11: 100. A stratified sample of 80 is required, but you want to ensure ≥ 5 from Year 11. How many from each year?
132.
A sample of 50 dogs at a vet has mean weight 18 kg, range 30 kg. The vet treats 1000 dogs/year. Estimate the population mean and discuss the spread.
133.
A polling firm surveys 1000 voters and reports candidate A at 51% support. Margin of error ±3%. What does this mean for confidence?
134.
A pet-owner survey has 60% response rate (out of 500 sent). 200 responded "yes, owns a pet". Estimate the percentage of pet owners in the surveyed population.
135.
A student conducts a survey at the school gate asking about phone use. Identify three types of bias possible and how to reduce each.
136.
A stratified sample of 100 from a population of 1000 is split: 600 women and 400 men. After sampling, the survey finds 60 women say "yes" and 30 men. Estimate the overall population proportion of "yes" responses.
137.
A class of 28 students completes a maths test. The teacher's class mean is 16/20 with range 7. The teacher reports "the school average is 16/20". Critique this claim.
138.
A survey of 200 students yields a 95% confidence interval of "30–40% prefer pizza". Interpret this interval correctly. What does **not** it mean?
139.
Design an unbiased question to estimate the proportion of students at your school who exercise daily, and state the sampling method.
140.
A teacher cleans a dataset of 200 marks. She notices 15 zeros, 1 value of 100, and 5 missing entries. Outline a cleaning strategy.
141.
A histogram has class widths and frequencies: width 4 → freq 12; width 4 → freq 20; width 8 → freq 24; width 4 → freq 4. (a) Densities for each. (b) Tallest bar.
142.
Construct an infographic comparison: round-trip Geneva-Nice CO₂ per person — Plane 80 kg, Train 15.3 kg, Car (solo) 165.4 kg. Express each as a percentage of the largest, and identify the best chart.
143.
A pie chart represents a budget of £24 000 with slices: Salaries 50%, Rent 25%, Supplies 15%, Other 10%. Find the £ amount and angle for each slice.
144.
A double bar chart compares mean test scores in 2024 and 2025 across 3 schools. School A: 60, 65; B: 70, 72; C: 80, 78. Compute the year-on-year change for each and identify the trend.
145.
A line chart of crime rates per 1000 over 5 years: 35, 30, 28, 26, 22. Find (a) the percentage decrease from year 1 to 5. (b) Annual rate of decrease (assuming linear).
146.
A pie chart has slices with angles 90°, 90°, 90°, 90°. (a) Is this misleading or fine? Explain. (b) What does the chart say about the data?
147.
A misleading bar chart shows sales: 2024 → 100; 2025 → 110, but the y-axis goes 95 to 115. Discuss why this is misleading and how to fix it.
148.
For 50 student scores, the relative frequencies are 0–20%: 0.04; 20–40%: 0.12; 40–60%: 0.32; 60–80%: 0.36; 80–100%: 0.16. Construct the frequency table.
149.
A line chart of monthly profits shows oscillations between 1000 and 1500 chf throughout the year, but the overall trend is flat. What charts/measures would best convey this?
150.
A pie chart and a bar chart present the same data of 4 categories (each 25%). Which is easier to read? Which is more accurate?
151.
For the reaction-distance dataset, compute the standard deviation (use the formula $\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$ ). Mean = 16.25, n = 20.
152.
A class's mean is 16.25 and median is 15.5. After removing the largest value (28), the new mean is 15.63 and new median is 15. Discuss the robustness of mean vs median.
153.
Class A: mean 16.25 cm, sample of 20. Class B: mean 14.50 cm, sample of 30. Find the combined mean of all 50 students.
154.
A linear model fits reaction time (ms) = $-8a + 270$ where $a$ is age, fitted to ages 8-16. (a) Predict age 12 reaction time. (b) Explain why the model gives -50 ms at age 40, and the kind of error involved.
155.
Two boxplots: A (5-num: 10, 15, 20, 25, 30) and B (10, 12, 20, 28, 30). Both have median 20, range 20. Compare shapes.
156.
A teacher's test marks are: 65, 70, 72, 75, 80, 85, 90. Find Q1, Q3, IQR, and the 1.5×IQR fences.
157.
A reverse-percentage problem: in a survey, 65% of pupils said they "feel positive about school." 84 pupils said they did not feel positive. (a) How many surveyed? (b) How many felt positive?
158.
A class of $n$ students has mean test score 65. A new student with score 80 joins; new mean is 66.5. Find $n$ .
159.
A 2-way table: 130 study French, 90 study Spanish, 50 both. School of 200. Find (a) only French, (b) only Spanish, (c) neither.
160.
A histogram has unequal-width bars: width 4 → freq 12; width 4 → freq 20; width 8 → freq 24; width 4 → freq 4. Find the densities and the modal bar.