These are original Marlbridge practice questions, not past exam questions. All data sets are fictional. The rubrics are a suggested Marlbridge rubric to help you check your work; they are not an official scoring guideline. Assumptions for every question: each histogram bin includes its left edge and not its right edge (so 25 is in the 25–30 bin); “most” means more than half; outliers are judged by eye and called possible outliers, because the numerical rules come in Topic 1.7; round percentages to 1 decimal place.
Question 1 (multiple choice · foundation)
The finishing times of 40 runners in a fictional 5 km fun run were grouped into a histogram with these frequencies:
| Time (minutes) | 20–25 | 25–30 | 30–35 | 35–40 | 40–45 | 45–50 | 50–55 |
|---|---|---|---|---|---|---|---|
| Number of runners | 2 | 5 | 12 | 9 | 6 | 4 | 2 |
Which best describes the shape of the distribution of finishing times?
- (A) Unimodal and skewed left
- (B) Unimodal and skewed right
- (C) Approximately symmetric
- (D) Approximately uniform
Answer and explanation
Answer: (B). The single peak is in the 30–35 minute bin (12 runners). Below the peak there are only two bins (2 and 5 runners); above it the counts fall slowly over four bins (9, 6, 4, 2). The longer tail is towards the larger times, so the distribution is skewed right.
- (A) names the skew after the side where the peak is. Skew is named after the longer tail.
- (C) ignores the uneven tails: two bins left of the peak, four to the right.
- (D) needs bars of about equal height. Here the bars range from 2 to 12, with one clear peak.
Question 2 (multiple choice · core)
For which of these variables would you most expect a distribution that is skewed left?
- (A) The number of brothers and sisters of each student in a large school
- (B) Scores out of 10 on a very easy quiz taken by a whole class
- (C) The number of minutes each customer waits in a queue at a busy bank
- (D) The number of emails each employee in a company receives in one day
Answer and explanation
Answer: (B). On a very easy quiz most students score 8, 9 or 10, close to the maximum. A few students score lower, which creates a long tail towards the smaller values. That is skewed left.
- (A) Most students have 0, 1 or 2 brothers and sisters and a few have many: a long right tail, so usually skewed right.
- (C) Waits cannot go below 0; most are short and a few very long, so usually skewed right.
- (D) A few employees receive very many emails, so the long tail is usually to the right.
Question 3 (multiple choice · core)
The stem-and-leaf plot shows the scores (out of 60) of 20 students on a fictional chemistry test. Key: 4 | 2 means 42 marks.
1 | 2
2 |
3 | 1 4 6 8 9
4 | 0 2 3 5 5 7 8 9
5 | 0 1 2 4 6 8Which description is best supported by the plot?
- (A) Skewed right, with a gap between 12 and 31 marks
- (B) Skewed left, with a gap from 13 to 30 marks and a possible outlier at 12 marks
- (C) Approximately symmetric, with no unusual features
- (D) Bimodal, with one cluster at 12 marks and another from 31 to 58 marks
Answer and explanation
Answer: (B). Most scores are in the 40s (8 students) and 50s (6 students). The values trail off towards smaller scores: 5 in the 30s, then a single score of 12. The longer tail is to the left, so the distribution is skewed left. No student scored from 13 to 30 marks, which is a gap, and 12 is 19 marks below the next smallest score (31), so it is a possible outlier.
- (A) gets the gap right but the skew wrong: the long tail points to smaller scores.
- (C) ignores the long left tail and the isolated score of 12.
- (D) treats one value as a cluster. A cluster is a group of values close together; a single far value is a possible outlier. There is only one main peak, in the 40s.
Question 4 (constructed response · core)
A farm in the fictional village of Brenmoor weighed 15 eggs laid on one morning. The masses, in grams, are:
51, 53, 55, 56, 57, 57, 58, 58, 58, 59, 59, 60, 61, 62, 65
Make a dot plot of the data (or picture one), then describe the distribution of egg masses. Include shape, centre, variability and unusual features, in context.
Worked solution and suggested Marlbridge rubric
Shape. The tallest stack is at 58 g (3 eggs), with stacks of 2 at 57 g and 59 g. The values below 58 g roughly mirror those above it (for example 51 and 65, 55 and 61). The distribution is unimodal and approximately symmetric.
Centre. With 15 values, the median is the 8th value in order: 58 g.
Variability. The masses range from 51 g to 65 g, a range of 14 g. Most of the eggs (11 of 15) are between 55 g and 61 g.
Unusual features. There are no obvious outliers or large gaps. The smallest (51 g) and largest (65 g) eggs are only 2 g and 3 g from their nearest neighbours.
In context: the masses of these 15 Brenmoor eggs are unimodal and approximately symmetric, centred at about 58 g, ranging from 51 g to 65 g, with no obvious outliers or gaps.
| Point | What earns it |
|---|---|
| 1 | Shape: unimodal and approximately symmetric (or “roughly symmetric, single peak”) |
| 1 | Centre: median 58 g (accept “about 58 g”; the mean, 57.9 g, also earns it) |
| 1 | Variability: range 14 g, or min 51 g and max 65 g, or “most between 55 and 61 g” |
| 1 | Unusual features: states there are no obvious outliers or gaps, and the whole answer is in context with units (egg masses, grams) |
Do not award point 1 for “perfectly symmetric”: real data are only approximately symmetric. Do not award point 4 if the answer never mentions eggs or grams.
Question 5 (constructed response · core)
A fictional technology helpline recorded the length of 60 calls, in minutes:
| Length (minutes) | 0–2 | 2–4 | 4–6 | 6–8 | 8–10 | 10–12 | 12–14 | 14–16 | 16–18 | 18–20 |
|---|---|---|---|---|---|---|---|---|---|---|
| Number of calls | 10 | 16 | 8 | 2 | 0 | 0 | 6 | 10 | 6 | 2 |
(a) Describe the shape of the distribution and any unusual features. (b) A manager claims that “at least one third of calls last 12 minutes or longer”. Does the table support this claim? Justify your answer with numbers. (c) Suggest one reason, in the context of a helpline, why the data might show this shape.
Worked solution and suggested Marlbridge rubric
(a) The distribution is bimodal, with peaks in the 2–4 minute bin (16 calls) and the 14–16 minute bin (10 calls). There are two clusters: 36 calls lasted less than 8 minutes and 24 calls lasted 12 to 20 minutes. They are separated by a gap: no call lasted from 8 to 12 minutes. There is no obvious single outlier.
(b) Calls of 12 minutes or longer: 6 + 10 + 6 + 2 = 24. That is 24 ÷ 60 = 40.0% of calls. One third of 60 calls is 20 calls (33.3%). Since 24 > 20, at least one third of these calls lasted 12 minutes or longer. The table supports the claim.
(c) The helpline may get two kinds of call: quick questions answered in a few minutes, and harder problems that need a long conversation. Mixing two types of call gives two clusters. (Accept any sensible two-group reason in context.)
| Point | What earns it |
|---|---|
| 1 | Bimodal, with both peaks or both clusters identified |
| 1 | Gap from 8 to 12 minutes identified, in context |
| 1 | Count 24 (or 40%) and a correct comparison with one third (20 calls or 33.3%), with the conclusion that the claim is supported |
| 1 | A reason in context that explains two groups of calls |
A conclusion with no numbers does not earn point 3.
Question 6 (explanation · stretch)
A fictional gym recorded how many times each of 20 members visited in March:
1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9, 11, 12, 22
A student wrote this description:
“The distribution is skewed left, because most of the data is on the left. The mean is 6. 22 is an outlier because there is a gap. The data are spread out.”
(a) Identify three problems with the student’s description and correct each one. (b) Write an improved description of the distribution in context.
Model answer and suggested Marlbridge rubric
(a) Any three of these:
- Skew direction. Most values are low, but the tail stretches right, up to 22 visits, so the distribution is skewed right. Skew follows the longer tail.
- Reason for the outlier. A gap alone does not make a value an outlier. 22 is a possible outlier because it is 10 visits above the next largest value (12); the gap from 13 to 21 visits is a separate feature.
- Vague spread. “Spread out” gives no numbers. Visits range from 1 to 22 (range 21), with 19 of the 20 members visiting 1 to 12 times.
- No context. It never says what was measured: gym visits in March by these 20 members.
(The mean is correct: 120 ÷ 20 = 6 visits. Topic 1.7 explains why the median, 4.5 visits, is a better typical value here.)
(b) The distribution of the number of gym visits in March by these 20 members is unimodal and skewed right, with a peak at 3 visits. The median is 4.5 visits. Visits range from 1 to 22, but 19 members visited between 1 and 12 times. There is a gap from 13 to 21 visits, and one member who visited 22 times is a possible outlier.
| Point | What earns it |
|---|---|
| 1 | First problem identified and corrected |
| 1 | Second problem identified and corrected |
| 1 | Third problem identified and corrected |
| 1 | Improved description with correct shape, a centre, a spread with numbers, the gap and possible outlier, all in context |
Do not accept “the mean is wrong”: it is correct.
Question 7 (constructed response · stretch)
A fictional bakery recorded the minute past the hour at which each of 60 customers arrived between 9:00 and 10:00 one morning:
| Minutes past 9:00 | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 |
|---|---|---|---|---|---|---|
| Number of customers | 11 | 9 | 10 | 12 | 8 | 10 |
(a) Describe the shape of the distribution. (b) The owner claims that “the first 10 minutes after 9:00 are much busier than the rest of the hour, so we need extra staff only then”. Use the data to evaluate this claim. (c) Describe how a histogram of the arrival times would look if the owner’s claim were true.
Model answer and suggested Marlbridge rubric
(a) The bars range only from 8 to 12 customers, with no clear peak. The distribution of arrival times is approximately uniform: customers arrived at a fairly steady rate.
(b) If arrivals were spread evenly, each 10-minute bin would hold 60 ÷ 6 = 10 customers (16.7%). The first 10 minutes had 11 customers (18.3%), only 1 more than an even share. The 30–40 minute bin had more (12 customers). The data do not support the claim: the first 10 minutes were not much busier, and were not even the busiest 10 minutes.
(c) One tall bar for 0–10 minutes and much shorter bars for the rest of the hour: a peak at the left with a long right tail, so skewed right.
| Point | What earns it |
|---|---|
| 1 | Approximately uniform, with a reason (bars all between 8 and 12, no clear peak) |
| 1 | Compares the first bin (11) with an even share (10) or with the other bins, using numbers |
| 1 | Notes that the 30–40 bin (12) is at least as busy and concludes that the claim is not supported |
| 1 | Describes a tall first bar with shorter later bars, named as skewed right (or peak on the left with a long right tail) |
How did you do?
- Q1 or Q2 wrong: re-read “Shape, part 1: symmetric or skewed?” in the study guide. Skew follows the tail.
- Q3 or Q6 wrong: revisit “Unusual features: outliers, gaps and clusters” and Worked example 1.
- Q4 incomplete: cover all four parts, with context and units.
- Q5 or Q7 wrong: work through Worked example 2 again, especially how a count from the graph is compared with the claim.
- Q7(a) wrong: revisit “Shape, part 2: how many peaks?”.
Then tick off the topic checklist.