Mathematics • Year 7 • Unit 4 • Lesson 12
Interpreting Graphs — Mixed Challenge
Bring together all six features (trend, peak, trough, cluster, gap, outlier), the effect of outliers on the mean, and graph interpretation in context. Then spot a flawed description and create your own data story.
1. Mixed problems
Each question uses different graph features. Be specific. 2 marks each
1.1 A line graph shows test averages over 5 years: 65, 68, 70, 72, 75. State the trend in one word and give the start and end values.
1.2 A dot plot shows shoe sizes: 5, 6, 6, 7, 7, 7, 8, 8, 9, 14. Identify the outlier and state which scale word best describes the rest of the cluster (e.g. "tightly clustered" or "spread out").
1.3 Calculate the mean of {58, 62, 64, 65, 66, 68, 70, 71, 72, 95}. Then recalculate WITHOUT the outlier. Briefly explain the effect of the outlier on the mean.
1.4 A line graph of monthly visitors to a museum (in thousands): Jan 8, Feb 7, Mar 9, Apr 12, May 14, Jun 15, Jul 15, Aug 15, Sep 13, Oct 11, Nov 9, Dec 7. Identify the plateau and state which months it covers.
1.5 A bar chart shows ice cream and umbrella sales each month. Both rise in summer. A friend says "ice cream sales cause umbrella sales because both rise together." State what is wrong with this reasoning in one sentence.
1.6 Describe one piece of data using the formula Feature → Value → Real-world meaning. Choose any of: a peak, trough or outlier on a graph of your choice (e.g. your own attendance, a sports score). Give all three parts.
2. Find the mistake
A student wrote the following description of a monthly rainfall graph (mm): Jan 12, Feb 18, Mar 24, Apr 30, May 100, Jun 28, Jul 25, Aug 22, Sep 20, Oct 18, Nov 16, Dec 14. Exactly one statement contains a clear error. 3 marks
Statement 1: The peak is in May at 100 mm.
Statement 2: The trough is in January at 12 mm.
Statement 3: The overall trend is decreasing because the values fall from May onwards.
Statement 4: May is not an outlier because it's the highest value, so it must just be the peak.
(a) Which statement contains the mistake?
(b) Explain why that statement is wrong in one or two sentences.
(c) Write the corrected statement.
Stuck? An outlier is a value FAR from the rest. The peak can ALSO be an outlier if it sits far above all the other months.3. Open-ended challenge — tell a data story
This question has many correct answers. Show your work clearly. 4 marks
3.1 You are asked to design a 12-month line graph that tells the "story" of a small business's profit. Your invented data set must include exactly four features:
- an overall increasing trend for the year,
- one clear peak month (e.g. Christmas trading),
- one plateau of at least two consecutive months,
- one outlier month caused by a one-off event (e.g. a fire, a viral product).
Provide: (i) a list of 12 monthly profit values (Jan to Dec) in $, (ii) a short label for each of the four features, (iii) a one-sentence "story" that explains the year for the business.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Trend in one word
Trend: increasing. Start = 65, end = 75 (over 5 years).
1.2 — Shoe sizes
Outlier: size 14 (far above the cluster of 5–9). The rest is tightly clustered between sizes 5 and 9.
1.3 — Outlier effect on mean
Mean with outlier = (58+62+64+65+66+68+70+71+72+95) ÷ 10 = 691 ÷ 10 = 69.1.
Mean without 95 = 596 ÷ 9 ≈ 66.2.
Effect: the outlier (95) pulls the mean upward by about 3 marks, making the class look stronger than most students actually scored.
1.4 — Plateau
Plateau: June, July and August — all three months sit at 15,000 visitors with no change.
1.5 — Correlation vs causation
Sample: Two things rising together does NOT prove one causes the other — both ice cream and umbrella sales rise in summer because of a common cause (the season / weather), not because of each other.
1.6 — Feature → Value → Meaning (sample)
Feature: peak attendance. Value: 28 students at the lunchtime debate on Thursday. Meaning: Thursday was the most popular debate day — possibly because it was a special "Australia in the world" theme that attracted extra students.
Marking: 1 for each part (feature name, specific value, real-world meaning).
2 — Find the mistake
(a) The mistake is in Statement 4.
(b) The peak CAN be an outlier. May's value of 100 mm is the peak AND it sits far above every other month (which all sit between 12 and 30 mm) — so it is BOTH the peak and an outlier.
(c) Corrected: May (100 mm) is both the peak AND an outlier — it is the highest value and it sits far above every other month, which range from 12 to 30 mm.
3 — Business profit story (sample design)
Values ($): Jan 1000, Feb 1100, Mar 1100, Apr 1100, May 1300, Jun 1400, Jul 1500, Aug 1600, Sep 5000, Oct 1700, Nov 1800, Dec 2500.
Trend: overall increasing from $1000 to $2500.
Peak: September at $5000.
Plateau: February, March, April (all $1100).
Outlier: September ($5000) — far above all other months.
Story: "The business grew steadily through the year. After a slow plateau in autumn, a viral social media post in September caused a one-off spike in sales, and the year finished strongly in December."
Marking: 1 for valid data, 1 for each of the four features clearly labelled, 1 for the one-sentence story.