Mathematics • Year 10 • Unit 4 • Lesson 10
Standard Deviation in the Real World
Apply Lesson 10's standard deviation to real Year 10 contexts: HSC trial cohorts, sports team consistency, share-price volatility, weather variability and supermarket pricing. Practise the lesson's golden rule: compare two data sets using BOTH centre (mean) and spread (SD), then write a one-sentence interpretation.
1. Word problems
Show working in your SD calculations. For interpretation questions, write a full sentence with specific numbers.
1.1 — Basketball shooting practice. Two players' practice scores out of 20 are: Player A: 14, 15, 14, 16, 15, 14. Player B: 10, 18, 13, 19, 11, 17.
(a) Find each player's mean and sample standard deviation.
(b) Which player is more consistent? Justify using BOTH measures (Lesson 10 Remember callout). 3 marks
1.2 — Two factories' product weights. Two muesli-bar factories produce bars labelled "40 g". A sample of 5 bars from Factory X: 39, 40, 41, 40, 40. Sample of 5 from Factory Y: 35, 45, 40, 38, 42.
(a) Find the mean and SD for each factory.
(b) Both factories average 40 g, but only one factory's machines are reliable. Which one, and why? 3 marks
1.3 — HSC trial across two cohorts. Two Year 12 cohorts sat the same Maths trial. The school reports: Cohort 2024 mean = 72, s = 8. Cohort 2025 mean = 75, s = 15.
(a) Which cohort scored higher on average?
(b) Which cohort had more variable results?
(c) A parent asks "did the 2025 cohort do better?" Write a two-sentence answer using both centre AND spread. 3 marks
1.4 — Sydney winter rainfall. Daily rainfall (mm) at a Sydney station for one week was: 0, 0, 5, 12, 0, 8, 3.
(a) Find the mean and the sample standard deviation correct to 2 d.p.
(b) The SD is large compared with the mean — what does that tell you about Sydney's winter rainfall pattern in one sentence? 3 marks
1.5 — Share prices. Two ASX shares closed at these prices ($) over 5 days. Share K: 50.20, 50.05, 50.30, 49.95, 50.10. Share L: 50.00, 48.50, 52.20, 47.80, 51.50.
(a) Find the mean and SD of each share.
(b) An investor wants a "low-risk" share — which one and why? (Investors use SD as a measure of "volatility".) 3 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A friend says "If two classes have the same mean test mark, the teaching must be equally good." Using Lesson 10's misconception card ("If two data sets have the same mean, they are identical" — wrong), write a four-sentence reply that (i) explains why same mean does not equal same performance, (ii) names the missing measure (standard deviation), (iii) gives a tiny made-up example of two 5-student classes with the same mean but very different SD that would change the verdict on the teaching, and (iv) finishes with one rule of thumb a Year 10 student can use to spot when "same mean" is being used to hide variability.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Basketball
Player A: Σ = 88, mean = 88 ÷ 6 ≈ 14.67. Deviations: −0.67, 0.33, −0.67, 1.33, 0.33, −0.67. Squared: 0.45, 0.11, 0.45, 1.78, 0.11, 0.45. Sum ≈ 3.35. s² = 3.35 ÷ 5 = 0.67. s ≈ 0.82.
Player B: Σ = 88, mean = 14.67. Deviations: −4.67, 3.33, −1.67, 4.33, −3.67, 2.33. Squared ≈ 21.8, 11.1, 2.8, 18.7, 13.5, 5.4. Sum ≈ 73.3. s² = 14.7. s ≈ 3.83.
(b) Same mean (14.67), but Player A's SD (0.82) is much smaller than Player B's (3.83) → Player A is more consistent; Player B is far more variable shot-to-shot.
1.2 — Muesli-bar factories
Factory X: mean = 200 ÷ 5 = 40 g. Deviations: −1, 0, 1, 0, 0. Squared: 1, 0, 1, 0, 0. Sum = 2. s² = 0.5. s ≈ 0.71 g.
Factory Y: mean = 200 ÷ 5 = 40 g. Deviations: −5, 5, 0, −2, 2. Squared: 25, 25, 0, 4, 4. Sum = 58. s² = 14.5. s ≈ 3.81 g.
(b) Both average 40 g, but Factory X has a much smaller SD (0.71 vs 3.81), so Factory X's machines are more reliable — its bars consistently weigh ~40 g, while Factory Y's vary by ±5 g.
1.3 — HSC trial cohorts
(a) Cohort 2025 has a higher mean (75 vs 72) — scored higher on average.
(b) Cohort 2025 has the larger SD (15 vs 8) — more variable results.
(c) The 2025 cohort scored higher on average (mean 75 vs 72), but their results were much more variable (SD 15 vs 8), so they have more very high AND very low scores than the 2024 cohort.
1.4 — Sydney rainfall
(a) Σ = 28, mean = 28 ÷ 7 = 4 mm. Deviations: −4, −4, 1, 8, −4, 4, −1. Squared: 16, 16, 1, 64, 16, 16, 1. Sum = 130. s² = 130 ÷ 6 ≈ 21.67. s ≈ 4.66 mm.
(b) The SD (4.66 mm) is larger than the mean (4 mm). Sydney's winter rainfall is highly variable from day to day — most days are dry but a couple deliver heavy falls.
1.5 — Share prices
Share K: mean = 250.60 ÷ 5 = $50.12. Deviations from 50.12: 0.08, −0.07, 0.18, −0.17, −0.02. Squared: 0.0064, 0.0049, 0.0324, 0.0289, 0.0004. Sum ≈ 0.0730. s² ≈ 0.0183. s ≈ $0.14.
Share L: mean = 250.00 ÷ 5 = $50.00. Deviations: 0, −1.50, 2.20, −2.20, 1.50. Squared: 0, 2.25, 4.84, 4.84, 2.25. Sum = 14.18. s² = 3.545. s ≈ $1.88.
(b) An investor wants Share K: similar mean but SD ≈ $0.14 vs Share L's $1.88, so K is much less volatile (low risk).
2.1 — Explain your thinking (sample response)
Same mean does NOT mean same performance — Lesson 10's misconception card says identical-mean data sets can be wildly different. The missing measure is the standard deviation, which tells us how spread out the marks are. Example: Class P scores {68, 69, 70, 71, 72} and Class Q scores {30, 50, 70, 90, 110} both have mean 70 — but P (SD ≈ 1.58) shows almost everyone passing around 70, while Q (SD ≈ 31.62) has some students failing badly and some flying high. Rule of thumb: always ask for the standard deviation alongside the mean — if it's missing, the headline is probably hiding the spread.
Marking: 1 mark naming the misconception, 1 for naming SD, 1 for a valid contrast example, 1 for clear rule of thumb.