Mathematics Standard • Year 11 • Module 4 • Lesson 12

Module Review — Skill Drill

Build fluency across the whole module: classifying data, summary statistics (mean, median, range, IQR), the 68-95-99.7 rule, scatter plots and the line of best fit.

Build · Skill Drill

1. Quick recall — formulas from the module

Answer each question in the space provided. 1 mark each

Q1.1 Complete the formulas:

Mean = ____________________ Range = ____________________ IQR = ____________________

Q1.2 Complete the outlier fences:

Lower fence = Q1 − ____ × IQR Upper fence = Q3 + ____ × IQR

Q1.3 68-95-99.7 rule:

Within 1 SD: ____% Within 2 SD: ____% Within 3 SD: ____%

Stuck? Revisit lesson § Module Formula Summary.

2. Worked example — full single-variable workout

Compute mean, median, range and IQR for the data set, then identify any outliers using the 1.5 × IQR rule.

Data (sorted). 12, 15, 18, 22, 25, 28, 32, 35, 38, 42.

Step 1 — Mean.

Sum = 12 + 15 + 18 + 22 + 25 + 28 + 32 + 35 + 38 + 42 = 267. Mean = 267 ÷ 10 = 26.7.

Step 2 — Median (n = 10 → average of 5th and 6th).

Median = (25 + 28) ÷ 2 = 26.5.

Step 3 — Range.

Range = 42 − 12 = 30.

Step 4 — Quartiles and IQR.

Lower half = {12, 15, 18, 22, 25} → Q1 = 18. Upper half = {28, 32, 35, 38, 42} → Q3 = 35. IQR = 35 − 18 = 17.

Step 5 — Outlier check (1.5 × IQR).

1.5 × 17 = 25.5. Lower fence = 18 − 25.5 = −7.5. Upper fence = 35 + 25.5 = 60.5. No values below −7.5 or above 60.5 → no outliers.

Conclusion. Mean = 26.7, median = 26.5, range = 30, IQR = 17. The mean and median are very close → distribution roughly symmetric.

3. Faded example — fill in the missing steps

Data: 8, 12, 15, 18, 22, 25, 28, 30, 35, 40. Fill the blanks. 4 marks

Step 1 — Mean:

Sum = ____. Mean = ____ ÷ 10 = ____

Step 2 — Median:

Average of the ____ and ____ values = ( ____ + ____ ) ÷ 2 = ____

Step 3 — Range:

Range = ____ − ____ = ____

Step 4 — Q1, Q3, IQR:

Q1 = ____, Q3 = ____, IQR = ____

Conclusion sentence: "The mean is ____ and the median is ____, so the distribution is approximately ____________."

Stuck? Revisit lesson § Module Formula Summary — sum, divide, find middle, then quartiles.

4. Graduated practice — module mix

This section blends every topic from the module. Show one line of working per part.

Foundation — naming and identifying (4 questions)

Q	Problem	Answer
4.1 1	"Hair colour" is categorical or numerical?
4.2 1	"Number of pets per household" is discrete or continuous?
4.3 1	For 7, 9, 11, 13, 15 the median is ____.
4.4 1	If r = −0.05, the linear correlation is ____________ (strength + direction).

Standard — typical HSC mix (6 questions)

Each part draws from a different lesson in the module.

4.5 Find the mean, median, range and IQR for: 5, 7, 8, 10, 12, 14, 18. 2 marks

4.6 A normal distribution has mean = 60, SD = 8. Between what values do 95% of data lie? 2 marks

4.7 Box plot summary: min = 12, Q1 = 20, med = 28, Q3 = 36, max = 60. Use the 1.5 × IQR rule to check whether 60 is an outlier. 2 marks

4.8 Compare two classes: Class A mean = 75, SD = 5. Class B mean = 75, SD = 12. Which is more consistent, and why? 2 marks

4.9 Find the equation of a line of best fit through (2, 40) and (6, 80). 2 marks

4.10 r = 0.78 between ice-cream sales and drowning incidents. State the correlation in plain English, then explain in one sentence why eating less ice cream would NOT reduce drownings. 2 marks

Extension — combining topics (2 questions)

4.11 A normally distributed test has mean = 70, SD = 8 (n = 400). (a) Estimate the number of students scoring above 86. (b) A student who scored 70 transfers to a class with mean = 70, SD = 4 and keeps the same raw score. Describe in one sentence how their relative standing in the new class differs from the old. 3 marks

4.12 A scatter plot of study hours (x, range 1–10 hours) vs trial mark (y) gives the line of best fit y = 6x + 40. (a) Predict the mark for 4 hours of study. (b) Predict for 20 hours of study and state the two problems with that prediction. 3 marks

Stuck on 4.11(b)? In the old class the student is right at the mean (z = 0); in a tighter class with the same mean they are still at the mean — their raw position is identical, but spread does not change because the score equals the mean.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (hair colour) I wrote categorical, not numerical.

For 4.2 (pets per household) I wrote discrete (count data), not continuous.

For 4.3 (median of 5 values) I picked the middle value (11), not the mean.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Formulas

Mean = sum of values ÷ number of values. Range = max − min. IQR = Q3 − Q1.

Q1.2 — Outlier fences

Lower fence = Q1 − 1.5 × IQR. Upper fence = Q3 + 1.5 × IQR.

Q1.3 — 68-95-99.7 rule

Within 1 SD: 68%. Within 2 SD: 95%. Within 3 SD: 99.7%.

Q3 — Faded example (8, 12, 15, 18, 22, 25, 28, 30, 35, 40)

Step 1: Sum = 233. Mean = 23.3.
Step 2: 5th and 6th values = 22 and 25; (22 + 25) ÷ 2 = 23.5.
Step 3: Range = 40 − 8 = 32.
Step 4: Lower half {8, 12, 15, 18, 22} → Q1 = 15. Upper half {25, 28, 30, 35, 40} → Q3 = 30. IQR = 15.
Conclusion: Mean = 23.3, median = 23.5 → approximately symmetric.

Q4.1

Categorical (a label, not a number).

Q4.2

Discrete numerical (counts cannot be fractional).

Q4.3

Median = 11 (middle value of 5 sorted values).

Q4.4

Very weak negative (|r| ≈ 0.05; essentially no relationship).

Q4.5 — Summary stats for 5, 7, 8, 10, 12, 14, 18

Sum = 74; mean = 74 ÷ 7 ≈ 10.57. Median = 10 (middle of 7 values). Range = 18 − 5 = 13. Lower half {5, 7, 8} → Q1 = 7; upper half {12, 14, 18} → Q3 = 14; IQR = 7.

Q4.6 — 95% range, mean 60 SD 8

60 ± 2(8) = 44 to 76.

Q4.7 — Is 60 an outlier?

IQR = 36 − 20 = 16. Upper fence = 36 + 1.5(16) = 36 + 24 = 60. 60 = upper fence (boundary case) — usually considered the limit, not an outlier. Any value > 60 would be.

Q4.8 — Class A vs Class B (same mean)

Class A is more consistent (SD 5 vs 12) — its scores cluster more tightly around 75.

Q4.9 — Line through (2, 40) and (6, 80)

m = (80 − 40) ÷ (6 − 2) = 10. 40 = 10(2) + b ⇒ b = 20. y = 10x + 20.

Q4.10 — Ice cream and drowning

r = 0.78 → strong positive linear correlation. Eating less ice cream would not reduce drowning because the lurking variable is hot weather: hot days drive both ice-cream sales and the number of swimmers (and therefore drowning incidents).

Q4.11 — Normal mean 70 SD 8 (n = 400)

(a) 86 = 70 + 2(8) = mean + 2 SD. Above 2 SD ≈ 2.5%. 2.5% of 400 = 10 students.
(b) A score of 70 equals the mean in both classes. Relative standing is identical in both (z = 0; 50th percentile). However, in the tighter class (SD 4), any deviation from 70 would translate to a much larger z-score, so the student's standing would shift dramatically if their mark changed.

Q4.12 — Line y = 6x + 40

(a) y = 6(4) + 40 = 64.
(b) y = 6(20) + 40 = 160. Two problems: (1) extrapolation — 20 hours is well outside the 1–10 hour data range, so the linear pattern is not guaranteed; (2) marks cap at 100, so 160 is impossible. Diminishing returns also apply.