Your weak spots

Insights load after your first practice round.

Module 4 · L12 of 12 ~25 min MS-S1 ⚡ +80 XP available

Module Review

You've now explored statistical analysis from data collection through to prediction. This review consolidates every concept from the module into a coherent framework, use it to identify gaps, strengthen understanding, and prepare for your examinations. This is your last lesson; make every XP count.

Final challenge, Without notes, can you recall: the 68-95-99.7 rule, how to compare two distributions, what $r$ means, and why correlation is not causation?

0/5QUESTS

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Before you review, rate your confidence

+5 XP warm-up

Rate your confidence in each key area before reviewing (1 = shaky, 5 = solid). This will guide where to focus your revision time.

auto-saved

Module formula and rule summary

+5 XP to read

Every formula and rule you need for Module 4 Statistics assessments:

Centre & Spread

Mean $= \bar{x} = \dfrac{\sum x}{n}$ · Median: middle value of ordered data · IQR $= Q_3 - Q_1$ · Range $= \text{max} - \text{min}$

Outlier fences

Lower fence: $Q_1 - 1.5 \times \text{IQR}$. Upper fence: $Q_3 + 1.5 \times \text{IQR}$. Values outside these fences are outliers.

Line of best fit

$y = mx + b$ where $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Substitute one point to find $b$.

68-95-99.7 rule (normal distribution): 68% within 1 SD · 95% within 2 SD · 99.7% within 3 SD · mean = median = mode.
Correlation coefficient $r$: $-1 \leq r \leq +1$ · $|r|$ measures strength · sign gives direction · $r \ne$ causation.

The complete Module 4 story

Statistical analysis builds in a logical sequence, each lesson extends the previous one:

L1–3 · Data foundations: Types of data (categorical/numerical, discrete/continuous) · collecting data (census vs sample, sampling bias) · displaying data (histograms, stem-and-leaf, frequency tables).

L4–6 · Describing distributions: Centre (mean, median, mode) · spread (range, IQR, standard deviation) · box plots, five-number summary, outlier detection with $1.5 \times \text{IQR}$ rule.

L7–8 · Comparing and modelling: Parallel box plots · comparing centre, spread, shape · normal distribution and the 68-95-99.7 rule.

L9–11 · Bivariate data: Scatter plots (direction, strength, form, outliers) · correlation coefficient $r$ and its limitations · line of best fit ($y = mx + b$) · interpolation vs extrapolation.

All key terms, module-wide glossary

MedianMiddle ordered value. Use for skewed data or when outliers are present.

IQR$Q_3 - Q_1$. Robust spread measure. Not affected by outliers.

Normal distributionSymmetric bell curve. Mean = median = mode. Apply 68-95-99.7 rule.

Correlation ($r$)Linear association between two variables. $-1 \leq r \leq +1$.

Lurking variableA third factor causing both variables to change, creates spurious correlation.

ExtrapolationPredicting beyond the observed data range. Unreliable.

The five pitfalls that cost marks in assessments

exam-critical

Pitfall 1

Using mean for skewed data

Always check for outliers first. If data is skewed or contains outliers, the median is a better measure of centre. The mean is pulled toward the tail.

Pitfall 2

Confusing range and IQR

Range uses extreme values and is affected by outliers. IQR uses only the middle 50% and is robust. Know when to use each.

Pitfall 3

Forgetting to order data

Median and quartiles require the data to be ordered first. Skipping this step leads to wrong answers every time.

Pitfall 4

Correlation = causation

The most common statistical error. Correlation shows association; causation requires experimental evidence. Always check for lurking variables.

Pitfall 5

Extrapolating without flagging it

Making a prediction beyond the data range without noting reliability risk. Always say: "This is extrapolation, the prediction may be unreliable."

Quick check: Data set: 8, 12, 15, 18, 22, 25, 28, 30, 35, 40. An outlier (value = 100) is added. Which measure of centre is MORE affected?

Examination tips, maximising your marks

exam-critical

We just saw five pitfalls that cost marks, using mean for skewed data, quoting one statistic, omitting context, reversing skew direction, and missing units. That raises a question: knowing the pitfalls is one thing, what concrete habits during the exam stop you from falling into them under time pressure? This card answers it → four exam technique rules: show all working, interpret every result in context, check skew language matches the tail direction, and always include units.

Show working: Method marks are awarded even when the final answer is wrong.
Interpret in context: Do not just calculate, explain what the number means in the given situation.
Compare systematically: Centre, spread, shape, always all three for comparison questions.
Use statistical language: "median", "IQR", "correlation", "interpolation", these words signal understanding.
Draw carefully: Label axes, use scales, mark key points (Q1, Q3, median, whiskers).
State reliability: For every prediction, say whether it is interpolation or extrapolation and whether it is reliable.

Five exam pitfalls: using mean for skewed data; quoting only one statistic; stating values without context; reversing skew direction; missing units or scale. For each, the fix is to check for outliers, compare both measures, and always interpret in context.

Pause, copy the four exam technique rules: (1) show all working for method marks; (2) interpret results in context, not just numbers; (3) use the correct statistical term (median not mean for skewed data); (4) check that skew direction language matches the actual tail into your book.

Data: 12, 15, 18, 22, 25, 28, 32, 35, 38, 42. (a) Find mean and median. (b) Add value 100, which measure changes more? (c) A test on this data has mean = 70, SD = 10. Find the range for the middle 68%.

Mean $= \dfrac{12+15+...+42}{10} = \dfrac{267}{10} = 26.7$
Median $= \dfrac{25+28}{2} = 26.5$ (average of 5th and 6th values)

Data already ordered. Mean ≈ median → roughly symmetric.

Fill the gap: A line through (2, 40) and (6, 80) has slope $m = $ . When $x = 4$, the predicted $y = $ .

Revision activities

Data: 12, 15, 18, 22, 25, 28, 32, 35, 38, 42. Find: mean, median, range, IQR. Is the distribution roughly symmetric?

A test has mean = 70, SD = 10 (normally distributed). What percentage score above 80? What percentage score below 50? Is a score of 95 unusual?

Box plot: min = 10, Q1 = 20, median = 35, Q3 = 50, max = 70. (a) Find IQR. (b) Are there any outliers (using 1.5 × IQR rule)? (c) Describe the shape.

Line through (2, 40) and (6, 80). Find the equation. Predict $y$ at $x = 4$ and at $x = 15$. Comment on reliability.

Match each situation to the best statistical approach:

Top 3 list: Name THREE things you must always do when comparing two distributions in an HSC answer.

Module reflection, how far you've come

You've covered 12 lessons of statistical analysis: from organising raw data to predicting outcomes with a line of best fit. The most important insight in the whole module is probably this: statistics is not just calculation, it is a way of thinking critically about data, uncertainty, and the claims people make from numbers. Every lesson has prepared you to be a better consumer and producer of statistical reasoning.

Look back at your self-assessment from Card 01. Which areas improved most? What still needs work?

auto-saved

Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence.

Short answer

ApplyBand 42 marks

SA 1. Data: 8, 12, 15, 18, 22, 25, 28, 30, 35, 40. (a) Calculate the mean, median, range, and IQR. (b) Add value 100. Recalculate the mean and median. Which measure is more robust? Explain. (2 marks)

auto-saved

AnalyseBand 53 marks

SA 2. Two schools: School A mean = 78, SD = 5, $n = 200$. School B mean = 75, SD = 12, $n = 200$. (a) Compare the distributions comprehensively. (b) A student with $z$-score = 1.5 at School A transfers to School B with the same raw mark. What is their new $z$-score at School B? (c) Which school would you recommend for a risk-averse student vs a risk-tolerant student? Justify with statistics. (3 marks)

auto-saved

📖 Answers (click to reveal)

Drill 1: Mean = 26.7; Median = 26.5; Range = 30; IQR = Q3 − Q1 = 35 − 18 = 17. Roughly symmetric (mean ≈ median). Drill 2: 80 = mean + 1 SD → 16% above; 50 = mean − 2 SD → 2.5% below. 95 is 2.5 SD above mean, unusual. Drill 3: IQR = 50 − 20 = 30. Fences: 20 − 45 = −25 and 50 + 45 = 95. Max = 70 < 95 and min = 10 > −25, no outliers. Roughly symmetric (median 35 is midpoint of IQR 20–50). Drill 4: m = 10, b = 20; y = 10x + 20. At x = 4: y = 60 (interpolation, reliable). At x = 15: y = 170 (extrapolation, unreliable).

SA 1 (2 marks): (a) Mean = 23.3; Median = (22 + 25)/2 = 23.5; Range = 32; Q1 = 15, Q3 = 30, IQR = 15 [1]. (b) New mean = (233 + 100)/11 = 30.3; new median = 22 (6th of 11). Median is more robust, changed by 1.5 vs mean changed by 7 [1].

SA 2 (3 marks): (a) School A: higher mean (78 vs 75), smaller SD (5 vs 12), better typical performance AND more consistent. School B: lower average with wide variation [1]. (b) Raw mark = 78 + 1.5 × 5 = 85.5. New z-score at B = (85.5 − 75)/12 = 0.875. Student drops from well above average to moderately above average [1]. (c) Risk-averse: School A, more predictable, most students score 68–88 (within 2 SD). Risk-tolerant: School B, chance of exceptional result (above 99) but also risk of poor one (below 51) [1].

Boss battle · The Statistics Grand Master

earn bronze · silver · gold

Five timed questions spanning all Module 4 topics. This is the final boss, your chance to prove you've mastered Statistical Analysis. Beat the boss to bank a tier.

⚔ Enter the final arena

Science Jump · final module challenge

Climb platforms reviewing all Module 4 concepts. Pool: lesson 12.

Mark module as complete

Tick when you've finished the practice and review for the entire module.

← Lesson 11 · Line of Best Fit End of Module 4

Module overview · Maths Standard · Module quiz