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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?
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Before you review, rate your confidence

+5 XP warm-up Rate your confidence in each key area before reviewing (1 = shaky, 5 = solid).

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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.

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Module formula and rule summary

+5 XP to read Every formula and rule you need for Module 4 Statistics assessments:

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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.
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The complete Module 4 story

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

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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.
Beyond the syllabus. This page reviews the old twelve-lesson Statistical Analysis module, so the last row above sits outside Data analysis (MST-11-08). Bivariate work, Scatter plots, measuring the strength of an association, predicting from a fitted straight line, lurking variables and causation, and the normal-distribution percentage rule in the row before it, is Year 12 material and will not be examined in Year 11. Review everything else on this page, because that is what you are assessed on: classifying data, sampling and bias, frequency tables and histograms, mean, median and mode, range and population standard deviation, quartiles and box plots, and comparing two data sets.
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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.
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The five pitfalls that cost marks in assessments

Quick check: Data set: 8, 12, 15, 18, 22, 25, 28, 30, 35, 40.

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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?

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Examination tips, maximising your marks

We just saw five pitfalls that cost marks, using mean for skewed data, quoting one statistic, omitting context, reversing skew...

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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%.

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

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Revision activities

Revision activities

Revision activities

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

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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?

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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.

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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:

  • Skewed distribution with outliers
  • Predicting within data range
  • r = 0.85 between two variables
  • Comparing two groups
  • Very strong positive, check causation
  • Centre, spread, AND shape
  • Use median, not mean
  • Interpolation, reliable

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

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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.

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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?

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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

Pick your answer, then rate your confidence.

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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)

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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)

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📖 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].