← Unit 4 Data and Probability Review

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MA5-DAT-C-01

Data and Probability Review

⏱ 25 min📚 Year 9📈
Think First

A survey of 100 students shows mean study time = 5 hours, median = 4 hours. What does this suggest? Is the sample representative if only Year 9 students were asked?

💡 Revision: Ensure you can calculate mean, median, mode, probability, and interpret data displays.

Learning Intentions

Know

  • Mean vs median
  • Sampling bias
  • Probability rules
  • Data interpretation

Understand

  • When to use each measure of centre
  • How bias affects conclusions
  • When to apply addition and multiplication rules

Can Do

  • Calculate and compare measures of centre
  • Identify bias in data collection
  • Solve multi-step probability problems
  • Interpret box plots and Venn diagrams
MeanMedianBiasUnionIntersection
Learn Phase
1

Choosing Measures of Centre

Which is best?

Guidelines for choosing:

  • Mean: Best for symmetric data without outliers
  • Median: Best for skewed data or data with outliers
  • Mode: Best for categorical data or finding most common value

Mean > Median suggests right skew (tail to the right).

Mean < Median suggests left skew (tail to the left).

2

Probability Rules Summary

Key formulas

Essential rules:

Probability Rules

$P(A') = 1 - P(A)$

$P(A cup B) = P(A) + P(B) - P(A cap B)$

$P(A ext{ then } B) = P(A) imes P(B|A)$

For independent events: $P(A cap B) = P(A) imes P(B)$

Mutually exclusive: $P(A cap B) = 0$

3

Data Collection Checklist

Evaluating studies

When evaluating a statistical study:

  1. Was the sample representative?
  2. Was the sampling method appropriate?
  3. Were questions unbiased?
  4. Was the sample size adequate?
  5. Are conclusions supported by the data?

Be sceptical of claims based on small, biased, or unrepresentative samples.

✓

Check Understanding

Try it yourself

In a class of 40, 25 passed Maths, 20 passed Science, and 12 passed both. Find P(passed at least one subject) and P(passed neither).

Worked Example

Data and Probability Review

1

Data: 2, 5, 7, 8, 10, 15, 20. Find mean, median, mode, range, IQR.

Mean = 67/7 ≈ 9.57

Median = 8

Mode = none

Range = 18

Q1 = 5, Q3 = 15, IQR = 10

2

A bag has 3 red, 4 blue, 3 green. Two drawn without replacement. Find P(both blue).

P(first blue) = 4/10

P(second blue) = 3/9

P(both blue) = 4/10 × 3/9 = 12/90 = 2/15

3

A survey asks leading questions at a shopping centre during work hours. Identify three sources of bias.

1. Leading questions (response bias)

2. Shopping centre location excludes non-shoppers (selection bias)

3. Work hours excludes working people (selection bias)

Common Misconceptions

✕

Mean is always the best measure. No — median is better for skewed data or data with outliers.

✕

P(A or B) = P(A) + P(B) always. Only for mutually exclusive events. Otherwise subtract the overlap.

✕

A large sample guarantees accurate results. A large biased sample is worse than a small representative one.

Your Turn

Practice — Data and Probability

Work through each question in your book or digitally. Answers are in the Questions phase.

1For data 4, 6, 8, 10, 12, 100, which measure of centre is most appropriate? Justify.
2A card is drawn from a deck, then a coin tossed. Find P(ace and heads).
3A school survey is conducted by asking the first 30 students arriving. Identify the bias.
Real-World Anchor

Social Science and Policy

The Australian Bureau of Statistics publishes guidelines for evaluating statistical claims. Understanding sampling bias and appropriate measures helps citizens critically evaluate media reports, political claims, and advertising that use statistics.

📓 Copy Into Your Books

▼

Measures

  • Mean - symmetric data
  • Median - skewed/outliers
  • Mode - categorical

Probability

  • Complement: 1 − P(A)
  • Addition: P(A)+P(B)−P(A∩B)
  • Multiplication: P(A) × P(B|A)

Bias check

  • Representative sample?
  • Appropriate method?
  • Unbiased questions?
  • Adequate size?
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Best for skewed data:
Mean > Median indicates:
P(A∪B) for mutually exclusive:
Asking only friends is:
Independent events: P(A∩B) =
Box plot shows min=5, Q1=10, med=15, Q3=20, max=30. IQR=
P(neither A nor B) =
A large biased sample is:
1Find mean, median, IQR for: 5, 8, 10, 12, 15, 18, 25. Describe the skew.
2Two dice rolled. Find P(sum is even) using a tree diagram or sample space.
3Explain why online polls may not represent public opinion.

Comprehensive Answers

1Mean, median, IQR for 5,8,10,12,15,18,25.
Mean=13.3, Median=12, Q1=8, Q3=18, IQR=10. Right skew (mean > median).
2P(sum even) with two dice.
18/36 = 1/2. (Both even or both odd: 3×3 + 3×3 = 18 outcomes).
3Why online polls unrepresentative.
Self-selection bias - only motivated people respond. Not random or representative.
MC 1Best for skewed.
Median. Answer: B
MC 2Mean > median.
Right skew. Answer: B
MC 3P(A∪B) mutually exclusive.
P(A)+P(B). Answer: A
MC 4Asking friends.
Convenience sampling. Answer: C
MC 5Independent P(A∩B).
P(A)×P(B). Answer: B
MC 6IQR from box plot.
20−10=10. Answer: B
MC 7P(neither A nor B).
1−P(A∪B). Answer: A
MC 8Large biased sample.
Worse than small representative. Answer: B
SA 1Stats and skew.
Mean=13.3, Median=12, IQR=10. Right skew.
SA 2P(sum even).
1/2.
SA 3Online polls.
Self-selection bias.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.