Module 4: Probability, Statistics & Data Handling
Master Averages, Histograms, Cumulative Frequency & Probability Trees
Probability and Statistics make up 15% of both Foundation and Higher papers. These topics offer predictable, high-scoring calculation and graph drawing marks.
1. Statistical Averages & Spread
| Measure | Definition | How to Calculate |
|---|---|---|
| Mean | Arithmetic average of all values. | Mean = ∑x / N |
| Median | Middle value when ordered. | Position = (N + 1) / 2 |
| Mode / Modal Class | Most frequently occurring value. | Look for highest frequency count. |
| Range | Measure of spread. | Max Value - Min Value |
| Interquartile Range (IQR) | Spread of middle 50% of data. | Upper Quartile (UQ) - Lower Quartile (LQ) |
2. Histograms & Cumulative Frequency
Histograms with Unequal Class Widths
Unlike standard bar charts, the area of each bar on a histogram represents frequency.
Frequency Density = Frequency ÷ Class Width
Frequency = Frequency Density × Class Width
When drawing a Cumulative Frequency graph, plot points at the UPPER CLASS BOUNDARY of each interval (not the midpoint!). Join points with a smooth S-shaped curve.
3. Probability & Tree Diagrams
Probabilities always lie between 0 (impossible) and 1 (certain).
- AND Rule (Independent Events): Multiply probabilities along branches: P(A and B) = P(A) × P(B).
- OR Rule (Mutually Exclusive Events): Add probabilities across branches: P(A or B) = P(A) + P(B).
- Complementary Rule: P(NOT A) = 1 - P(A).
Conditional Probability without Replacement
If picking 2 counters from a bag containing 5 red and 4 blue (total 9):
- First pick Red: P(R₁) = 5/9.
- Second pick Red: P(R₂ | R₁) = 4/8 = 1/2.
- P(Both Red) = 5/9 × 4/8 = 20/72 = 5/18.
Frequently Asked Questions
Frequency Density = Frequency ÷ Class Width. On a histogram, the AREA of each bar represents the frequency, not the height!
Step 1: Find the midpoint (x) of each class interval. Step 2: Multiply each midpoint by its frequency (f × x). Step 3: Sum the (f × x) column. Step 4: Divide total (f × x) by total frequency (∑f).
When items are taken 'without replacement', the total number of remaining items decreases by 1 for the second pick (e.g. 5/10 becomes 4/9). Always update both numerator and denominator!