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IGCSE MATH: STATISTICS
STATISTICS LEARNER SUCCESS NOTES
Cambridge IGCSE Mathematics (0580)
TABLE OF CONTENTS
- 1. Measures of Central Tendency (Mean, Median, Mode)
- 2. Measures of Spread (Range, Interquartile Range)
- 3. Scatter Diagrams and Correlation
- 4. Line of Best Fit
- 5. Stem-and-Leaf Diagrams
- 6. Frequency Tables and Grouped Data
- 7. Cumulative Frequency Diagrams
- 8. Histograms and Frequency Density
- 9. Pie Charts and Sector Angles
- 10. Box-and-Whisker Plots
- 11. Probability from Data
1. MEASURES OF CENTRAL TENDENCY
MEAN (Average)
Definition: Sum of all values divided by the number of values.
Formula: Mean = Σx ÷ n | Mean (frequency) = Σ(f·x) ÷ Σf | Mean (grouped) = Σ(f × midpoint) ÷ Σf
Example 1: 7, 9, 20, 3, 9
Mean = (7+9+20+3+9) ÷ 5 = 48 ÷ 5 = 9.6
Example 2 (frequency table):
| Number of pets | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Frequency | 1 | 2 | 3 | 5 | 7 | 3 | 3 |
Mean = (0×1 + 1×2 + 2×3 + 3×5 + 4×7 + 5×3 + 6×3) ÷ 24 = 84 ÷ 24 = 3.5 pets
Example 3 (grouped data):
| Height (h cm) | 2 < h ≤ 6 | 6 < h ≤ 10 | 10 < h ≤ 13 | 13 < h ≤ 17 |
|---|---|---|---|---|
| Frequency | 23 | 47 | 45 | 38 |
Midpoints: 4, 8, 11.5, 15 Σ(f×mid) = 23×4 + 47×8 + 45×11.5 + 38×15 = 1555.5 Σf = 153 → Mean = 1555.5 ÷ 153 ≈ 10.2 cm
MEDIAN (Middle Value)
Definition: The middle value when data is arranged in order.
Position: (n+1)÷2 th value.
Example 1: 7,9,20,3,9 → order: 3,7,9,9,20 → median = 9 Example 2: 0.3,0.4,1.2,0.8,1.1,2.1 → order: 0.3,0.4,0.8,1.1,1.2,2.1 → median = (0.8+1.1)/2 = 0.95 kg
MODE (Most Common)
Definition: The value that appears most frequently.
Example: 7,9,20,3,9 → mode = 9 (twice)
Modal class: class with highest frequency (grouped data).
When to use each measure
- Mean: No extreme outliers; all values used; numerical & fairly symmetric.
- Median: Outliers present; skewed data; middle typical value.
- Mode: Categorical data; most common item; discrete.
2. MEASURES OF SPREAD
RANGE
Range = Highest value – Lowest value
Data: 7,9,20,3,9 → range = 20–3 = 17
INTERQUARTILE RANGE (IQR)
IQR = Q3 – Q1
Quartile positions: Q1 = ¼(n+1)th, Q3 = ¾(n+1)th (or from cumulative frequency: n/4 and 3n/4).
Example (n=16): Q1 = 4th value, Q3 = 12th value; IQR = Q3–Q1
3. SCATTER DIAGRAMS AND CORRELATION
Types: Positive (upward), Negative (downward), Zero/No correlation, Strong/Weak.
Drawing: Crosses (×), label axes, do NOT join points.
4. LINE OF BEST FIT
Straight line through scatter: equal points above/below, passes through mean point.
Prediction: Interpolation (within data) – reliable; Extrapolation (outside) – less reliable.
5. STEM-AND-LEAF DIAGRAMS
Shows individual values, shape, median, mode. Key is essential.
Masses (kg): 0.3,0.4,1.2,0.8,1.1,2.1,1.7,1.8,1.2,2.3,0.7,1.1 0 | 3 4 7 8 1 | 1 1 2 2 7 8 2 | 1 3 Key: 0|3 = 0.3 kg
Median: 12 values → between 6th & 7th = (1.1+1.2)/2 = 1.15 kg
Mode: 1.1 & 1.2 (bimodal)
Back-to-back stem-and-leaf
Girls | | Boys
3 1| 15 | 8 9
6 5 3 2| 20 | 3 5 6 9
9 8 6 3 0 | 25 | 0 2 3 8
Key: 2|20|3 = 22 min (girls), 23 min (boys)
6. FREQUENCY TABLES AND GROUPED DATA
Class width = upper boundary – lower boundary
Midpoint = (lower + upper) ÷ 2
Modal class = class with highest frequency
Midpoint = (lower + upper) ÷ 2
Modal class = class with highest frequency
Finding median class: total n ÷ 2, add frequencies until surpass position.
7. CUMULATIVE FREQUENCY DIAGRAMS
Running total; plot upper boundaries; join with smooth curve.
| Height (h cm) | h ≤ 6 | h ≤ 10 | h ≤ 13 | h ≤ 17 | h ≤ 20 |
|---|---|---|---|---|---|
| Cumulative freq | 23 | 70 | 115 | 153 | 185 |
Reading: median = n/2, Q1 = n/4, Q3 = 3n/4, percentiles = (p/100)×total.
Example: total = 200 → 30th percentile position = 0.30×200 = 60 → read value from axis.
8. HISTOGRAMS AND FREQUENCY DENSITY
Frequency Density = Frequency ÷ Class Width
Bars touch; area = frequency.
| Mass (m grams) | 496 < m ≤ 500 | 500 < m ≤ 504 | 504 < m ≤ 508 | 508 < m ≤ 510 |
|---|---|---|---|---|
| Frequency | 16 | 74 | 104 | 6 |
| Class width | 4 | 4 | 4 | 2 |
| Freq density | 4 | 18.5 | 26 | 3 |
9. PIE CHARTS AND SECTOR ANGLES
Sector angle = (frequency ÷ total) × 360°
| Flavour | Vanilla | Strawberry | Chocolate | Mango |
|---|---|---|---|---|
| Sold | 6 | 8 | 9 | 7 |
| Angle | 72° | 96° | 108° | 84° |
Check total = 360°; reverse: frequency = (angle/360°) × total.
10. BOX-AND-WHISKER PLOTS
Five-number summary: Min, Q1, Median, Q3, Max.
|----[======|======]----| ↑ ↑ ↑ ↑ ↑ Min Q1 Med Q3 Max
IQR = Q3–Q1; median line inside box; whiskers to min/max.
Interpretation: symmetry, spread, compare medians and IQRs.
11. PROBABILITY FROM DATA
P(event) = frequency of event ÷ total frequency
Eggs: total 300, Very large = 81 → P = 81/300 = 27/100 = 0.27
From cumulative frequency: P(X > k) = (total – CF at k) / total.
QUICK REFERENCE FORMULAS
| Measure | Formula |
|---|---|
| Mean (raw) | Σx ÷ n |
| Mean (frequency) | Σ(f·x) ÷ Σf |
| Mean (grouped) | Σ(f·midpoint) ÷ Σf |
| Median position | (n+1)/2 th value |
| Range | Max – Min |
| IQR | Q3 – Q1 |
| Q1 position | n/4 (CF diagram) |
| Q3 position | 3n/4 (CF diagram) |
| Class width | Upper – lower boundary |
| Midpoint | (lower+upper)/2 |
| Frequency density | Frequency ÷ class width |
| Sector angle | (freq/total) × 360° |
| nth percentile (CF) | position = (n/100)×total |
| Probability | favorable / total |
EXAM TECHNIQUES AND COMMON ERRORS
General tips
- Show all working – method marks.
- Use appropriate accuracy (2–3 sf), include units.
- Check if answer makes sense.
Common errors
- Grouped mean: use midpoint, not boundaries.
- Median: order data first; even n = average of two middles.
- Cumulative frequency: plot at upper boundaries; smooth curve.
- Histogram: freq density when unequal widths; bars touch.
- Pie chart: angles sum to 360°.
- Scatter: do not join points; line of best fit balanced.
- Stem-and-leaf: ordered leaves; include key.
PRACTICE QUESTIONS WITH SOLUTIONS
Question 1: Mean and Median
Data: 7, 9, 20, 3, 9
(a) Mean = 48/5 = 9.6 (b) Median = 9 (c) Remove 9 (median stays 9, range 20-3=17)
Question 2: Grouped Data Mean
| Height (h cm) | 2 < h ≤ 6 | 6 < h ≤ 10 | 10 < h ≤ 13 | 13 < h ≤ 17 |
|---|---|---|---|---|
| Frequency | 23 | 47 | 45 | 38 |
Midpoints: 4,8,11.5,15 Σ(f×mid)=1555.5, Σf=153 → mean ≈ 10.2 cm
Question 3: Pie Chart
Walk:12, Car:7, Bicycle:9, Bus:4 → total 32. Walk angle = (12/32)×360° = 135°.
Question 4: Cumulative Frequency
n=80, median=35, Q1=25, Q3=45 → IQR=20. CF at 48 =72 → >48 = 8 students.
Question 5: Frequency Density
| Mass (g) | 0<m≤0.5 | 0.5<m≤1.5 | 1.5<m≤3 |
|---|---|---|---|
| Freq | 20 | 18 | 9 |
Class widths: 0.5, 1, 1.5 → Freq density: 40, 18, 6
Question 6: Scatter & line of best fit
Positive correlation; at 5.5h → estimate ≈ 38 marks.
Question 7: Stem-and-leaf
0 | 3 4 7 8 1 | 1 1 2 2 7 8 2 | 1 3 Key: 1|2 = 1.2 kg → median = 1.15 kg
FINAL CHECKLIST FOR EXAM SUCCESS
- ✓ Know all formulas by heart
- ✓ Practise all diagram types: scatter, stem, box, CF, histogram, pie
- ✓ Show working – method marks
- ✓ Use ruler for straight lines, protractor for pie
- ✓ Label axes, include keys, check angles sum to 360°
- ✓ Read questions twice – underline command words
- ✓ Manage time: easy questions first
Good luck with your Statistics exam! 📊📈
End of Statistics Learner Success Notes