Statistics
Page 5: Measures of Centre – Part A (Mean)
📊 What Are Measures of Centre?
Measures of centre are values describing the "centre" or "middle" of a dataset.
The three main measures of centre:
📊
Mean
📍
Median
🏆
Mode
📊 Arithmetic Mean – Raw Data
Mean = sum of all values divided by number of values
\(\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}\)
✏️ Example 1:
Scores of 5 students: 70, 85, 90, 75, 80
\(\bar{x} = \frac{70 + 85 + 90 + 75 + 80}{5} = \frac{400}{5} = 80\)
Mean: 80
💡 Notation:
- \(\bar{x}\) (x-bar) = sample mean
- \(\mu\) (mu) = population mean
⚖️ Weighted Mean – Frequency Table
When data are in a frequency table, use theMean weighted mean:
\(\bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} = \frac{\sum (x_i \cdot f_i)}{n}\)
✏️ Example 2: number of siblings of 30 students
| No. of siblings (x) | Frequency (f) | x · f |
|---|---|---|
| 0 | 3 | 0 × 3 = 0 |
| 1 | 12 | 1 × 12 = 12 |
| 2 | 10 | 2 × 10 = 20 |
| 3 | 4 | 3 × 4 = 12 |
| 4 | 1 | 4 × 1 = 4 |
| Total | n = 30 | Σ(xf) = 48 |
\(\bar{x} = \frac{48}{30} = 1.6\)
Mean: 1.6 siblings
📊 Mean for Grouped Data (Continuous)
When data are in classes, use theclass midpoint (\(x_i\)) is used as a representative:
\(\bar{x} = \frac{\sum (x_i \cdot f_i)}{n}\)
where \(x_i\) = class midpoint
✏️ Example 3: scores of 40 students
| Class | Midpoint (xᵢ) | Frequency (fᵢ) | xᵢ · fᵢ |
|---|---|---|---|
| 50-59 | 54.5 | 4 | 218 |
| 60-69 | 64.5 | 8 | 516 |
| 70-79 | 74.5 | 12 | 894 |
| 80-89 | 84.5 | 10 | 845 |
| 90-99 | 94.5 | 6 | 567 |
| Total | n = 40 | 3040 | |
\(\bar{x} = \frac{3040}{40} = 76\)
Mean: 76
⚠️ Note:
This is Mean an estimate (class midpoints used) - The exact values within each class are unknown, so the class midpoint is used.
📋 Properties of the Mean
| Property | Explanation | Example |
|---|---|---|
| Adding a constant | Add k to every value → mean increases by k | \(\bar{x} = 80\), add 5 → \(\bar{x}_{new} = 85\) |
| Multiplying by a constant | Multiply every value by k → mean is multiplied by k | \(\bar{x} = 80\), multiply by 2 → \(\bar{x}_{new} = 160\) |
| Sum of deviations | Sum of deviations from the mean = 0 | \(\sum (x_i - \bar{x}) = 0\) |
| Sensitive to outliers | Extreme values affect the mean | 10, 10, 10, 100 → mean = 32.5 |
⚖️ General Weighted Mean
✏️ Example 4: computing the final grade
Exam (60%): 85, Assignment (25%): 90, Attendance (15%): 100
\(\bar{x} = \frac{85 \times 60 + 90 \times 25 + 100 \times 15}{60 + 25 + 15}\)
\(\bar{x} = \frac{5100 + 2250 + 1500}{100} = \frac{8850}{100} = 88.5\)
Final grade: 88.5
✏️ Example 5: Mean of two groups
Class A: 25 students, mean 78
Class B: 30 students, mean 82
What is the overall mean?
\(\bar{x} = \frac{25 \times 78 + 30 \times 82}{25 + 30} = \frac{1950 + 2460}{55} = \frac{4410}{55} = 80.18\)
Overall mean: 80.18
Note: not (78+82)/2 = 80, because the groups are of different sizes!
💡 Exam Tips
Raw data: \(\frac{\sum x}{n}\)
Table: \(\frac{\sum xf}{n}\)
Grouped: use the class midpoint
📝 Summary – Page 5
\(\bar{x} = \frac{\sum x_i \cdot f_i}{n}\)
For grouped data: \(x_i\) = class midpoint