IB MYP 4-5 Maths- Correlation, quantitative handling, using technology- Study Notes - New Syllabus
IB MYP 4-5 Maths- Correlation, quantitative handling, using technology – Study Notes
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- Correlation, quantitative handling, using technology
IB MYP 4-5 Maths- Correlation, quantitative handling, using technology – Study Notes – All topics
Correlation: Quantitative Handling
Correlation: Quantitative Handling
Correlation measures the strength and direction of a relationship between two quantitative variables. It does not imply causation, only association.
Types of Correlation:
| Type | Description | Scatter Plot Pattern |
|---|---|---|
| Positive | As one variable increases, the other increases |
Points rise from left to right |
| Negative | As one variable increases, the other decreases |
Points fall from left to right |
| No Correlation | No predictable relationship |
Points scattered randomly |
Quantitative Measures of Correlation:
1. Pearson’s Correlation Coefficient (\( r \))
Pearson’s correlation coefficient measures the strength and direction of the linear relationship between two continuous variables.
Formula:
\( r = \dfrac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sqrt{\sum (x_i – \bar{x})^2 \cdot \sum (y_i – \bar{y})^2}} \)
Alternative Formula (Using sums):
\( r = \dfrac{n\sum xy – (\sum x)(\sum y)}{\sqrt{\left[n\sum x^2 – (\sum x)^2\right]\left[n\sum y^2 – (\sum y)^2\right]}} \)
Interpretation of \( r \):
- \( r = +1 \): Perfect positive correlation
- \( r = -1 \): Perfect negative correlation
- \( r = 0 \): No linear correlation
- \( 0 < |r| < 0.3 \): Weak, \( 0.3 \le |r| < 0.7 \): Moderate, \( |r| \ge 0.7 \): Strong
Spearman’s Rank Correlation (\( \rho \) or \( r_s \))
Spearman’s rank correlation measures the strength and direction of a monotonic relationship between two variables using their ranks (used when data is ordinal or not normally distributed).
Formula:
\( r_s = 1 – \dfrac{6\sum d^2}{n(n^2 – 1)} \)
Where:
- \( d \) = difference between ranks of each pair
- \( n \) = number of data pairs
Comparison:
- Pearson’s r: For continuous, normally distributed data (linear relationship).
- Spearman’s rank: For ranked/ordinal data or non-linear relationships.
Using Technology (Calculator & Excel):
Steps on Casio Scientific Calculator (fx-991ES, fx-100MS, etc.):
- Press MODE → Select STAT.
- Choose 2-VAR (for two-variable statistics).
- Enter X-values in the first column and Y-values in the second column.
- Press AC, then SHIFT → STAT → Reg → r.
- The calculator displays the correlation coefficient \( r \).
Steps in Excel/Google Sheets:
- Enter X-values in column A and Y-values in column B.
- In a new cell, type:
=CORREL(A1:A5, B1:B5)and press Enter. - Excel displays the correlation coefficient \( r \).
Example:
The table shows hours studied (X) and test scores (Y) for 5 students:
| X | Y |
|---|---|
| 2 | 50 |
| 4 | 60 |
| 6 | 70 |
| 8 | 80 |
| 10 | 90 |
▶️Answer/Explanation
Step 1: Enter data in calculator under 2-VAR mode.
Step 2: Find \( r \) using SHIFT → STAT → Reg → r.
Step 3: \( r = 1.0 \) (Perfect positive correlation).
Using Excel: Use =CORREL(A1:A5,B1:B5) → Output = 1.0
Example:
The table shows temperature (°C) and ice cream sales:
| Temperature | Sales |
|---|---|
| 20 | 100 |
| 25 | 150 |
| 30 | 200 |
| 35 | 250 |
▶️Answer/Explanation
Calculator: Enter data in 2-VAR → r ≈ 1.0.
Excel: CORREL() → Output ≈ 1.0.
Interpretation: Strong positive correlation between temperature and sales.
Example:
The table shows the marks of 5 students in Math (x) and Science (y):
| x (Math) | y (Science) |
|---|---|
| 10 | 20 |
| 20 | 40 |
| 30 | 50 |
| 40 | 70 |
| 50 | 90 |
▶️Answer/Explanation
Step 1: Compute sums: \( n = 5, \sum x = 150, \sum y = 270, \sum xy = 9800, \sum x^2 = 5500, \sum y^2 = 16600 \)
Step 2: Apply formula: \( r = \dfrac{5(9800) – (150)(270)}{\sqrt{\left[5(5500) – (150)^2\right]\left[5(16600) – (270)^2\right]}} \)
\( r = \dfrac{49000 – 40500}{\sqrt{(27500 – 22500)(83000 – 72900)}} \)
\( r = \dfrac{8500}{\sqrt{5000 \times 10100}} \approx \dfrac{8500}{\sqrt{50,500,00}} \approx 0.95 \)
Interpretation: Strong positive correlation.
Example:
The table shows ranks of 6 students in Math and English:
| Student | Math Rank | English Rank |
|---|---|---|
| A | 1 | 2 |
| B | 2 | 3 |
| C | 3 | 1 |
| D | 4 | 4 |
| E | 5 | 5 |
| F | 6 | 6 |
▶️Answer/Explanation
Step 1: Compute \( d = \text{Math Rank} – \text{English Rank} \) and \( d^2 \):
A: -1 → 1, B: -1 → 1, C: 2 → 4, D: 0 → 0, E: 0 → 0, F: 0 → 0
\( \sum d^2 = 6 \)
Step 2: Apply formula: \( r_s = 1 – \dfrac{6(6)}{6(6^2 – 1)} = 1 – \dfrac{36}{6(35)} = 1 – \dfrac{36}{210} = 1 – 0.1714 = 0.8286 \)
Interpretation: Strong positive correlation between ranks.