IB Mathematics AI SL Linear correlation of bivariate data MAI Study Notes - New Syllabus

IB Mathematics AI SL Linear correlation of bivariate data MAI Study Notes

LEARNING OBJECTIVE

Linear correlation of bivariate data.

Key Concepts:

Bivariate Data
Correlation Coefficients
Linear Regression

MAI HL and SL Notes – All topics

REGRESSION

We have a list of paired data. For example:

We assume that:
x is the independent variable (explanatory variable).
y is the dependent variable (response variable).

We can plot these points on a scatter diagram:

A parameter r, called the correlation coefficient (Pearson’s product-moment correlation coefficient), measures the strength and direction of this relationship. It ranges from:
$ -1 \leq r \leq 1 $

r ≈ 1: Strong positive linear relationship.
r ≈ -1: Strong negative linear relationship.
r ≈ 0: No linear relationship.

There is also a regression line (line of best fit) of the form:
$ y = ax + b $
This line minimizes the sum of the squared differences between the observed and predicted values.

USE OF GDC (Casio CFX)
To find a, b, and r:
1. MENU → STAT
2. Enter x-values in List 1 and y-values in List 2.
3. CALC → REG → X → aX + b

Example

Given:

a = 9.83 (slope)
b = 23.1 (y-intercept)
r = 0.99 (very strong positive correlation)

Find Equation of line.

▶️Answer/Explanation

Solution:

Thus, the regression line is: $ y = 9.83x + 23.1 $

Predicting Values
Interpolation: Predicting within the range of data.
For x = 18:
$ y = 9.83 \times 18 + 23.1 ≈ 200 $
Extrapolation: Predicting outside the range (less reliable).
For x = 40:
$ y = 9.83 \times 40 + 23.1 ≈ 416 $

Note: To predict x from y, we need a different regression line (x on y).

CHARACTERISTICS OF THE REGRESSION LINE

1. Passes through the point $(\bar{x}, \bar{y})$:
$\bar{x} = \text{mean of } x = 19.7$
$\bar{y} = \text{mean of } y = 216.9$
The line passes through (19.7, 216.9).

2. Splits points into two halves:
~50% of points lie above, ~50% below the line.

CHARACTERISTICS OF THE CORRELATION COEFFICIENT (r)

The correlation between x and y is characterised according to the value of r as follows: