ACJC 2026 Correlation and Linear Regression Lecture Notes

1 18 CORRELATION AND LINEAR REGRESSION SYLLABUS • Concepts of scatter diagram, correlation coefficient and linear regression • Understand that bivariate data consists of the values of 2 variables (independent and dependent variables) obtained from the same sample, expressed as ordered pairs; • Use a graphic calculator to plot the scatter diagram for a set of bivariate data to determine if there is a linear relationship between the 2 variables; • Understand that the correlation coefficient is a measure of the fit of a scatter diagram to a linear model • Calculation and interpretation of the product moment correlation coefficient and of the equation of the least squares regression line • Calculate the product moment correlation coefficient for a set of bivariate data using a graphic calculator, and relate the value (in particular, values close to −1, 0 and 1) to the appearance of the scatter diagram; • Understand that a high correlation between 2 variables does not necessarily imply one directly causes the other; • Understand the concepts of linear regression and ‘least squares’ with reference to the scatter diagram; • Use a graphic calculator to find the equation of the least squares regression line, and interpret its slope and intercept (A different line of regression will be obtained if we interchange the independent and dependent variables) • Interpolation and extrapolation

ACJC 2025/26 H2 Mathematics (9758) 2 • Understand the concepts of extrapolation and interpolation of data, and use the appropriate regression line to make prediction or estimate a value in practical situations • Use of square, reciprocal or log transformation to achieve linearity. • Use an appropriate transformation to linearise a set y a bx=+ of bivariate data to fit the regression model. CONTENTS 1 Introduction ...................................................................................... 3 2 Bivariate Samples ............................................................................. 3 3 Independent and Dependent Variables ............................................. 3 4 Scatter Diagrams .............................................................................. 4 5 Linear Correlation ............................................................................ 7 5.1 Calculating the value of Product Moment Correlation Coefficient, r ........................................................................... 7 5.2 Properties of r.......................................................................... 9 6 Linear Regression ........................................................................... 14 6.1 Least Squares Regression Line of y on x .............................. 15 6.2 Least Squares Regression Line of x on y .............................. 16 7 Estimation Using Least Squares Regression Lines ........................ 21 8 Linear Transformation .................................................................... 29 9 Miscellaneous Examples ................................................................ 32 Annex A: Gradient of Least Squares Regression Line .......................... 38 Annex B: Relationship between Correlation Coefficient and Least Squares Regression Lines ..................................................... 39

18 Correlation and Linear Regression 3 LECTURE 1 Lesson Outline • Introduction to bivariate samples with independent and dependent variables • Plot the scatter diagram using a graphing calculator • Calculate and interpret the linear product moment correlation coefficient 1 INTRODUCTION Previous chapters have focused on developing methods and models for a single random variable. However, there are many data sets that provide information about two variables and we are keen to find out the relationship between these variables, such as petrol consumption of a car and the speed at which it is driven, height of father and height of son when 18, or a student’s grade in mock examination and in national examination. In this chapter, we will study the correlation between two variables and in particular, to investigate whether there is linear relationship between two variables, which will then allow for possible predictions. 2 BIVARIATE SAMPLES Bivariate samples consist of members which have measurements of two different characteristics available. For example, we measure the heights and the weights of students in a particular class in ACJC. This type of data is called bivariate (as opposed to univariate) data. 3 INDEPENDENT AND DEPENDENT VARIABLES Independent variables and dependent variables refer to values that change in relationship to each other. The independent variables (controlled variables) are those that are deliberately manipulated to invoke a change in the dependent variables. The dependent variables (response variables) are those that are observed to change in response to the independent variables.

ACJC 2025/26 H2 Mathematics (9758) 4 Independent variable, x Dependent variable, y Number of people in a lift. Total weight of people in the lift. Amount of fertilizer used per unit field area. Weight of crop yield per unit field area. Hours of sleep the night before an examination. Examination score. 4 SCATTER DIAGRAMS Observations of variables X and Y from a bivariate sample can be considered as ordered pairs or coordinate pairs 1 1 2 2( , ),( , ),...,( , ) nnx y x y x y . These can then be plotted against axes that represent the characteristics measured. This diagram is called a scatter diagram. Conventionally, the independent variable is plotted on the horizontal axis while the dependent variable is plotted on the vertical axis. Real life examples of scatter diagrams: Source: https://www.mathsisfun.com/data/scatter-xy-plots.html Source: https://data36.com/scatter-plot-pandas-matplotlib/

18 Correlation and Linear Regression 5 Possible scatter diagrams and relationships Linear Relationship Curvilinear Relationship No Relationship The scatter diagram provides an overview of the relationship between variables. Often, given a data set, we draw the scatter diagram to observe whether there is linear relationship between the variables. When drawing a scatter diagram, • the axes must be labelled • the extreme values (minimum and maximum values) of each variable on the horizontal and vertical axes must be indicated on the diagram. The scatter diagram is also useful in identifying outliers, which are data points that appear to be quite different from the trend shown by the other data points in the diagram, and can be removed to provide a more accurate picture of the relationship between the variables.

ACJC 2025/26 H2 Mathematics (9758) 6 Example 1 The yield (per hectare) of a crop, c, is believed to depend on the May rainfall, m. Records for nine regions are kept for the average values of c and m, and these are recorded below. m 14.7 10.4 18.8 13.1 14.9 13.8 16.8 11.8 12.2 c 8.3 10.1 15.2 6.4 11.8 12.2 13.4 11.9 9.9 Sketch a scatter diagram for the data. [N1990/part] Solution 1) Press STAT and select 1: Edit… to access the lists. 2) Enter the data for variables m and c in the existing list names L1 and L2 for the independent and dependent variables respectively. c is the dependent variable (y-axis) and m is the independent variable (x-axis). 3) To plot, press [2ND] [STATPLOTS], turning on 1:Plot1. 4) Press [2ND] [1], [2ND] [2] to key in L1, L2 for the respective Xlist and Ylist. 5) To view the scatter diagram, press ZOOM and select 9:ZoomStat.

18 Correlation and Linear Regression 7 6) A scatter diagram with data points is obtained. To read the coordinates of each point on the scatter diagram, use TRACE. Copy the scatter diagram from the GC, label both axes, and the extreme x and y values. ■ Note • From the sca