Linear Correlation

This page is all about looking for correlations in bivariate data (to see if there is a relationship between x and y variables). You will learn how to calculate r, Pearson’s product-moment correlation coefficient and how to interpret it. We will look at y on x regression as well as x on y regression using them to consider interpolation and extrapolation.


Key Concepts

On this page, you should learn about

  • linear correlation of bi-variate data
  • scatter diagrams
  • y on x regression line
  • x on y regression line
  • interpolation and extrapolation
  • Pearson' product-moment correlation coefficient, \(r\)

Summary

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Test Yourself

Here is a quiz that gets you to estimate the r value for different graphs


START QUIZ!

Here is a quiz that tests your understanding of linear correlation


START QUIZ!

Exam-style Questions

Question 1

The world record times in seconds for the women’s 100m sprint from 1970 onwards are given below

Use your calculator to write down

a) \(\bar{x}\) , the mean year

b) \(\bar{y}\) , the mean time

c) \(r\), Pearson’s product-moment correlation coefficient

The equation of the regression line y on x is y = ax + b

d) Find the values of a and b for these data

e) Show that \((\bar{x},\bar{y})\) lies on this line

f) Use the regression line to estimate the world record time in 2024


Hint

Full Solution

 

Question 2

The table below shows the test scores of SL students following a six-week revision period using studyib.

a) Work out r, Pearson’s correlation coefficient

The y on x regression line is y = ax + b

b) Find a and b

c) A student scores 80 marks before revision. Use the regression line to estimate the score after revision

The x on y regression line is x = cx + d

d) Find c and d

e) A student scores 90 marks after revision. Use the regression line to estimate the score before revision

f) Find the point of intersection of the y on x and the x on y regression lines


Hint

Full Solution

 

Question 3

The following table shows the hand spans and the heights of eight basketball players on a team

The relationship between x and y can be modelled by the x on y line of regression x = ay + b

  • Find the values of a and b
  • Write down the correlation coefficient
  • Another basketball player is 193cm tall. Use this regression line to estimate the handspan of this player.

Another player is 180cm tall. Use this regression line to estimate the handspan of this player.


Hint

Full Solution

 

Question 4

The following table shows the age in days (x) and length of new-born babies in cm (y).

The relationship between the variables is modelled by the regression line with equation y = ax + b

a) Find the values of a and b

b) Write down the correlation coefficient

c) Use your equation to estimate the length of a baby that is 40 days old

d) Use your equation to estimate the length of a baby that is 150 days old

e) Use your equation to estimate the age of a child that is 54 cm long.


Hint

Full Solution

 

MY PROGRESS

How much of Linear Correlation have you understood?