2.01 Bivariate data and line of best fit

In the previous chapter, we learnt to describe the relationship between numerical variables. If there is a relationship, we can recognise when it is linear or non-linear, strong or weak, and positive or negative.

We can draw a line, by eye, that best fits the points in a scatter plot. The images below show some examples of a line of best fit. The line of best fit does not necessarily go through the data points. It is positioned so that it minimises the overall distance from the points.

Constructing a line of best fit in this way can be very useful because the line summarises all of the data points in a way that allows us to predict the value of the response (dependent) variable that corresponds to a given value of the explanatory (independent) variable, or vice-versa. However, it takes a certain amount of judgement to choose the position and angle of the line. Constructing the line by eye is not very reliable, especially if the scatter plot does not show a strong relationship.

Fortunately, there are mathematical calculations that we can use to determine a line of best fit with more consistency. In this lesson, we will learn to construct a least-squares regression line.

The least-squares regression line is a straight line, so it can be represented by a linear function, in gradient-intercept form: y=mx+c, where y is the predicted value of the response (dependent) variable, x s our explanatory (independent) variable, m is the gradient of our line, and c is the vertical intercept (or y-intercept) of our line.

It is common practice to use the symbol \hat{y} (which is pronounced "y-hat"), instead of y, as the variable for the predicted value of the response variable.

Unless we are working with a very small data set, the calculations to determine the least-squares regression line are too tedious to do by hand. In this course, we will use technology for these calculations.

Examples

Now that we can determine the equation of the least-squares regression line, we are able to make important observations about the original data. When we are analysing data, it is important that we consider the context.

In the calculator examples above, y represents the vocabulary (the response variable) and x represents the number of pages read per day (the explanatory variable).

We found the least squares regression line in the form y=mx+c, to be y=14.87x+30.26.

The m value displayed is the gradient of the least-squares regression line. Since this is a positive number, it indicates that there is a positive relationship between the variables. In the context of this example, this tells us that for each additional page of reading per day, a child's vocabulary increases by approximately 15 words.

The c value provided is the vertical intercept of the line of best fit and, in the context of this example, tells us that a child that does no reading would have a vocabulary of approximately 30 words.

Given a least squares squares regression line of the form y=mx+c:

The m value shows the gradient:

• if the gradient is positive, when the explanatory increases by 1 unit, the response variable increases by m units.

• if the gradient is negative, when the explanatory increases by 1 unit, the response variable decreases by m units.

The c value shows the vertical intercept (also known as the y-intercept):

• when the explanatory variable is 0, the value of the response variable is c.

When we interpret the vertical intercept, we need to consider if it makes sense for the explanatory variable to be zero and the response variable to have a value indicated by c.

Examples

The correlation coefficient was introduced in the previous chapter as a measure that tells us the strength of a relationship between two variables. It is denoted by the letter r. The sign of r also tells us the direction of the relationship.

A perfect positive correlation has a value of 1. That means that if we graphed the variables on a scatter plot, it would show that all the data points lie exactly on a straight line with a positive gradient.

A perfect negative correlation has a value of -1 in which case all data points lie exactly on a straight line with a negative gradient.

A value of 0 indicates that there is no relationship between the variables.

To be more descriptive about the relationship between variables, we can further divide up the line to indicate other values with descriptions like "weak", "moderate" and "strong", in the figure below.

The strength of correlation depends on the size of the r value, so we can ignore the positive or negative sign:

• if the size is less than 0.5, then we have a weak correlation;

• if the size of the correlation is greater than 0.8, then we have a strong correlation.

• if the size is between 0.5 and 0.8, then we a moderate correlation.

Examples

Example 3

Given the following data:

x	1	4	7	10	13	16	19
y	4	4.25	4.55	4.4	4.45	4.75	4.2

a

Calculate the correlation coefficient and give your answer to two decimal places.

Worked Solution

Create a strategy

Use technology to find the correlation coefficient.

Apply the idea

Using the Statistics mode in your calculator, enter each x-value along with its y-value into a data table on your calculator then find the linear regression.

Look for the correlation coefficient (r):r=0.47

Watch question walkthrough

b

Choose the best description of this correlation.

A

Weak negative

B

Strong positive

C

Moderate negative

D

Weak positive

E

Strong negative

F

Moderate positive

Worked Solution

Create a strategy

Use the figure below to identify the best description of the correlation:

Apply the idea

The value of r=0.47 is positive, and between 0 and 0.5. So there is a weak positive correlation between the variables.

The correct answer is D.

Watch question walkthrough

When we discuss the coefficient of determination or the value of r^2, we can already tell that it must be related to the value of the correlation coefficient (r) and something to do with measuring the relationship between two variables.

r^2 tells us the proportion of the response variable (y) that can be explained by the variation in the explanatory variable (x).

For example, if r^2=0.92 then we can say that 92\% of the variation in the response variable is explained by the variation in the explanatory variable.

Alternatively, we can say that 8\% of the variation in the response variable is not explained by the variation in the explanatory variable.

Unlike the correlation coefficient, the coefficient of determination is always a positive value, so it does not indicate if there is a positive or negative relationship between variables.

The closer r^2 is to 1, the more the variation in the response variable is explained by the variation in the explanatory variable.

If we already have the value of r, we can square the value to get r^2.

So if r=0.8, then r^2=0.64, and if r=-0.9, then r^2=0.81.

If we don't already have the value of r, our calculators will calculate it for us. In fact, the r^2 value is given on the same screen as the r value.

Just as with r values when two variables have a strong relationship, that is r^2 is close to 1, we cannot say that one variable causes change in the other variable.

Examples

Example 4

A linear association between two data sets is such that the coefficient of determination, r^2, is 0.80.

a

Calculate the correlation coefficient if the relationship is negative. Give your answer to four decimal places.

Worked Solution

Create a strategy

Take the negative square root of the coefficient of determination.

Apply the idea

Remember that the relationship in this example is negative.

\displaystyle r	\displaystyle =	\displaystyle -\sqrt{0.80}	Take the negative square root of r^2
	\displaystyle =	\displaystyle -0.8944	Evaluate

b

Hence choose the option that best describes the strength of the relationship.

A

Weak

B

Moderate

C

Strong

Worked Solution

Create a strategy

Use the figure below to identify the best description of the correlation:

Apply the idea

The correlation coeffficient is close to -1, so there is a strong relationship.

The correct answer is C.

Example 5

A linear association between two data sets is such that the correlation coefficient is -0.72.

What proportion of the variation can be explained by the linear relationship? Give your answer to the nearest percent.

Worked Solution

Create a strategy

The proportion of variation is also the same as the coefficient of determination. So, we square the correlation coefficient.

Apply the idea

\displaystyle r^2	\displaystyle =	\displaystyle (-0.72)^2	Square -0.72
	\displaystyle =	\displaystyle 0.52	Evaluate and round up

So 52\% of the variation can be explained by the linear relationship.

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Example 6

The heights (in \text{cm}) and the weights (in \text{kg}) of 8 primary school children is shown on the scattergraph below.

110

115

120

125

130

135

140

145

\text{Height}

45

50

55

60

\text{Weight}

a

Calculate the value of the coefficient of determination. Give your answer to two decimal places.

Worked Solution

Create a strategy

Use the linear regression function on your calculator.

Apply the idea

Using the Statistics mode, enter each x-coordinate along with its y-coordinate into a data table on your calculator then find the linear regression.

Look for the coefficient of determination (r^2):r^2=0.93

Watch question walkthrough

b

Calculate the value of the correlation coefficient. Give your answer to two decimal places.

Worked Solution

Create a strategy

Take the square root of the coefficient of determination.

Apply the idea

\displaystyle r	\displaystyle =	\displaystyle \sqrt{0.93}	Take the square root of 0.93
	\displaystyle =	\displaystyle 0.96	Evaluate

Reflect and check

Or we could use the same procedure as in part (a), and look for the correlation of coefficient (r):r=0.96

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c

What percentage of the variation in weight is accounted for by the height of the child? Give your answer to the nearest whole percent.

Worked Solution

Create a strategy

Convert the coefficient of determination into a percentage.

Apply the idea

We have already found a value for the coefficient of determination (as a decimal).

\displaystyle 0.93	\displaystyle =	\displaystyle 0.93 \times 100\%	Multiply by 100\%
	\displaystyle =	\displaystyle 93\%	Evaluate

d

Consider these two comments on the claim “The weight of a child is primarily influenced by their height.”

Which do you think is most correct?

A

This claim is valid and is supported by the strong relationship between the two variables.

B

While this claim is supported by a strong relationship between the two variables, we cannot state the causality as there may be other factors inluencing the outcome.

Worked Solution

Create a strategy

Consider your answers to previous parts of this question and choose the statement that you think is most correct.

Apply the idea

The claim describes a strong relationship as the coefficient of determination, 0.93, is close to 1. But a high coefficient of determination (or correlation) value does not necessarily imply causation.

So the correct answer is B.