The Linear Correlation Coefficient

Description

Read this discussion on linear correlation. You will learn what the linear correlation coefficient is, how to compute it, and what it tells us about the relationship between two variables x and y.

The Linear Correlation Coefficient

Learning Objective

To learn what the linear correlation coefficient is, how to compute it, and what it tells us about the relationship between two variables $x$ and $y$ .

Figure 10.3 "Linear Relationships of Varying Strengths" illustrates linear relationships between two variables

$x$ and

$y$ of varying strengths. It is visually apparent that in the situation in panel (a),

$x$ could serve as a useful predictor of

$y$ , it would be less useful in the situation illustrated in panel (b), and in the situation of panel (c) the linear relationship is so weak as to be practically nonexistent. The linear correlation coefficient is a number computed directly from the data that measures the strength of the linear relationship between the two variables

$x$ and

$y$ .

Figure 10.3 Linear Relationships of Varying Strengths

Definition

The linear correlation coefficient for a collection of

$n$ pairs

$(x,y)$ of numbers in a sample is the number

$r$ given by the formula

$r = \dfrac{SS_{xy}}{\sqrt{SS_{xx} \cdot SS_{yy}}}$

where

$SS_{xx}=Σx^2−\dfrac{1}{n}(Σx)^2$ ,

$SS_{xy}=Σxy−\dfrac{1}{n}(Σx)(Σy)$ ,

$SS_{yy}=Σy^2−\dfrac{1}{n}(Σy)^2$

The linear correlation coefficient has the following properties, illustrated in Figure 10.4 "Linear Correlation Coefficient ": The value of $r$ lies between −1 and 1, inclusive.
The sign of $r$ indicates the direction of the linear relationship between $x$ and $y$ :
1. If $r < 0$ then $y$ tends to decrease as $x$ is increased.
2. If $r > 0$ then $y$ tends to increase as $x$ is increased.
The size of $|r|$ indicates the strength of the linear relationship between $x$ and $y$ :
1. If $|r|$ is near 1 (that is, if $r$ is near either 1 or −1) then the linear relationship between $x$ and $y$ is strong.
2. If $|r|$ is near 0 (that is, if $r$ is near 0 and of either sign) then the linear relationship between $x$ and $y$ is weak.

Figure 10.4 Linear Correlation Coefficient R

Pay particular attention to panel (f) in Figure 10.4 "Linear Correlation Coefficient ". It shows a perfectly deterministic relationship between

$x$ and

$y$ , but

$r=0$ because the relationship is not linear. (In this particular case the points lie on the top half of a circle)

Example 1

Compute the linear correlation coefficient for the height and weight pairs plotted in Figure 10.2 "Plot of Height and Weight Pairs".

Solution:

Even for small data sets like this one computations are too long to do completely by hand. In actual practice the data are entered into a calculator or computer and a statistics program is used. In order to clarify the meaning of the formulas we will display the data and related quantities in tabular form. For each

$(x,y)$ pair we compute three numbers:

$x^2$ ,

$xy$ , and

$y^2$ , as shown in the table provided. In the last line of the table we have the sum of the numbers in each column. Using them we compute:

	$x$	$y$	$x^2$	$xy$	$y^2$
	68	151	4624	10268	22801
	69	146	4761	10074	21316
	70	157	4900	10990	24649
	70	164	4900	11480	26896
	71	171	5041	12141	29241
	72	160	5184	11520	25600
	72	163	5184	11736	26569
	72	180	5184	12960	32400
	73	170	5329	12410	28900
	73	175	5329	12775	30625
	74	178	5476	13172	31684
	75	188	5625	14100	35344
$Σ$	859	2003	61537	143626	336025

$S S_{x x}=\Sigma x^{2}-\frac{1}{n}(\Sigma x)^{2}=61537-\frac{1}{12}(859)^{2}=46.91 \overline{6}$

$S S_{x y}=\Sigma x y -\frac{1}{n}(\Sigma x)(\Sigma y)=143626-\frac{1}{12}(859)(2003)=244.58 \overline{3}$

$S S_{y y}=\Sigma y^{2}-\frac{1}{n}(\Sigma x)^{2}=336025-\frac{1}{12}(2003)^{2}=1690.91 \overline{6}$

so that

$r=\frac{S S_{v}}{\sqrt{S S_{w} S S_{y}}}=\frac{244.58 \overline{3}}{\sqrt{(46.91 \overline{6})(1690.91 \overline{6})}}=0.868$

The number

$r=0.868$ quantifies what is visually apparent from Figure 10.2 "Plot of Height and Weight Pairs": weights tends to increase linearly with height (

$r$ is positive) and although the relationship is not perfect, it is reasonably strong (

$r$ is near 1).

This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor.

Key Takeaways

The linear correlation coefficient measures the strength and direction of the linear relationship between two variables $x$ and $y$ .
The sign of the linear correlation coefficient indicates the direction of the linear relationship between $x$ and $y$ .
When $r$ is near 1 or −1 the linear relationship is strong; when it is near 0 the linear relationship is weak.

Site:	Saylor Academy
Course:	MA121: Introduction to Statistics
Book:	The Linear Correlation Coefficient

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Date:	Sunday, June 4, 2023, 11:08 PM

The Linear Correlation Coefficient

Description

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