16.2 – Pearson Correlation Coefficient

Before we start, let us revise a little bit on standard deviation. We all know that the standard error s is given by the formula

In this chapter, we will be dealing with 2 variables, and thus, we need to specify whether the standard error is for the values of x or y. To make the difference, we put a subscript x or y to indicate which variable it refers to. So over here, we have

 
the standard errors for x and y respectively. We denote the variances of variables x and y as

  
Note that s_xx and s_x² mean the same thing, it is just a different notation for some books. With this information in mind, we shall now introduce the covariance, which is defined by the formula

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PEARSON’S PRODUCT-MOMENT CORRELATION COEFFICIENT

The correlation coefficient is a statistic which provides the information on how strong the relationship of 2 variables is. Pearson’s product-moment correlation coefficient, also known as Pearson correlation coefficient or product-moment correlation coefficient, is a numerical value between –1 and 1 inclusive, which indicates the linear degree of scatter. It is represented by the formula

where, –1 ≤ r ≤ 1.

When r → 1, it indicates strong positive correlation, which means the regression line has a positive gradient, or y increases as x increases. Similarly, as r → –1, it indicates the presence of strong negative correlation. If r = 1 or r = –1, The points lie exactly on a straight line, and we say that they have perfect positive / negative correlation.

However, when r = 0, it does not necessarily mean that there is no correlation. It might indicate that the variables x and y are independent of each other. Besides, it might also indicate that the variables x and y have a non-linear relationship. Take a look at the diagram below:

Sorry but the dots are ugly. This diagram represents a quadratic function. The variables do have a quadratic relationship, but however, its correlation coefficient r = 0. This is just an example of how r = 0 fail to explain anything. On the other hand, having r close to zero only approximates that the data is positively linear correlated. Take a look at the diagram below.

This diagram has a very high r, about 0.7 to 0.8. But however, it doesn’t mean that the data is highly positively linear correlated. It might mean that there isn’t a relationship after all.

r is independent of the units used in the relation, and is very useful in determining the correlation of a 2 variables. Evaluating r can be tedious if you make use of the definitions of s_x and s_y. So here is the best way to calculate r:

Some other common formulas to find r are:

Besides, there is also this Big S format, whereby

and using this convention, the formula for r is
 

I would suggest that you keep to the ‘small s format’. In order to teach you how to find r efficiently using the calculator, consider the example below.

Calculate the value of the p-m correlation coefficient for the data in the following table. Comment on your answers.

Let’s make use of the calculator’s functions. Using your CASIO fx-570MS, press the mode button, and select REG mode. There will many kinds of REG mode, so you press ‘1’ for Lin mode (which means ‘linear’).

Now, to input the data, you press [x-value] [, button] [y-value] [DT button]. So you should type in 5, 4.3 and the DT button for the first readings. Now the screen should display

[n=                     ]
[                       1]

Continue typing every data, and press the AC button when you are done. Now you press SHIFT + S-SUM. You will be able to get lots of data from here: Σx², Σx, n, Σy², Σy and Σxy. These are the useful information you needed for your r (you need these to show your workings). But there’s a better one, press SHIFT + S-VAR. You get to find the values of x̅, xσn (s_x), y̅, yσn (s_y), and in fact, r itself! The only thing you can’t get is s_xy (what a pity). So using your calculator, you find that the answer is

r = 0.93, it is a strong positive correlation.

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That’s all for this section. With enough knowledge, we will go into the next and very last section, which will be on Regression Lines. ☺

16.1 – Scatter Diagrams

A scatter diagram is a diagram produced when pairs of values are plotted, to determine the relationship between 2 variables. Usually a scatter diagram contains bivariate data, which is data connecting 2 variables, x and y. Using the usual convention, x is the independent variable (explanatory variable), where it is controlled by the user who is analysing the situation. y on the other hand, is the dependent variable (response variable), it is the variable that is influenced by the previous one. I believe you learned this in your form 1 Science already.

In a scatter diagram, the independent variable is represented by the x-axis, while the dependent variable is on the y-axis. Basically, a scatter diagram is just a normal graph, with lots of dots on it. Suppose you want to analyse the relationship between the temperature of a chemical mixture, with its yield of a new compound. You started the experiment with various temperatures, and after a fixed time, you measure the yield of the new compound (precipitate). And you plot them in a graph like the one below.

Having drawn a scatter diagram, you can then look for a mathematical relationship between the variables x and y. This relation of y = f(x) is known as the regression function. The scatter diagram above shows a positive linear relationship between the data, but with a large dispersion. You can also find a line of best fit, or regression line to make things clearer. Other kinds of relationship between 2 data are:

For the data in diagrams (a) and (b), we say that there is linear correlation between the data. Diagram (d) shows that there is no correlation between the data, meaning that x and y are independent of one another.

Mathematically, there may appear to be a relationship between two data, but sometimes in reality, there isn’t any relationship. For example, you want to prove that the ears of a spider are on its legs. So you experiment it by putting it on the table, and shout at it and calculate its reaction time. Then you repeat your experiment by cutting its legs one by one. When all the legs are cut, it can’t hear your shout and therefore doesn’t move, so you have wrongly concluded that its ears are grown on its legs!

The appearance of a mathematical relationship doesn’t imply that there is a casual relationship. An increase in one variable does not necessarily cause an increase, or decrease, in the other variable.

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Now that you understand scatter diagrams, we shall proceed to learn the relationship of a correlation coefficient with a scatter diagram. We will learn more about the regression lines in the last post. ☺