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**mean the same thing, it is just a different notation for some books. With this information in mind, we shall now introduce the

_{x}^{2}**covariance**, which

**is defined by the formula**

__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

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**

**. So here is the best way to calculate**

_{y}**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: **Σx ^{2}, Σx, n, Σy^{2}, 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**, and in fact,

_{x}), y̅, yσn (s_{y})**r**itself! The only thing you can’t get is

**s**(what a pity). So using your calculator, you find that the answer is

_{xy}r = 0.93, it is a strong positive correlation.

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**. ☺

Hi ,

ReplyDeleteAs we know y = mx + b is the equation of line where m is the slope and b is the intercept .So if it will be good to find the y = mx + b calculator then it will be quit beneficial to find these .