# Product Moment Correlation Coefficient

## Product Moment Correlation Coefficient Revision

**Product Moment Correlation Coefficient**

The **Product Moment Correlation Coefficient** (**PMCC** or sometimes just r) is a number that tells you** how correlated your data are**. It is **always between** -1 and 1.

- If it is
**positive**then there is**positive correlation**, with stronger positive correlation being closer to 1. - If it is
**negative**then there is**negative correlation**, with stronger negative correlation being closer to -1. - If it is 0 then there is
**no correlation**– a number**close to**0 is usually an indicator that**variables are not correlated**too.

You can calculate the **Product Moment Correlation Coefficient** on your calculator – it will be in the **statistical functions**.

**Note: **Linear transformations of x and y, such as S=ax+b, **do not affect the Product Moment Correlation Coefficient**.

Make sure you are happy with the following topics before continuing.

**Hypothesis Testing with the Product Moment Correlation Coefficient**

When measuring variables that are not correlated, it is possible that **some correlation might arise by chance**. A **Pr****oduct Moment Correlation Coefficient** **hypothesis test** is a test to see if variables really are **correlated**.

The **population parameter** is \rho, the **product moment correlation coefficient** of the **population**.

The **test statistic** is r, the **product moment correlation coefficient** of the **sample** (the data we have).

**The hypotheses are always the same:**

H_{0}: \rho=0

H_{1}: \rho>0\text{ or }\rho<0 (one tail)

H_{1}: \rho\neq 0 (two tail)

In your **formula booklet** there will be a **table against which you test** your test statistic. For example, if we had a **significance level** of 0.05 and a **sample size** of 6, we would compare our** test statistic** to the value 0.7293.

**Example 1: Calculating the PMCC**

Calculate the **Product Moment Correlation Coefficient** from the following data:

**[1 mark]**

To do this on a scientific calculator you should go into **statistics mode** – there will usually be a button called **MODE** or **MENU** from where it can be accessed. This will present you with multiple options, one of which will read like a regression line (such as **A+BX**). Upon pressing it, you should be given a table that you can put the values in. Then, the calculator will give you key statistics from the table, including the **PMCC**. These can be accessed by pressing the **STAT** button after entering the values into the tables. Doing all of this gives:

**Example 2: PMCC Hypothesis Test**

Are the following two variables **correlated** to 5\% **significance**?

**[6 marks]**

H_{0}: \rho=0, H_{1}: \rho\neq 0

Test statistic is PMCC of the variables in the table, which is r=-0.3587

Significance level: \alpha=0.05 but two tailed test so we use 0.025

Critical value from table is 0.6319>0.3587.

Do not reject H_{0}. Insufficient evidence to suggest the variables are correlated.

## Product Moment Correlation Coefficient Example Questions

**Question 1: **Match these statements to the PMCC values.

a) Strong positive correlation.

b) Weak negative correlation.

c) No correlation.

d) Strong negative correlation.

e) Weak positive correlation.

- -0.9
- 0.3
- -0.01
- 0.875
- -0.5

**[5 marks]**

a) Strong positive correlation means close to 1, which corresponds to 4: 0.875

b) Weak negative correlation corresponds to a negative number that is not close to -1, which is 5: -0.5

c) No correlation means close to 0, which is 3: -0.01

d) Strong negative correlation means a number close to -1 which is 1: -0.9

e) Weak positive correlation corresponds to a positive number that is not close to 1, which is 2: 0.3

**Question 2:** Find the product moment correlation coefficient of the following data.

**[2 marks]**

To do this on a scientific calculator you should go into statistics mode – there will usually be a button called MODE or MENU from where it can be accessed. This will present you with multiple options, one of which will read like a regression line. Upon pressing it, you should be given a table that you can put the values in. Then, the calculator will give you key statistics from the table, including the PMCC. Doing all of this gives:

r=-0.3975**Question 3: **Tally feeds her fish varying amounts of food and monitors the amount of time they spend resting during the day. She collects 100 data points and finds a correlation of r=0.2156. She dismisses this as “clearly not correlated”. Perform a hypothesis test at the 5\% level to find if Tally should have dismissed her findings.

**[3 marks]**

H_{0}:\rho=0

H_{1}:\rho\neq 0

Test statistic is PMCC, which is r=0.2156.

Significance level: \alpha=0.05 but two tailed test so we use 0.025

Critical value from the table is 0.1966<0.2156

Reject H_{0}. Sufficient evidence to suggest Tally’s results are correlated.