# IB Class Notes

## Calculating Correlation

Aim: Here we will learn how to calculate coefficient of correlation ($r$) by hand with the help of EXCEL.

As a reminder, coefficient of correlation ($r$) tells us whether two sets of data, say rate of chemical reaction and temperature, are linearly related or not.

### What should my value of r be?

1. If your data points rise from left to the right of the graph ($x$ and $y$ increases together) then your $r$ will be positive.

2. Your $r$ should be negative if the data points slope downward from left to the right of the graph. That is, as $x$ increases then value of $y$ decreases and vice versa.

3. Perfect positive linear correlation is +1 and perfect negative linear correlation is -1.

4. No linear correlation is when $r = 0$ .

5. Thus, the value of $r$ must always lie between -1 and +1, i.e. -1 ≤ $r < 1$
If your value of calculated $r$ is not in this range then you probably have made a mistake and should re-check your workings.

6. Coefficient of correlation ($r$) is calculated as
$r = \large \frac{\sum{xy} -\frac{1}{n}\sum{x} \sum{y} }{ \sqrt{ \left( \sum{x^2} -\frac{1}{n}(\sum{x})^2 \right) \left(\sum{y^2}-\frac{1}{n}(\sum{y} )^2 \right) } }$ Product-moment correlation coefficient is another name for $r$. Another way to express $r$ is
$r = \large \frac{S_{xy}}{S_xS_y}$
where
$S_{xy} = \sum{xy} - \frac{1}{n}\sum{x}\sum{y}$ ,
$S_x = \sqrt{ \left( \sum{x^2} -\frac{1}{n}(\sum{x})^2 \right) }$, and
$S_x = \sqrt{ \left( \sum{x^2} -\frac{1}{n}(\sum{y})^2 \right) }$

 Note that $S_{xy} = ns_{xy}$ or $n$Covariance of $x$ and $y$. Distinguish the use of capital $S$ from small $s$. The covariance of $x$ and $y$, $s_{xy} = \frac{\sum{xy} - \large \frac{\sum{x}\sum{y}}{n} }{n}$ Similarly $S_x = ns_x$ or $n$standard deviation of $x$. Standard deviation of $x$, $s_x = \sqrt{ \frac{\left[ \sum{x^2} - \large \frac{(\sum{x})^2}{n} \right]}{n} }$ and $S_y = ns_y$ or $n$standard deviation of $y$. Standard deviation of $y$, $s_y = \sqrt{ \frac{\left[ \sum{y^2} - \large \frac{(\sum{y})^2}{n} \right]}{n} }$

Email KokMing Lee
Li Po Chun United World College of Hong Kong,
10 Lok Wo Sha Lane, Sai Sha Road,
Shatin, New Territories, Hong Kong SAR.