Correlation of continuous variables: Pearson’s test

What is the correlation of 2 continuous variables called?

Bivariate data

E.g. rate constant and temp

E.g. height and mass

What do we use to summarise quantitative bivariate data?

Scatter diagram

What does the scatter diagram show us?

• Distribution of the data

• Relationship between the 2 variables

Pearson’s correlation coefficient

Used for continuous variables from a normal distribution

r =

sample value correlation coefficient

(Correlation in the sample size measured)

ρ =

Population value

(Correlation in the whole population)

S_{xx =}

S_yy =

S_xy =

equation for r

What values can r be?

r can take values from -1 to +1

r = +1

• Increase in x = increase in y

• Perfect positive correlation

r = -1

• perfect negative correlation

• Increase in x = decrease in y

r = 0

no correlation

If S_xy is positive…

r will also be positive

Example:

Calculate the Pearson’s correlation coefficient for the height (y) / mass (x) data

1. Calculate S_xx, S_yy and S_xy

2. Calculate r

r = 211.0 / √386.0 × 150.83

r = 0.874

3. Correlation

• r is close to 1

• So we know there is a positive correlation

4. Evaluation

• Small sample size, so how significant is this positive correlation?

• So should conduct a hypothesis test to calculate the Pearson’s population correlation coefficient

Why do we conduct the hypothesis test?

To assess correlation in the whole population (ρ) » Pearson’s population correlation coefficient

Formula for t values in hypothesis test for population correlation coefficient

Degrees of freedom in hypothesis test

number of pairs of data - 2

Example: Hypothesis Test to calculate Pearson’s population correlation coefficient

Mass / height data, r = 0.874

Is there a significant positive correlation?

1. State hypotheses

H_0: there is no relationship between the 2 variables

H₁: there is a significant positive correlation between the 2 variables

2. Calculate t value

t = 0.874 x √(6 - 2 / 1- 0.874²)

t = 3.597

3. Look up critical t value in table

• 5% significance level

• One-sided test

• Degrees of freedom: 4

• t = 2.13

4. If calculate t > tabulated t, reject H₀ and accept H₁

5. Conclusion and assumptions made

• There is good evidence to suggest at the 5% significance level that there is a positive correlation between mass and height

• Assumption: mass and height are normally distributed

Interpretations of r / ρ