Experiments with Two Variables

One of the simplest experiments we can conduct involves:

• An Independent Variable which is changed throughout the experiment, normally called
  

• A Dependent Variable which is measured throughout the experiment, normally called
  

The reason behind experiments is usually to determine whether or not influences .

Example

One possible experiment could be:

• - how much chilli we put in a meal

• - a rating out of for how much we enjoyed the meal

Each meal would have a different amount of chilli and rating. We can think of this as a pair of numbers:

After conducting the experiment multiple times, we get a sequence of pairs of numbers, which is a bivariate dataset.

Types of Correlation

The main types of correlation are inspired by elementary functions. Some examples are below.

Linear Correlation

Quadratic Correlation

Exponential Correlation

Trigonometric Correlation

In this course we place most of our emphasis on linear correlation, and normally group all other types as “non-linear”.

No Correlation

Verbally Describing Linear Correlation

Describing Direction

The direction of a linear correlation can either be:

• Positive - if tends to increase as increases

• Negative - if tends to decrease as increases

From the scatterplot we can use the approximate gradient of the trend:

Describing Strength

The better the dataset approximates a line, the stronger the linear correlation:

Calculating Linear Correlation

Pearson’s Correlation Coefficient is a number in the range that describes both the direction and strength of a linear correlation.

Finding Pearson’s Correlation Coefficient

The video below is a good demonstration on how you can find Pearson’s Correlation Coefficient with a calculator:

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Describing Direction

The direction is described by the sign of the coefficient, so if:

• then the linear correlation is positive

• then the linear correlation is negative

• then there is no linear correlation

Describing Strength

The strength is described by the magnitude of the coefficient . A rough guide as to how we can interpret the magnitude is:

If we have:

• then the dataset forms a perfect line with positive gradient

• then the dataset forms a perfect line with negative gradient

• then the dataset only approximates a line

Describing Linear Correlation

We can use both of the above ideas to completely describe a linear correlation with Pearson’s Correlation Coefficient. Some examples are:

• A dataset with has a positive, moderate linear correlation

• A dataset with has a negative, weak linear correlation

• A dataset with has a positive, strong linear correlation

Understanding Covariance

Covariance

The covariance between two variables is given by:

• If then the direction of correlation is positive

• If then the direction of correlation is negative

How Covariance gives Direction

To understand how the covariance formula works, we need to first find the point of averages :

Considering the term, we know that if:

• then the term is positive

• then the term is negative

• then the term is
  

Considering the term, we know that if:

• then the term is positive

• then the term is negative

• then the term is
  

We can now consider the product term if we remember that:

• The product of two positive numbers is positive

• The product of two negative numbers is positive

• The product of a positive and negative number is negative

So the total sum will be positive if most of the data points tend to be in positive regions. Similarly, it will be negative if most of the data points tend to be in negative regions.

So this means that:

• If the linear correlation is positive most data points tend to be in the positive regions covariance is positive

• If the linear correlation is negative most data points tend to be in the negative regions covariance is negative

Understanding Pearson’s Correlation Coefficient

Pearson’s Correlation Coefficient

We obtain this coefficient by scaling the covariance:

Expanding this formula gives:

The Effect of Scaling

We

A couple of fleshed out examples :)