Question

how do third variables or extraneous variables show that correlation does not equal causation?

Answer 1

Correlation does not imply causation. Correlation tells us how strongly the pair of variables x and y are linearly related and change together. It does not tell us why and how behind the relationship.

The third variable or extraneous variable z which can influence both x and y separately can make it such and x and y are cause of one for the other, whereas in actual fact it is not so.

For example,

let

x = sale of ice cream

y = death due to drowning.

We can note a high positive correlation between x = sale of ice cream and y = death due to drowning. The actual fact is x is not the cause of y.

The reason is the presence of third variable z = hot weather which influences both x and y.

Since it very hot, ice cream sale increases.

Because of hot weather, people go into water games, causing death due to drowning.