• Correlation → measures how two variables move together, but does not prove one causes the other.

• Causation → one variable directly causes changes in another.

This graph shows a correlation between practice attendance and performance levels. As shown, the data does not prove higher performance levels are achieved by attending more practices.

For swimming, the longer you are underwater, less energy is used to complete the lap. This graph shows how underwater time directly changes energy output in swimmers.

In this graph, a hidden third variable called “Swim Speed” is taken into account. Swim speed can affect both energy output and underwater time alike.

set.seed(123)

underwater_time = sample(5:40, 25, replace = TRUE)

energy_output = underwater_time * 4 + rnorm(25, mean = 0, sd = 10)

plot( underwater_time, energy_output, main = “Underwater Time vs Energy Output”, xlab = “Underwater Time (seconds)”, ylab = “Energy Output (calories)”, col = “gold”, pch = 19)

\[\text{Correlation}(X,Y)\neq\text{Causation}(X,Y)\]

\(X\not\Rightarrow Y\)

\(r = \frac{sum (x_i - \bar{x})(y_i - \bar{y})} {\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\)

\(P(Y \mid X) \neq P(Y \mid do(X))\)

- Observing X is NOT the same as intervening to change X.