Step 1

If we suspect linear relationship between two variables, then for a given set of data, we can evaluate the value of Pearson correlation coefficient without having any assumption. And we can have a sense of with what intensity the variables are linearly related.

Step 2

But if we would like to test the significance of a given correlation value for a set of observations, and we would like to infer whether the data really gives sufficient evidence in favor of significant correlation or linear relationhip between considered variables, then for the construction of test statistics, assumption of normality is necessary.

If we suspect linear relationship between two variables, then for a given set of data, we can evaluate the value of Pearson correlation coefficient without having any assumption. And we can have a sense of with what intensity the variables are linearly related.

Step 2

But if we would like to test the significance of a given correlation value for a set of observations, and we would like to infer whether the data really gives sufficient evidence in favor of significant correlation or linear relationhip between considered variables, then for the construction of test statistics, assumption of normality is necessary.