Question

How do we determine whether a continuous variable is normally distributed?

Answer 1

In order to determine whether the continuous variable is normally distributed, we usually follow the following two steps:

(1) Graphical Check: A commonly used procedure is to draw the 'normal probability plot', also called a 'Q-Q plot' (or 'quantile-quantile' plot). Roughly speaking, the points on a normal probability plot should lie in a straight line. I have attached two Q-Q plots, one with normal random variable and other with exponential random variables to understand how the Q-Q plots looks for normal and non normal data.

Notice here that in the first plot all the points are placed along the diagonal line whereas in the second plot they are not.

(2) Normality Test: The Shapiro–Wilk test is a test of normality. The Shapiro–Wilk test tests the null hypothesis that a sample came from a normally distributed population. In R, there is a built in command to perform this test, given by,

shapiro.test(x)

where x is the sample. If p value is less than 0.05, then the sample is not from normal distribution. I have attached the screenshot of the R code for this test. Hope it helps.