Shapiro Wilk normality test in R

The Shapiro Wilk test is a test used to check the normality of the data, specially for small data sets with 50 samples or less. In R, the shapiro.test function performs this test for a numeric vector of values.

Hypothesis

The Shapiro-Wilk test checks for normality of the data. The null hypothesis (\(H_0\)) is that the distribution of the population is normal, while the alternative hypothesis is that the distribution of the population is not normal:

• \(H_0\): The distribution of the population is normal.
• \(H_1\): The distribution of the population is NOT normal.

Check for normality and interpretation

Example with non-normal data

Consider a sample data set with 30 values drawn from an exponential distribution:

# Sample data
set.seed(3)
x <- rexp(30)

# Histogram and density
hist(x, freq = FALSE, col = "white")
lines(density(x), lwd = 2, col = "red")

If you want to check if your data is normal you can input your data to the shapiro.test function, as shown below.

# Sample data
set.seed(3)
x <- rexp(30)

# Shapiro-Wilk test
shapiro.test(x)

	Shapiro-Wilk normality test

data:  x
W = 0.83304, p-value = 0.0002792

The function will return a formatted list of class "htest" with the name of your data, the value of the Shapiro-Wilk statistic (W) and the p-value.

In order to interpret the result you need to compare the p-value with a significance level (\(\alpha\)) which is a threshold to determine if a test is statistically significant or not. Significance levels usually are 0.1 (for 90% of confidence), 0.05 (for 95% of confidence), 0.01 (for 99% percent of confidence).

As the p-value (0.000279) is lower than the usual significance levels (0.1, 0.05, and 0.01) we reject the null hypothesis of normality.

Example with normal data

Now, consider some sample data drawn from a normal distribution:

# Sample data
set.seed(5)
x <- rnorm(30)

# Histogram and density
hist(x, freq = FALSE, col = "white")
lines(density(x), lwd = 2, col = "red")

If you apply the Shapiro-Wilk test for this data you will get the following output:

# Sample data
set.seed(5)
x <- rnorm(30)

# Shapiro-Wilk test
shapiro.test(x)

	Shapiro-Wilk normality test

data:  x
W = 0.95084, p-value = 0.178

The p-value is 0.178, greater than the usual significance levels (0.1, 0.05, and 0.01), so we don’t have enough evidence to reject the null hypothesis of normality.