The Shapiro Wilk test checks if the normal distribution model fits the observations. It is usually the most powerful test for the normality.

The test uses only the right-tailed test. When performing the test, the W statistic is only positive and represents the difference between the estimated model and the observations. The bigger the statistic, the more likely the model is not correct. The left-tailed may represent a value that is too small, the W statistic can't be too small.

3. Calculate the p-value from the SW tables.

Since you use tables the W will be between two values: W1 and W2, and the P-value between the relevant two p-values: P

Calculate the approximate p-value using a linear ratio.

$$ p-value=p_1+\frac{(W-W_1)}{W_2-W_1}*(P_2-P_1)$$

Compare to other tests the Shapiro Wilk test has a good power to reject the normality, but like any other test, it needs to have a sufficient sample size.

Like any other test, the Shapiro Wilk Test power depends on the effect size the test is expected to identify. When the distribution is similar to the normal distribution, the effect size is small and large sample size is required.

When the distribution is different than the normal distribution, the effect size is large and small sample size is required.

The following chart shows the power of the Shapiro-Wilk test to reject the normality assumption for the chi-square distribution data.

When the distribution is similar to the normal distribution the effect size that the Shapiro Wilk test needs to recognize is small.

When the distribution is different than the normal distribution the effect size that the Shapiro Wilk test needs to recognize is small.

In the following examples, there is some focus on the distribution symmetric, but this is only one parameter

The following chart was created with R simulation. Degree of freedom: 2, 5, 10, 20, 30, 60.

Sample size (n): 2 - 200.

Significance level (α): 0.05.

The χ^{2}(60) distribution is quite symmetrical, **skewness = 0.3651** (√(8/60)), very close to zero. The effect size the Shapiro Wilk test needs to recognize is **small**, hence you need to have a large sample size of **440** (out of the chart scale) to gain the power of **0.8**. In this case, the chance to reject the normality assumption is 80%.

The χ^{2}(10) distribution is less symmetrical, **skewness = 0.8944** ( √(8/10)), so you need to have a smaller sample size of **77** to gain the power of 0.8.

The χ^{2}(5) distribution is less symmetrical, **skewness = 1.2649** ( √(8/5)), so you need to have a smaller sample size of **41** to gain the power of **0.8**.

The χ^{3}(3) distribution is not symmetrical, **skewness = 1.6329** ( √(8/3)), So you need to have a smaller sample size of **26** to gain the power of **0.8**. In this case, the chance to reject the normality assumption is 80%.

**P-value = 0.0007**

The following chart was created with R simulation. Degree of freedom: (1,1), (5,2), (10,10), (30,10), (50,50), (100,100).

Sample size (n): 2 - 200.

Significance level (α): 0.05.

The F(50,50) distribution is moderate skewed, **skewness = 0.9217** ( √(8/3)), looks very similar to the normal distribution. Hence only a large sample size of **90** to gain the power of **0.8**. In this case, the chance to reject the normality assumption is 80%.

The F(5,2) distribution is not symmetrical, **skewness = 1.6329** ( √(8/3)), doesn't look like a normal distribution.

So you need to have a small sample size of **9** to gain the power of **0.8**.

The following chart was created with R simulation. The uniform distribution is symmetrical but doesn't look like the normal distribution.