# Bro. Brown's statistics reference pages:

# The Shapiro-Wilk test of normality

**Purpose:** To test whether a population is a Normal population.

**Requirements:**

1) The data are numeric and ratio.

2) The sample size is at least 3 (that's "three").

3) The sample is a random sample (no specific sampling method is required).

**Hypotheses:**

The null hypothesis can be expressed as (e.g.)

H:_{0}The population of individual measurements is normalH:_{0}The data come from a normal distributionH:_{0}The data come from a plausibly normal distributionH:_{0}The data are normal

The alternative hypothesis is

H:_{a}Not H, which can be expressed as (e.g.)_{0}

H:_{a}The population of individual measurements isnotnormal

H:_{a}The datado notcome from a normal distribution

H:_{a}The datado notcome from a plausibly normal distribution

H:_{a}The data arenotnormal

**Test Statistic:** The calculation by hand of the Shapiro-Wilk test statistic (usually called *W*)* * is best left to students who have had a course in Linear Algebra. Suffice it to say that this statistic is always between 0 and 1, and that the smaller this statistic is, the less plausible the normality of the population.

In 200-level Statistics courses at BYU-Idaho, we always get our Shapiro-Wilk statistic and its P-value from SPSS.

**Notes:** The Shapiro-Wilk test measures the difference between the shape of your data's histogram and the shape of the Normal distribution having the same mean and standard deviation. It can detect skewing, outliers, and subtle departures from Normality in terms of the precise shape of the hump.

**SPSS instructions for testing Normality:** (Not presently available.)

**Related topics:**