Menu location: Analysis_Parametric_Shapiro Wilk.
This function is a (semi/non)parametric analysis of variance that detects a broad range of different types of departure from normality in a sample of data.
StatsDirect requires a random sample of between 3 and 5000 data for its Shapiro-Wilk test.
The null hypothesis of the test is that the sample is taken from a normal distribution, thus P < 0.05 for W rejects this supposition of normality. You should not use any of the parametric methods with samples for which W is significant.
Most authors agree that this is the most reliable test for non-normality for small to medium sized samples (Conover, 1999; Shapiro and Wilk, 1965; Royston, 1982a, 1982b, 1995). Please do not assume that the result of this test is clear evidence of normality or non-normality, it is just one piece of evidence that can be helpful. Other information, such as the type of distribution found from larger samples of this type, might also be important.
StatsDirect adjusts the Shapiro-Wilk test for censored data. You should use this adjustment when any of your data are censored (i.e. for some observations you only know that the value of X is >Y1 or <Y0 for some "censoring point" Y). You must know how many observations are censored, if not then you are dealing with a truncated distribution which requires different techniques (Verrill and Johnson, 1988). Note that you need enter only the total number of censored data and not censoring information for each data point, the actual data points that you enter should all be uncensored (Royston, 1995).
Example
Test workbook (Parametric worksheet: Penicillin).
Consider the following 30 penicillin yields.
penicillin
|
0.0987 |
0.0000 |
|
0.0533 |
-0.0026 |
|
0.0293 |
-0.0036 |
|
0.0246 |
-0.0042 |
|
0.0200 |
-0.0114 |
|
0.0194 |
-0.0139 |
|
0.0191 |
-0.0222 |
|
0.0180 |
-0.0333 |
|
0.0172 |
-0.0348 |
|
0.0132 |
-0.0363 |
|
0.0102 |
-0.0363 |
|
0.0084 |
-0.0402 |
|
0.0077 |
-0.0583 |
|
0.0058 |
-0.1184 |
|
0.0016 |
-0.1420 |
To test these data for non-normality using StatsDirect you must first prepare them in a workbook column. Alternatively, open the test workbook using the file open function of the file menu. Then select the Shapiro-Wilk test from the parametric methods section of the analysis menu. Select the column marked "Penicillin" when prompted for data and enter 0 as the number of censored data.
For this example:
Shapiro-Wilk W test for non-normality
Sample name: Penicillin
Uncensored data = 30
Censored data = 0
Mean = -0.007033
Standard deviation = 0.0454
Squares about mean = 0.059774
W = 0.892184
P = 0.0054
Sample unlikely to be from a normal distribution
Here the test statistic was clearly significant at P = 0.05 which rejects the null hypothesis that these data are from a normal distribution. In fact, these data were from a 2 by 5 factor grouping experiment.
N.B. Do NOT use this test to say that your data are "normally distributed", this assertion is quite wrong. The Shapiro-Wilk test provides evidence for certain types of "non-normality" it does NOT guarantee "normality".