Shapiro-Wilks significant, all other indicators NORMAL

#1
Hello everyone,

I am checking my assumptions before running an ANCOVA, one of which is that the DV is normally distributed across all levels of the IV. MY IV is 3 groups (Ns of 32, 31, and 32) and my DV is a continuous variable from 1 - 5. Before splitting into groups, it looks normal and the skew statistic is -.25. However, when I split into groups and run the Shapiro Wilk test, all three groups come back significant - indicating non-normality. Shapiro Wilk is recommended for samples < 50, which I have when looking at separate groups.

Here is where it gets weird. Visually, the data look normal using Q-Q plots and histograms (perhaps a bit pointy, which may be the issue). In addition, when I use the skew statistics to calculate this on my own (Z Skewness= Skewness-0 / SE Skewness and Z Kurtosis= Kurtosis-0 / SE Kurtosis), both values come back UNDER 1.96.

An example:
Group 1
skew = -.502, SE skew = .421;
-.502 / .421 = -1.19
kurtosis = .131, SE kurtosis = .821.
; .131 / .821 = .159.

However, Shapiro Wilk = .899, p < .05

The same is true in all 3 groups.

Needless to say, I would prefer not to transform if I don't have to, but I am unable to find any resources or citations stating why this would occur and whether it's ok to go with my hand calculations over the Shapiro-Wilk test.

Any insight is very much appreciated!

Thank you all!
 
#2
Hey all, since I got no hits on here, I decided not to use Shapiro-Wilks. All the resources I found online intimated that you can use whichever indicator you prefer, so I am going with a combination of the hand calculations and the guidelines set by Bulmer (1979) of -1 to 1 for acceptable skew.
 

Karabiner

TS Contributor
#3
The general assumption for analyses of small
samples with procedures from the general linear
model is that the residuals from the model are
from a normally distributed population. But with
n > 90 you do not have a small sample. So the
results should be valid regardless of the distribution
(cf. central limit theorem).

With kind regards

K.