College of Education
and Health Professions
The Normal Distribution
Chapter 3
Assumption of Normality
• Many statistical tests (t, ANOVA) assume that the sampling
distribution is normally distributed.
• This is a problem, we don’t have access to the sampling
distribution.
• But, according to the central limit theorem if the sample data
are approximately normal, then the sampling distribution will
be normal.
• Also from the central limit theorem, in large samples (n > 30)
the sampling distribution tends to be normal, regardless of
the shape of the data in our sample.
• Our task is to decide when a distribution is approximately
normal.
A Z Score is a Standardized Statistic
95.0% of the scores fall between a Z of -1.96 to +1.96
97.4% of the scores fall between a Z of -3.00 to +3.00
99.9% of the scores fall between a Z of -3.30 to +3.30
How do we decide if a distribution is approximately normal?
99.9% of the scores fall between a Z of
-3.30 to +3.30
Normality Statistics are not Reliable for Large Samples
Z > 1.95 is significant at p < .05
Z > 2.58 is significant at p < .01
Z > 3.29 is significant at p < .001
Significance tests for normality: Kolomogorov-Smirnov, ShapiroWilk, skew, kurtosis, should not be used in large samples
(because they are likely to be significant even when skew and
kurtosis are not too different from normal.
Small Data Sets (n<20) and Normality?
-.91
-.04
-.28
-.36
-1.86
-2.99
-.32
1.63
-.19
-.32
Small data sets are adversely affected by occasional
extreme scores, even when the extreme score is
less than a Z of 3.3.
Approximately Normal?
We will use the following to determine if a distribution
is approximately normal:
1. Q-Q Plot values should lie close to the 45 line.
2. Distribution should be similar in shape to the
normal curve.
3. Skew & Kurtosis should be reasonably close to 0.
4. Data points with a Z score > +3.3 or < -3.3 will be
considered as outliers and removed.
Settings to Superimpose Normal Curve on Histogram
1
Histogram with Normal Curve
Distribution may be
leptokurtic (peaked)
Positively skewed?
Need to run Explore.
Explore
Settings
We are 95% sure that the
true mean lies between
58.721 and 61.7085.
The distribution is
positively skewed.
The distribution is
leptokurtic.
This may be an outlier, we
can use the descriptives
command to generate Z
scores.
Don’t use the K-S test for
normality.
If the Shapiro-Wilk is less
than 0.05 the data may
not be normal. But, it is
not a perfect test.
For small data sets (n <
100) if S-W has a p <
0.001 the data may not
be normal.
The test is unreliable for
large data sets n > 100.
The circles should fall on the 45 degree line. For this data set
the ends are deviating from the line, again suggesting a
problem with normality.
Case number 282 has a star, indicating that it is an extreme score.
An extreme value, E, is defined as a value that is smaller (or larger) than 3 boxlengths. We need to convert the data to Z scores to examine the Z for case 282.
Check this box to generate Z
scores for any variables in the
Variable(s) box.
The output is shown on the
next slide, it places the Z score
in your Data Sheet by adding a
Z in front of the variable(s)
selected.
The new column
ZReactionTime
contains the Z scores
for ReactionTime.
The Z score for case 282 is
4.944. Since the value is
much greater than our
arbitrary cutoff of 3.3.
We will delete this data
point then re-run Explore.
Copy the column, rename
the new variable
RTimeTrimmed, then
delete case 282.
The top descriptive
output is the original
data set.
Here is the data set with
case 282 removed. Look
at the changes in the
95% CI, Skew and
Kurtosis.
The Shapiro-Wilk is
significant,
indicating there may
be a problem with
normality.
Looks like case 84
and 100 may be the
cause.
We need to
generate Z scores
again.
The Q-Q plot and the box plot both suggest a problem. We need to run Z
scores to look at case 100 & 84.
Compute Z scores for the
RTimeTrimmed variable.
The new variable will be in
the data sheet labeled
ZRTimeTrimmed
Both case 84 &
100 have a Z
score above
3.3, our
arbitrary
cutpoint.
We can delete
them and then
run Explore.
Copy column
RTimeTrimmed,
make a new
variable
RTimeTrimmed
2
The original data set is
on the top.
RTimeTrimmed2
now has 3 data
points that have
been deleted.
We still may
have
normality
problems.
Looking better, run Z scores again on
RTimeTrimmed2, check cases 235 & 290.
The Z score for case 235 is 3.17, and for case 290 is 2.92. They are both
below our arbitrary cutoff of 3.3 for a Z score. The distribution is now
approximately normally distributed.