Checking for normality (Normal distribution)

Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA
Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course
Checking for Normality
(Normal Distribution)
Dr. S. A. Rizwan, M.D.,
Public Health Specialist,
Saudi Board of Preventive Medicine,
Riyadh, Kingdom of Saudi Arabia.
Nov 2019 1

When is non-normality a problem?
• Non-normality can be a problem when the sample size is small.
• Highly skewed data create problems.
• Highly leptokurtic data are problematic, but not as much as skewed
data.
• Normality becomes a serious concern when there is “activity” in
the tails of the data set.
– Outliers are a problem.
– “Clumps” of data in the tails are worse
Nov 2019 2

Ways to check for normality
1. Thumb rules
– Mean & Range & SD
– Skewness and kurtosis
– Compare mean, median and
mode
– Trimmed mean
– Outliers
2. Graphs
– Histogram with theoretical
normal curve
– QQ plot
– Box plot and outlier detection
– Stem and leaf plot
3. Formal statistical tests
– W/S test
– Jarque-Bera test
– Shapiro-Wilks test
– Kolmogorov-Smirnov test
– D’Agostino test
– Grubbs and Dixon test (for
outliers)
4. Comparing non-parametric
and parametric test results
Nov 2019 3

1. THUMB RULES
Nov 2019 4

Thumb rules
• SD less than half of mean -> may be normally
distributed
• Calculate the observed mean minus the
lowest possible value (or the highest possible
value minus the observed mean), and divide
this by the standard deviation.
– A ratio less than 2 suggests skew (Altman 1996). If
the ratio is less than 1 there is strong evidence of
a skewed distribution.
Nov 2019 5

Thumb rules
• Skewness
– <-1 or >1, highly skewed.
– -1 to -0.5 or 0.5 to 1, moderately skewed.
– -0.5 to 0.5, approximately symmetric
• Kurtosis
– For normally distributed data the value is 3, theoretically
– Statistical packages adjust this to zero
– Platykursis < 0 and leptokursis > 0
– Extremely non-normal distributions may have high positive or
negative kurtosis values, while nearly normal distributions close
to 0
– Kurtosis is positive if the tails are “heavier” and negative if the
tails are “lighter”
Nov 2019 6

Skewness and kurtosis
Nov 2019 7

Thumb rules
• In a normal distribution, mean=median=mode
• 5% Trimmed Mean
– lower and upper 5% of values deleted
– If the value of the 5% trimmed mean is very
different from the mean, this indicates outliers
• Examine the lowest and highest 5 extreme
values to look for outliers
Nov 2019 8

2. GRAPHS
Nov 2019 9

NORMAL CURVE
For each mean and standard
deviation combination a theoretical
normal distribution can be
determined
This distribution is based on the
proportions shown
Nov 2019 10

This theoretical normal distribution can then be
compared to the actual distribution of the data.
• Are the actual data statistically different than
the computed normal curve?
Theoretical normal
distribution calculated
from amean of 66.51 anda
standard deviation of
18.265.
Theactual data
distribution that hasa
mean of 66.51 and a
standard deviation of
18.265.
Nov 2019 11

Comparison of histograms
Nov 2019 12

Q-Q plots
• Q-Q plots display the observed values against normally distributed data
(represented by the diagonal line).
• Normally distributed data fall along theline.
Nov 2019 13

Example Q-Q plots
• The Q-Q and detrended Q-Q plots
show systematic deviations from
normality
• Notice that the overall shape of
the detrended plot is parabolic
(U-shaped).
• Also notice that the deviations
from normality are relatively
large: the y-axis of the detrended
normal q-q plot indicates that the
deviations range in magnitude
from -0.2 to 1.2.
Nov 2019 14

Beware!
• Graphical methods are
typically not very useful
when the sample size is
small.
• These data do not ‘look’
normal, but they are not
statistically different than
normal.
Nov 2019 15

Stem and leaf plot
• Frequency
– This is the frequency of the
leaves
• Stem
– It is the number in the 10s
place of the value of the
variable
• Leaf
– It is the number in the 1s
place of the value of the
variable.
– The number of leaves tells
you how many of these
numbers is in the variable
Nov 2019 16

Parts of a boxplot
a) Maximum score unless there are
values more than 1.5 times the
interquartile range above Q3, in
which, it is the third quartile plus
1.5 times the interquartile range
b) Third quartile (Q3) or 75th
percentile.
c) Median (q2) or 50th percentile.
d) First quartile (q1) or 25th percentile.
e) Minimum score unless there are
values less than 1.5 times the
interquartile range below Q1, in
which case, it is the first quartile
minus 1.5 times the interquartile
range.
Nov 2019 17

Comparison of box plots
Nov 2019 18

Example of outliers in boxplot
Nov 2019 19

3. FORMAL STATISTICAL TESTS
Nov 2019 20

Background of tests
• Statistical tests for normality are more precise since actual
probabilities are calculated.
• Tests for normality calculate the probability that the sample
was drawn from a normal population.
• The hypotheses used are:
– Ho: The sample data are not significantly different than a
normal population.
– Ha: The sample data are significantly different than a normal
population.
Nov 2019 21

Background of tests
• Typically, we are interested in finding a difference between groups.
When we are, we ‘look’ for small probabilities.
• If the probability of finding an event is rare (less than 5%) and we
actually find it, that is of interest.
• When testing normality, we are not ‘looking’ for a difference.
• In effect, we want our data set to be NO DIFFERENT than normal.
We want to accept the null hypothesis.
• So when testing for normality:
– Probabilities > 0.05 mean the data are normal.
– Probabilities < 0.05 mean the data are NOT normal.
Nov 2019 22

W/S test
• A fairly simple test that requires only the sample
standard deviation and the data range.
• Should not be confused with the Shapiro-Wilk test.
• Based on the q statistic, which is the ‘studentized’
(meaning t distribution) range, or the range expressed
in standard deviation units. Tests kurtosis.
whereq istheteststatistic,w istherangeof thedataands is the
standarddeviation.
s
q 
w
Nov 2019 23

Range constant,
SD changes
Range changes,
SD constant
Nov 2019 24

Village
Population
Density
Aranza 4.13
Corupo 4.53
SanLorenzo 4.69
Cheranatzicurin 4.76
Nahuatzen 4.77
Pomacuaran 4.96
Sevina 4.97
Arantepacua 5.00
Cocucho 5.04
Charapan 5.10
Comachuen 5.25
Pichataro 5.36
Quinceo 5.94
Nurio 6.06
Turicuaro 6.19
Urapicho 6.30
Capacuaro 7.73
Standard deviation (s)=0.866
Range(w) = 3.6
n =17
0.866
3.6
s
qCritical Range  3.06 to 4.31
 4.16q 
q 
w
The W/S test uses a critical range. IF the calculated value falls WITHIN the range, then
accept Ho. IF the calculated value falls outside the range then reject Ho.
Since 3.06 < q=4.16 < 4.31, then we accept Ho.
W/S test
Nov 2019 25

Table for W/S
test
• Since we have a critical
range, it is difficult to
determine a probability
range for our results.
• Therefore we simply
state our alpha level.
• The sample data set is
not significantly
different than normal
(W/S4.16, p > 0.05).
Nov 2019 26

Jarque–Bera Test
• A goodness-of-fit test of whether sample data have the
skewness and kurtosis matching a normal distribution.
• Where x is each observation, n is the sample size, s is
the standard deviation, k3 is skewness, and k4 is
kurtosis.
Nov 2019 27

Example dataset for JB test
Village
Population
Density
Mean
Deviates
Mean
Deviates3
Mean
Deviates4
Aranza 4.13 -1.21 -1.771561 2.14358881
Corupo 4.53 -0.81 -0.531441 0.43046721
SanLorenzo 4.69 -0.65 -0.274625 0.17850625
Cheranatzicurin 4.76 -0.58 -0.195112 0.11316496
Nahuatzen 4.77 -0.57 -0.185193 0.10556001
Pomacuaran 4.96 -0.38 -0.054872 0.02085136
Sevina 4.97 -0.37 -0.050653 0.01874161
Arantepacua 5.00 -0.34 -0.039304 0.01336336
Cocucho 5.04 -0.30 -0.027000 0.00810000
Charapan 5.10 -0.24 -0.013824 0.00331776
Comachuen 5.25 -0.09 -0.000729 0.00006561
Pichataro 5.36 0.02 0.000008 0.00000016
Quinceo 5.94 0.60 0.216000 0.12960000
Nurio 6.06 0.72 0.373248 0.26873856
Turicuaro 6.19 0.85 0.614125 0.52200625
Urapicho 6.30 0.96 0.884736 0.84934656
Capacuaro 7.73 2.39 13.651919 32.62808641
12.595722 37.433505
x  5.34
s  0.87
Nov 2019 28

Jarque–Bera Test
Nov 2019 29

Jarque–Bera Test
• The Jarque-Bera statistic can be compared to the χ2 distribution (table)
with 2 degrees of freedom (df or v) to determine the critical value at an
alpha level of 0.05.
• The critical χ2 value is 5.991. Our calculated Jarque-Bera statistic is 4.12
which falls between 0.5 and 0.1, which is greater than the critical value.
• Therefore we accept Ho that there is no difference between our
distribution and a normal distribution (Jarque-Bera χ2, 0.5 > p > 0.1).
Nov 2019 30

Non-Normally Distributed Data
a. Lilliefors SignificanceCorrection
Kolmogorov-Smirnov
a
Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Average PM10 .142 72 .001 .841 72 .000
KS and Shapiro-Wilk test
• Remember that LARGE probabilities denote normally
distributed data
NormallyDistributedData
*. Thisisalowerboundofthetruesignificance.
a.LillieforsSignificanceCorrection
Kolmogorov-Smirnov
a
Shapiro-Wilk
AsthmaCases .069 72 .200* .988 72 .721
Nov 2019 31

NormallyDistributedData
*. Thisisalowerboundofthetruesignificance.
a.LillieforsSignificanceCorrection
In SPSS output above the probabilities are greater
than 0.05 (the typical alpha level), so we accept Ho
that these data are not different from normal.
Kolmogorov-Smirnov
a
Shapiro-Wilk
AsthmaCases .069 72 .200* .988 72 .721
Nov 2019 32

Non-Normally Distributed Data
a. Lilliefors SignificanceCorrection
Kolmogorov-Smirnov
a
Shapiro-Wilk
Average PM10 .142 72 .001 .841 72 .000
• In the SPSS output above the probabilities are less than
0.05 (the typical alpha level), so we reject Ho that these
data are significantly different from normal.
• Important: As the sample size increases, normality
parameters becomes MORE restrictive and it becomes
harder to declare that the data are normally distributed.
• So for very large data sets, normality testing becomes
less important.
Nov 2019 33

D’Agostino Test
• A very powerful test for departures from normality.
• Based on the D statistic, which gives an upper and lower
critical value.
• where D is the test statistic, SS is the sum of squares of the
data and n is the sample size, and i is the order or rank of
observation X. The df for this test is n (sample size).
• First the data are ordered from smallest to largest or largest
to smallest.
Nov 2019 34

0.26050
4.53 (39) 4.69 ...(17 9) 7.73
• Df = n = 17
• If the calculated value falls within the critical range, accept
Ho.
• Since 0.2587 < D = 0.26050 < 0.2860; accept Ho.
• The sample data set is not significantly different than normal (D0.26050, p >
0.05).
212
1.46410
Village
Population
Density i
Mean
Deviates2
Aranza 4.13 1 1.46410 (4.13  5.34)2
1.
Corupo 4.53 2 0.65610
SanLorenzo 4.69 3 0.42250 n 1 17 1
Cheranatzicurin 4.76 4 0.33640   9
Nahuatzen 4.77 5 0.32490 2 2
Pomacuaran 4.96 6 0.14440 T (i 9)X1
Sevina 4.97 7 0.13690 T  (19) 4.13  (2 9)
Arantepacua 5.00 8 0.11560
Cocucho 5.04 9 0.09000 T  63.23
Charapan 5.10 10 0.05760
Comachuen 5.25 11 0.00810
Pichataro 5.36 12 0.00040 D  63.23 
Quinceo 5.94 13 0.36000 (173
)(11.9916)
Nurio 6.06 14 0.51840 D  0.2587,0.286
Turicuaro 6.19 15 0.72250 Critical
Urapicho 6.30 16 0.92160
Capacuaro 7.73 17 5.71210
Mean =5.34 SS=11.9916
Nov 2019 35

Usethe nextlower
nonthe tableif your
samplesizeisNOT
listed.
Nov 2019 36

Normality Test Statistic Calculated Value Probability Results
W/S q 4.16 >0.05 Normal
Jarque-Bera χ2 4.15 0.5 >p >0.1 Normal
Kolmogorov-Smirnov D 0.2007 0.067 Normal
Shapiro-Wilk W 0.8827 0.035 Non-normal
D’Agostino D 0.2605 >0.05 Normal
Comparative performance of the tests
• Different normality tests produce vastly different
probabilities.
• This is due to where in the distribution (central, tails) or what
moment (skewness, kurtosis) they are examining.
Nov 2019 37

Normality tests & sample sizes
• Notice that as the
sample size increases,
the probabilities
decrease.
• In other words, it gets
harder to meet the
normality assumption as
the sample size
increases since even
small departures from
normality are detected.
Nov 2019 38

Which normality test to use?
• W/S:
– Simple, but effective.
– Not available in SPSS.
• Jarque-Bera:
– Tests for skewness and kurtosis
very effective.
– Not available in SPSS.
• D’Agostino:
– Powerful omnibus (skewness,
kurtosis, centrality) test.
– Not available in SPSS
• Kolmogorov-Smirnov:
– Not sensitive to problems in
the tails.
– For datasets>50.
• Shapiro-Wilks:
– Doesn't work well if several
values in the data set are the
same.
– Works best for <50 datasets,
but can be used with larger
data sets.
– Probably the better test in SPSS
Nov 2019 39

Obs
15
7
6
6
5
5
5
4
4
3
Ho:Thesuspectedoutlier isnot different than the sampledistribution.
Ha:Thesuspectedoutlier isdifferent than the sampledistribution.
• Thecritical value for ann =10from Grubbsmodified t table (G
table)at anα =0.05is2.18.
• Since2.671>2.18, reject Ho.
• Thesuspected outlier isfrom asignificantly different sample
population (GMax,2.671,p <0.01).
Grubbs test for outlier
Nov 2019 40

Grubbs test for outlier
Nov 2019 41

• df = n
• where xn is the suspected outlier, xn-1 is the next ranked observation, and x1
is the last ranked observation.
• Ho: The suspected outlier is not different than the sample distribution.
• Ha: The suspected outlier is different than the sample distribution.
• Thecritical value for ann =10from VermaandQuiroz-Ruiz
expanded Dixon table atan α =0.05is0.4122.Since0.6667>
0.4122,reject Ho.
• Thesuspected outlier isfrom asignificantly different
sample population (Q0.6667, p <0.005).
Obs
15
7
6
6
5
5
5
4
4
3
Dixon test for outlier
Nov 2019 42

Dixon test for outlier
Nov 2019 43

Requirements for outlier tests
• The data are from a normal distribution
• There are not multiple outliers (3+)
• The data are sorted with the suspected outlier first.
• If 2 observations are suspected as being outliers and both
lie on the same side of the mean, this test can be
performed again after removing the first outlier from the
data set.
• Caution must be used when removing outliers. Only
remove outliers if you suspect the value was caused by an
error of some sort, or if you have evidence that the value
truly belongs to a different population.
• In small sample size, extreme caution should be used when
removing any data.
Nov 2019 44

Tests ofNormality
a. Lilliefors Significance Correction
Kolmogorov-Smirnov
a
Shapiro-Wilk
Age .110 1048 .000 .931 1048 .000
Tests of Normality
Kolmogorov-Sm irnov
a
Shapiro-Wilk
TOTAL_VALU .283 149 .000 .463 149 .000
Tests ofNormality
*. This is a lower bound of the true significance.
Kolmogorov-Smirnova
Shapiro-Wilk
Z100 .071 100 .200* .985 100 .333
A combination of approaches (SPSS)
Nov 2019 45

4. COMPARE PARAMETRIC AND
NON PARAMETRIC TESTS
Nov 2019 46

Comparing tests
• It is advisable to first check for normality or your data
distribution
• If it is normally distributed, then use parametric tests
• However, it is better to also do non-parametric tests, because
they do not make modeling assumptions, and if you get
similar results, report your approach in the methods section
• If you get dissimilar results, then also report both to ensure
transparency, but give proper justification of which you think
is better and why
• Beware! By using multiple tests, the alfa error will be inflated
so use this strategy only when necessary
Nov 2019 47

Take home messages
• All methods may not agree with each other
• If you perform a normality test, do not ignore the
results.
• If the data are not normal, use non-parametric
tests.
• If the data are normal, use parametric tests.
• If you have groups of data, you MUST test each
group for normality
• Decision must be taken on a case by case basis
Nov 2019 48

THANK YOU
Kindly email your queries to sarizwan1986@outlook.com
Nov 2019 49

Checking for normality (Normal distribution)

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