QMETH Note 1: Testing for Normality
As Sharon applies a normal distribution for forecasting the rate of return of a stock
that has attracted her attention for investment, a natural question has occurred to her:
how can she find out whether the rate of return of a stock actually follows a normal
distribution? Although normality of a stock’s rate of return is accepted within the
investment profession, Sharon wonders if she can confirm the validity of the practice. In
this section, we learn tools to check if the normal distribution is a reasonable model for a
variable.
Histogram
A simple method is to collect data on the rate of return of a stock for a reasonable
number of periods, construct a histogram, and compare the shape with the normal
distribution that has the same mean and the standard deviation as the sample mean and
the sample standard deviation of the data, respectively. Fortunately, several convenient
statistical packages are available for drawing both the histogram with the normal curve
superimposed. As an example, the figure below shows a histogram with a normal curve
for recent 61 observations of the monthly rate of return of Exxon. It is produced by
SPSS.
20
10
Std. Dev = .05
Mean = .014
0
N = 61.00
-.125
-.075
-.100
-.025
-.050
.025
.000
.075
.050
.125
.100
.175
.150
The histogram uses only 61 observations, whereas the normal curve superimposed
depicts the histogram using infinitely many observations. Therefore, sampling errors
should show up as gaps between the two curves. The graph shows that the distribution of
1
the rate of return of Exxon appears not markedly different from a normal distribution, at
least in the middle part.
Normal probability plot
The procedure described above is easy to understand, but is not effective for revealing
a subtle but systematic departure of the histogram from normality. A better graphical
check of normality is a normal probability plot. The plot can be easily developed using
Excel and we describe the process below.
The first step is to sort the data from the lowest to the highest. Let n be the number of
observations. Then, the lowest observation, denoted as x(1) is the (1/n) th quantile of the
data. A quantile times 100 is the percentile, so x(1) is also the (1/n) x 100 th percentile of
the data. With this convention, however, the largest observation becomes the 100
percentile of the data, which presents a problem, as the 100th percentile of a normal
distribution is infinity, the value that can never be assumed in observation. A suggested
choice is to define the i-th largest observation, x(i) as the (i/(n+1))th quantile, or the
(i/(n+1)) x 100 percentile of the data. In the Excel worksheet on the next page both
choices are computed for comparison. The next step is to determine for each observation
the corresponding quantile of the normal distribution that has the same mean and the
standard deviation as the data. The following Excel function is a convenient way to
determine the normal (i/(n+1) th quantile, denoted as x’(i).
x’(i) = NORMINV(i/(n+1), sample mean, sample standard deviation).
i/(n+1)
x’(i)
This value is the expected quantile if the data come from a normal distribution. x(i)
2
should be close to x’(i) if the normality of the distribution is true. The quantile of the
normal distribution -- with the mean and the standard deviation equaling the sample
mean and the sample standard deviation, respectively -- are computed in the column
with the heading “Expected.”
A normal probability plot is a scatterplot of the data vs. the expected quantiles. The
plot is shown below. If the data indeed come from a normal distribution, then the
scatterplot should deviate in a random fashion from the reference line. Note that the 45
degree line serves as a convenient reference line for detecting a systematic departure
from normality.
0.2000
0.1500
0.1000
0.0500
0.0000
-0.1 -0.0 0.00 0.05
000 -0.0500
500
00
00
0.10
00
0.15
00
0.20
00
-0.1000
-0.1500
We show below two normal probability plots, one for a sample of 60 independent
observations from a uniform distribution over the interval (0, 1) and the other for a
sample of 60 independent observations from an exponential distribution with mean 0.5.
Probabilities at both tails of a uniform distribution are significantly larger than a normal
distribution. On the other hand, an exponential distribution is skewed. These departures
from the normality are quite common, and the two plots show how each departure can be
detected by using the normal probability plots.
3
Normal Probability Plot of Data From a Uniform Population Distribution
The plot on the right is a normal probability plot of observations from a uniform
distribution. The plot has an elongated S shape.
1.0
0.9
0.8
U
n
i
f
o
r
m
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
expunif
Normal Probability Plot of Data From an Exponential Population Distribution
The plot on the right is a normal probability plot of observations from an exponential
distribution. The plot is convex.
2
E
x
p
o1
n
0
0
1
expexpo
4
2
Jarque-Bera Statistic
A normal probability plot test can be inconclusive when the plot pattern is not clear.
In such case it is useful to compute a few numbers that measure non-normality. The
asymmetry of the distribution is measured by the skewness which is the third central
moment of the distribution:
x   

3 E 
  
3
The sample skewness is evaluated as follows:
ˆ3 

3
1 n xi  x 

 
n i 1  ˆ 
where:
n
1
2
ˆ   xi  x 
n i 1
The skewness  3 is 0 for a symmetric population, as can been seen from the formula.
ˆ 3 is significantly different from 0, then one can infer
Therefore, the sample skewness 
that the population distribution is unlikely to be symmetric and hence not normal.
Another number that can be used to check the normality of the distribution is the
fourth central moment of the distribution, called the kurtosis  4 :
x   

4  E 
  
4
The sample kurtosis is computed as follows:
4
x  x 
ˆ 4  1  
 i



ˆ 
n i 1 
n
The kurtosis measures the amount of the tail probabilities of the distribution and equals 3
ˆ 4 is significantly
for a normal population distribution. Therefore, if the sample kurtosis 
5
different from 3, then one can infer that the population distribution is unlikelty to be a
normal distribution.
ˆ 4 as follows:
ˆ 3 and 
The Jarque-Bera Statistic combines the two measures 
n  2 1
2
ˆ 3   
ˆ 4  3 

JB 

6 
4
For a large number of observations, JB higher than 6 suggests that the population
distribution is unlikely to be a normal distribution. The Excel worksheet below illustrates
the computation of these statistics for the rate of return of the Weyerhaeuser stock for
recent 60 months.
Month
Weyerhaeus er
0.27020
0.09449
0.08873
-0.02731
-0.10706
0.02551
0.02139
0.08088
-0.05442
-0.27098
-0.06645
0.10320
-0.00968
0.13487
-0.10145
-0.02065
-0.02000
0.11735
-0.08037
-0.02010
-0.00513
0.01753
0.02051
0.01005
0.08657
-0.04167
0.02899
0.10986
0.01709
-0.07143
0.16018
-0.01181
-0.05976
-0.06610
0.00459
0.00913
-0.10679
0.00000
0.06154
-0.04155
0.11735
-0.06849
-0.04216
-0.15026
-0.02439
-0.07875
0.07586
0.12179
0.06514
0.00543
0.04324
0.11606
0.14085
-0.11934
0.05794
-0.01339
57
0.00452
58
0.00631
59
-0.14932
60
0.17021
AVERAGE
0.00931
STDEVP
0.09164
z-sc ore
2.8469
0.9295
0.8667
-0.3996
-1.2699
0.1768
0.1318
0.7810
-0.6955
-3.0586
-0.8267
1.0246
-0.2073
1.3701
-1.2086
-0.3269
-0.3198
1.1789
-0.9787
-0.3209
-0.1575
0.0898
0.1223
0.0081
0.8431
-0.5563
0.2147
1.0973
0.0850
-0.8810
1.6464
-0.2305
-0.7537
-0.8229
-0.0515
-0.0019
-1.2669
-0.1016
0.5699
-0.5550
1.1789
-0.8490
-0.5617
-1.7413
-0.3677
-0.9609
0.7262
1.2275
0.6092
-0.0423
0.3703
1.1649
1.4354
-1.4039
0.5307
-0.2477
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
-0.0522
-0.0328
-1.7310
1.7558
0.00000
JB=
(z-sc ore)
23.0741
0.8031
0.6509
-0.0638
-2.0478
0.0055
0.0023
0.4764
-0.3364
-28.6127
-0.5649
1.0756
-0.0089
2.5722
-1.7656
-0.0349
-0.0327
1.6386
-0.9373
-0.0331
-0.0039
0.0007
0.0018
0.0000
0.5992
-0.1721
0.0099
1.3211
0.0006
-0.6839
4.4625
-0.0122
-0.4282
-0.5572
-0.0001
0.0000
-2.0333
-0.0010
0.1851
-0.1709
1.6386
-0.6120
-0.1772
-5.2795
-0.0497
-0.8873
0.3830
1.8495
0.2261
-0.0001
0.0508
1.5806
2.9572
-2.7669
0.1494
-0.0152
3
(z-sc ore)
65.6902
0.7464
0.5641
0.0255
2.6004
0.0010
0.0003
0.3721
0.2339
87.5141
0.4670
1.1021
0.0018
3.5242
2.1339
0.0114
0.0105
1.9318
0.9173
0.0106
0.0006
0.0001
0.0002
0.0000
0.5052
0.0957
0.0021
1.4496
0.0001
0.6025
7.3469
0.0028
0.3227
0.4586
0.0000
0.0000
2.5760
0.0001
0.1055
0.0949
1.9318
0.5196
0.0995
9.1929
0.0183
0.8526
0.2782
2.2702
0.1378
0.0000
0.0188
1.8412
4.2447
3.8843
0.0793
0.0038
4
-0.0001
0.0000
-5.1869
5.4131
-0.03913
1.43903
0.0000
0.0000
8.9787
9.5046
3.75464
0.3 0000
0.2 0000
0.1 0000
0.0 0000
-0.2 000
-0.1 000
0.0 000
-0.1 0000
-0.2 0000
-0.3 0000
6
0.1 000
0.2 000
0.3 000