# Testing for Normality By Using Plot

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```					 Testing for Assumptions (listed in 6.2) of the Disturbance
of the Population Regression

Assumption                     Plot                 Statistic
(look for key patterns)    (summarize key
patterns)

a. Linearity           Residual vs. Predicted            NA

E( e | Xs) = 0

Residual vs. Predicted      Breusch - Pagan
b. Constant Variance

Var(e | Xs) = 

Histogram               Skewness

Normal plot (Q-Q or P-P)        Kurtosis
c. Normality
Jarque – Bera

(See Appendix I)         (See Appendix II)

d. Independence        Line plot of the residual   Durbin - Watson

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Appendix I: Testing for Normality By Using a Q-Q Plot
A natural question in applying a normal distribution is: how can we test whether the
the data actually come from a normal distribution? A simple method is to 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 stock rate of return of Exxon.

20

10

Std. Dev = .05
Mean = .014
0                                                                                                   N = 61.00
-.125           -.075           -.025          .025          .075          .125          .175
-.100           -.050           .000          .050          .100          .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
the rate of return of Exxon appears not markedly different from a normal distribution at
least in the middle part.
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 in 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 100 percentile of a normal

2
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)
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
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.

3
0.2000

0.1500

0.1000

0.0500

0.0000
- 0.1 - 0.0 0.00    0.05   0.10       0.15   0.20
000 - 0.0500 00
500          00     00         00     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 is 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.

Normal Probability Plot of Data From a Uniform Distribution

The plot on the right is a normal
probability plot of observations from a
uniform distribution. The plot has an                       1.0
elongated S shape.                                          0.9
0.8
U   0.7
n   0.6
i
f   0.5
o   0.4
r
0.3
m
0.2
0.1
0.0
0.0    0.5      1.0

expunif

4
Normal Probability Plot of Data From an Exponential 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   2
expexpo

5
Appendix II: Testing for Normality By Using a 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
 3 E       
  

The sample skewness is evaluated as follows:

3
1 n xi  x 
3 
ˆ           ˆ 
n i 1   

where:

n
ˆ 1
  xi  x 
2

n i 1

The skewness 3 is 0 for a symmetric population, as can been seen from the formula.
Therefore, the sample skewness  is significantly different from 0, then one can infer
ˆ3
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 the
fourth central moment of the distribution, called the kurtosis  4 :

x   
4
4  E       
  

The sample kurtosis is computed as follows:

4
1 n x i x 
4 
ˆ           ˆ 
n i 1   

The kurtosis measures the amount of the tail probabilities of the distribution and equals 3
for a normal population distribution. Therefore, if the sample kurtosis 4 is significantly
ˆ

6
different from 3, then one can infer that the population distribution is unlikelty to be not
normal.
The Jarque-Bera Statistic combines the two measures  and 4 as follows:
ˆ3     ˆ

n         1         2 
JB  ˆ 3   4  3 
2
       ˆ
6         4           

For a large number of observations, JB higher than 6 suggests that the population
distribution is unlikely to be normal.
The worksheet below illustrates the computation of these statistics for the rate of
return of the Weyerhaeuser stock for recent 60 months.

7
Computing the Skewness, the Kurtosis and the Jarque-Bera Statistc for Recent 60
Monthly Return of the Weyerhaeuser Stock

Month    Weyerhaeuser              z-score        (z-score) 3        (z-score) 4
1      0.27020                  2.8469         23.0741            65.6902
2      0.09449                  0.9295           0.8031             0.7464
3      0.08873                  0.8667           0.6509             0.5641
4    -0.02731                 -0.3996          -0.0638              0.0255
5    -0.10706                 -1.2699          -2.0478              2.6004
6
7
8      0.02551
0.02139
0.08088                  0.1768
0.1318
0.7810           0.0055
0.0023
0.4764             0.0010
0.0003
0.3721
9
10
11    -0.05442
-0.27098
-0.06645                 -0.6955
-3.0586
-0.8267          -0.3364
-28.6127
-0.5649              0.2339
87.5141
0.4670
12
13
14      0.10320
-0.00968
0.13487                  1.0246
-0.2073
1.3701           1.0756
-0.0089
2.5722             1.1021
0.0018
3.5242
15
16
17    -0.10145
-0.02065
-0.02000                 -1.2086
-0.3269
-0.3198          -1.7656
-0.0349
-0.0327              2.1339
0.0114
0.0105
18
19
20      0.11735
-0.08037
-0.02010                   1.1789
-0.9787
-0.3209            1.6386
-0.9373
-0.0331              1.9318
0.9173
0.0106
21
22
23    -0.00513
0.01753
0.02051                -0.1575
0.0898
0.1223         -0.0039
0.0007
0.0018             0.0006
0.0001
0.0002
24
25
26      0.01005
0.08657
-0.04167                   0.0081
0.8431
-0.5563            0.0000
0.5992
-0.1721              0.0000
0.5052
0.0957
27
28
29      0.02899
0.10986
0.01709                  0.2147
1.0973
0.0850           0.0099
1.3211
0.0006             0.0021
1.4496
0.0001
30
31
32    -0.07143
0.16018
-0.01181                 -0.8810
1.6464
-0.2305          -0.6839
4.4625
-0.0122              0.6025
7.3469
0.0028
33
34
35    -0.05976
-0.06610
0.00459                -0.7537
-0.8229
-0.0515          -0.4282
-0.5572
-0.0001              0.3227
0.4586
0.0000
36
37
38      0.00913
-0.10679
0.00000                -0.0019
-1.2669
-0.1016            0.0000
-2.0333
-0.0010              2.5760
0.0001
39
40
41      0.06154
-0.04155
0.11735                  0.5699
-0.5550
1.1789           0.1851
-0.1709
1.6386             0.1055
0.0949
1.9318
42
43
44    -0.06849
-0.04216
-0.15026                 -0.8490
-0.5617
-1.7413          -0.6120
-0.1772
-5.2795              0.5196
0.0995
9.1929
45
46
47    -0.02439
-0.07875
0.07586                -0.3677
-0.9609
0.7262         -0.0497
-0.8873
0.3830             0.0183
0.8526
0.2782
48
49
50      0.12179
0.06514
0.00543                  1.2275
0.6092
-0.0423            1.8495
0.2261
-0.0001              2.2702
0.1378
0.0000
51
52
53      0.04324
0.11606
0.14085                  0.3703
1.1649
1.4354           0.0508
1.5806
2.9572             0.0188
1.8412
4.2447
54
55
56    -0.11934
0.05794
-0.01339                 -1.4039
0.5307
-0.2477          -2.7669
0.1494
-0.0152              3.8843
0.0793
0.0038
57      0.00452                -0.0522          -0.0001              0.0000
58      0.00631                -0.0328            0.0000             0.0000
59    -0.14932                 -1.7310          -5.1869              8.9787
60      0.17021                  1.7558           5.4131             9.5046
AVERAGE      0.00931                0.00000        -0.03913             3.75464
STDEVP      0.09164                      JB=        1.43903

0.30 000

0.20 000

0.10 000

0.00 000
-0.200 0   -0.100 0      0.00 00    0.10 00   0.20 00     0.30 00

-0.100 00

-0.200 00

-0.300 00

8

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