Tests on a true normal distribution

Using the tests in a normal distribution in which it is expected to get true from results. Notice that since the data is randomly generated false-negative results may happen.

Visual check using histogram

Visual check using boxplot

Visual check using quantile-quantile plot (Q-Q Plot)

Statitistical normality check 

It is a normal distribution - D'Agostino and Pearson's (DP) test ( 0.593 > 0.05 ).
It is a normal distribution - Lilliefors test ( 0.42 > 0.05 ).
It is a normal distribution - Shapiro-Wilk (SW) test ( 0.484 > 0.05 ).
It is a normal distribution - Anderson-Darling (AD) test ( 0.784 > 0.401 ).

 4 in 4 tests says it is likely to be a normal distribution ( alpha = 0.05 )

Testing an approximate normal distribution

This next distribution is closelly normal. In order to get less false-negative results alpha value is being set to a lower value.

It is NOT a normal distribution - D'Agostino and Pearson's (DP) test ( 0.029 NOT > 0.05 ).
It is a normal distribution - Lilliefors test ( 0.436 > 0.05 ).
It is NOT a normal distribution - Shapiro-Wilk (SW) test ( 0.042 NOT > 0.05 ).
It is a normal distribution - Anderson-Darling (AD) test ( 0.784 > 0.613 ).

 2 in 4 tests says it is likely to be a normal distribution ( alpha = 0.05 )

Testing against an exponential distribution

This next distribution is not normal at all, no true results are expected

It is NOT a normal distribution - D'Agostino and Pearson's (DP) test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Lilliefors test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Shapiro-Wilk (SW) test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Anderson-Darling (AD) test ( 0.784 NOT > 47.222 ).

 0 in 4 tests says it is likely to be a normal distribution ( alpha = 0.05 )

Testing against an random chisquare distribution with 5 degrees of freedom

This next distribution is not normal at all, no true results are expected

It is NOT a normal distribution - D'Agostino and Pearson's (DP) test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Lilliefors test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Shapiro-Wilk (SW) test ( 0.0 NOT > 0.05 ).
It is NOT a normal distribution - Anderson-Darling (AD) test ( 0.784 NOT > 22.743 ).

 0 in 4 tests says it is likely to be a normal distribution ( alpha = 0.05 )

Developed by Cleber Jorge Amaral.

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