Normality Test

 

 

Normality tests are used to determine how likely a variable is normally distributed.

 

An informal approach to testing normality is to compare a histogram of the sample data to a normal probability curve, or a quantile-quantile plot (QQ plot) of the standardized data against the standard normal distribution.

 

In QQ plot, the correlation between the sample data and normal quantiles (a measure of the goodness of fit) measures how well the data can be modeled by a normal distribution. For normal data the points plotted in the QQ plot should fall approximately on a straight line, indicating high positive correlation. These plots are easy to interpret and also have the benefit that outliers are easily identified.

 

This function performs univariate normality tests, which include the Anderson–Darling test, the Cramér–von-Mises criterion, the Lilliefors test for normality (itself an adaptation of the Kolmogorov–Smirnov test), the Pearson's chi-square test, and the Shapiro–Francia test.

 

In comparisons of power, it is found Anderson-Darling test is one of the best Empirical distribution function statistics for detecting most departures from normality. The only statistic close was the Cramér–von Mises test statistic. It may be used with small sample sizes n ≤ 25. Very large sample sizes may reject the assumption of normality with only slight imperfections, but industrial data with sample sizes of 200 and more have passed the Anderson–Darling test.

 

Below is a sample input window:

 

 

Below is a sample output:

              

 Normality Test

 

         Shapiro-Francia normality test

 

data:  SBP

W = 0.8744, p-value < 2.2e-16

 

                

 Histogram of SBP

      low upp med.point frequency     percent

 [1,]  80 100        90        15 0.094339623

 [2,] 100 120       110       267 1.679245283

 [3,] 120 140       130       331 2.081761006

 [4,] 140 160       150       109 0.685534591

 [5,] 160 180       170        37 0.232704403

 [6,] 180 200       190        21 0.132075472

 [7,] 200 220       210        12 0.075471698

 [8,] 220 240       230         2 0.012578616

 [9,] 240 260       250         1 0.006289308

 

         Shapiro-Francia normality test

 

data:  DBP

W = 0.9094, p-value < 2.2e-16

 

                

 Histogram of DBP

      low upp med.point frequency    percent

 [1,]  50  60        55       163 2.05031447

 [2,]  60  70        65       282 3.54716981

 [3,]  70  80        75       283 3.55974843

 [4,]  80  90        85        40 0.50314465

 [5,]  90 100        95        16 0.20125786

 [6,] 100 110       105         3 0.03773585

 [7,] 110 120       115         4 0.05031447

 [8,] 120 130       125         2 0.02515723

 [9,] 130 140       135         1 0.01257862

[10,] 140 150       145         1 0.01257862

 

Empower also outputs following plots: