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Load required R packages

library("dplyr")
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library("ggpubr")
## Loading required package: ggplot2

Store the data in the variable my_data

my_data <- ToothGrowth

Check your data We start by displaying a random sample of 10 rows using the function sample_n()[in dplyr package].

Show 10 random rows:

set.seed(1234)
dplyr::sample_n(my_data, 10)
##     len supp dose
## 1  21.5   VC  2.0
## 2  17.3   VC  1.0
## 3  27.3   OJ  2.0
## 4  18.5   VC  2.0
## 5   8.2   OJ  0.5
## 6  26.4   OJ  1.0
## 7  25.8   OJ  1.0
## 8   5.2   VC  0.5
## 9   6.4   VC  0.5
## 10  9.4   OJ  0.5

Assess the normality of the data in R We want to test if the variable len (tooth length) is normally distributed.

Case of large sample sizes If the sample size is large enough (n > 30), we can ignore the distribution of the data and use parametric tests.

The central limit theorem tells us that no matter what distribution things have, the sampling distribution tends to be normal if the sample is large enough (n > 30).

However, to be consistent, normality can be checked by visual inspection [normal plots (histogram), Q-Q plot (quantile-quantile plot)] or by significance tests].

Visual methods Density plot and Q-Q plot can be used to check normality visually.

Density plot: the density plot provides a visual judgment about whether the distribution is bell shaped.

library("ggpubr")
ggdensity(my_data$len, 
          main = "Density plot of tooth length",
          xlab = "Tooth length")

Q-Q plot: Q-Q plot (or quantile-quantile plot) draws the correlation between a given sample and the normal distribution. A 45-degree reference line is also plotted.

library(ggpubr)
ggqqplot(my_data$len)

It’s also possible to use the function qqPlot() [in car package]:

library("car")
## Warning: package 'car' was built under R version 4.3.2
## Loading required package: carData
## Warning: package 'carData' was built under R version 4.3.2
## 
## Attaching package: 'car'
## The following object is masked from 'package:dplyr':
## 
##     recode
qqPlot(my_data$len)

## [1] 23  1

As all the points fall approximately along this reference line, we can assume normality.

Normality test Visual inspection, described in the previous section, is usually unreliable. It’s possible to use a significance test comparing the sample distribution to a normal one in order to ascertain whether data show or not a serious deviation from normality.

There are several methods for normality test such as Kolmogorov-Smirnov (K-S) normality test and Shapiro-Wilk’s test.

The null hypothesis of these tests is that “sample distribution is normal”. If the test is significant, the distribution is non-normal.

Shapiro-Wilk’s method is widely recommended for normality test and it provides better power than K-S. It is based on the correlation between the data and the corresponding normal scores.

Note that, normality test is sensitive to sample size. Small samples most often pass normality tests. Therefore, it’s important to combine visual inspection and significance test in order to take the right decision.

The R function shapiro.test() can be used to perform the Shapiro-Wilk test of normality for one variable (univariate):

shapiro.test(my_data$len)
## 
##  Shapiro-Wilk normality test
## 
## data:  my_data$len
## W = 0.96743, p-value = 0.1091

From the output, the p-value > 0.05 implying that the distribution of the data are not significantly different from normal distribution. In other words, we can assume the normality.