Normality and Transformations

https://bit.ly/41MCWbv

Camilo G.

ca.garcia2@uniandes.edu.co

Ada A.

a.acevedoa2@uniandes.edu.co

Ronald D.

r.diazf@uniandes.edu.co

What is normality?

When the frequencies of a random variable \(X\) cluster around a central value, it is said that it follows a normal distribution.

In summary a data set variable that appears to follow a normal distribution display three properties:

Most values clustered around the average.
Extreme values are less frequent, but not impossibles.
Distribution display are quite symmetric from the mean.

Normality assessment

The Q-Q plot

Is my data really normal? Let’s see the quntile-quantile (Q-Q) plot:

ggplot(adelie, aes(sample = bill_length_mm)) +
    geom_qq() +
    geom_qq_line()

The Shapiro-Wilk normality test:

In the Shapiro-Wilk normality test, normality is the null hypothesis. The alternative is that data do not follow a normal distribution. Therefore, if \(p\)-value \(\geq \alpha\) there is no evidence against normality.

shapiro.test(adelie$bill_length_mm)


    Shapiro-Wilk normality test

data:  adelie$bill_length_mm
W = 0.99289, p-value = 0.6848

Transformations

The Log-Normal distribution

By applying a log transformation to a log-normal distribution, we can go back to a normal distribution

Other transformations

Transformation	Function	R command	Use
Logarithmic to the right	\(x'=\ln{(x)}\)	`log(x)`	Proportions or skewed to the right
Arcosin	\(x'=\arcsin{(\sqrt{x})}\)	`asin(sqrt(x))`	Proportions or percentages
Square root	\(x'=\sqrt{x+\frac{1}{2}}\)	`sqrt(x+1/2)`	Counts
Exponential	\(x'=e^{x}\)	`exp(x)`	skewed to the left
Reciprocal	\(x=\frac{1}{x}\)	`1/x`	skewed to the right

Informative plots

library(ggpubr)

Histograms

gghistogram(penguins,
    x = "body_mass_g",
    add = "mean", 
    rug = TRUE,
    color = "species", 
    fill = "species",
    palette = c(
        "#00AFBB", 
        "#E7B800", 
        "#FC4E07"
    )
)

Box and Violin plots

ggboxplot(penguins,
    x = "species", 
    y = "body_mass_g",
    color = "species",
    add = "jitter", 
    shape = "species",
    palette = c(
        "#00AFBB", 
        "#E7B800", 
        "#FC4E07"
    )
)

Violin plots (with stats)

ggviolin(penguins,
    x = "species", 
    y = "body_mass_g",
    color = "species",
    shape = "species",
    add = "boxplot", 
    palette = c(
        "#00AFBB", 
        "#E7B800", 
        "#FC4E07"
    )
)

Scatter plot

ggscatter(penguins,
    x = "bill_length_mm",
    y = "body_mass_g",
    color = "black", 
    shape = 21, 
    size = 3,
    add = "reg.line", 
    conf.int = TRUE, 
    cor.coef = TRUE,
        add.params = list(
        color = "blue", 
        fill = "lightgray"
        ),
    cor.coeff.args = list(
        method = "pearson", 
        label.x = 30, 
        label.sep = "\n"
        )
)