The interpretation of graphical methods requires a good experience of the scientist to avoid misinterpretations [4] , otherwise, it is better to rely on numerical and statistical methods [4] .

The Stem-and-leaf plot is a sort of mixture between a table and a histogram; it uses individual values for numerical data like in tables, it also shows frequency of observations, and the highest and lowest values of the distribution, like in graphics [5] [9] [10] . For each value, there is a “leaf”, which is the last digit of a value, and each “leaf” is attached to the previous digits called “stem” of the same value [9] [10] . To construct the stem-and-leaf plot, the researcher should follow these steps [9] [10] :

1) Determination of the leaf and the stem for each value.

2) Vertical arrangement of the stems in an ascending or descending order.

3) Separation of the leaves from the stems by drawing a vertical line between them.

4) Grouping the leavers to form bars like histogram for each common stem.

The histogram (frequency distribution) plots the observed values in the horizontal axis against their frequency in the vertical axis [5] . In a normal distribution, the graph is bell-shaped curve and symmetric about the mean, and any departure from the bell-shaped curve is a departure from the normal distribution [4] [6] . Histogram can also give us an insight into data holes and outliers [5] .

In a boxplot, the median is a horizontal line inside the box, the Interquartile Range (IQR, range between the first and third quartile) are the length of the box and the whiskers (line extending from the top and bottom of the box) represent the minimum and maximum values [4] [5] . In a normal distribution, the median line is in the center of the box, the whiskers are symmetrical and slightly longer than the parts of the box [4] [5] .

The P-P graph (probability-probability plot) is used to evaluate how well the sample’s distribution conforms to a theoretical distribution (e.g. normal distribution) by plotting two sets of cumulative probability (observed and theoretical) against each other [4] [5] . In a normal distribution, the plotted points follow a straight line and any departures from this straight line indicate departures from a normal distribution [4] .

The Q-Q graph (quantile-quantile plot) (quantiles are values that split a data set into equal portions) is used to evaluate how well the sample’s distribution conforms to a theoretical distribution (e.g. normal distribution) by plotting two sets of quantiles (observed and theoretical) against each other [4] - [6] . In a normal distribution, the plotted points follow a straight line and any departures from this straight line indicate departures from a normal distribution [4] [6] . More information can be obtained from the angle between the straight line and the horizontal axis: if the angle is 45˚, the two compared distributions are identical, if the angle is inferior to 45˚, the distribution of the horizontal axis is more dispersed than the distribution of the vertical axis. But if the angle is superior to 45˚, the distribution of the vertical axis is more dispersed than the distribution of the horizontal axis [6] .

The detrended Q-Q plot shows the differences in values between the sample’s distribution and the theoretical distribution (e.g. normal distribution) [10] . In a normal distribution, the plotted points follow the horizontal line with the value zero on “y” axis and any departures from this horizontal line indicate departures from a normal distribution [10] .

According to the central limit theorem, when the sample size is larger than 30 or 40 individuals, the sample’s distribution tends to be a normal distribution, regardless of the shape of the data, and this implies that the researcher can use parametric tests [5] [6] .

The equality of values between the Mode, Median and Mean is used to assess normality:

$\text{Mode}=\text{Median}=\text{Mean}$

In a normal distribution, the mode, median and mean are nearly equal and any departure from the equality between these three values indicates a departure from a normal distribution [4] .

The Difference between the Mean and the Median is used to assess normality:

$\text{Difference}=\text{Mean}-\text{Median}$

In a normal distribution, the difference’s value is close to zero, but in a nonnormal distribution, the difference’s value is far from zero [6] . The higher the difference’s value is, the more nonnormal the distribution is [6] . If the mean is superior to the mode, the distribution is skewed to the right, but if the mean is inferior to the mode, the distribution is skewed to the left [6] .

The Mean and Standard Deviation (SD) can be resumed in the Coefficient of Variation [4] :

$\text{CV}=\frac{\text{Standard}\text{deviation}}{\text{Mean}}\times 100$

In a normal distribution, the standard deviation is less than half the mean (CV < 50%) but in a nonnormal distribution, the standard deviation is more than half the mean (CV > 50%) [2] [4] [11] .

Skewness is a measure of the asymmetry of a sample’s distribution, while Kurtosis is a measure of the peakedness of a sample’s distribution [4] [8] . In calculation, the researcher has to use the “excess kurtosis”, which is “Excess kurtosis = Kurtosis-3” [8] . In a nonnormal distribution, a positive skew value (positive skewness) indicates that the tail on the right side of the distribution is longer than the left side, and a negative skew value (negative skewness) indicates that the tail on the left side of the distribution is longer than the right side [8] . A positive “excess kurtosis” is called leptokurtic distribution (meaning high peak), and a negative “excess kurtosis” is called platykurtic distribution (meaning flat-topped curve) [8] ( Table 2 ).

${\text{Z}}_{\text{Skewness}}=\frac{\text{Skewness}\text{value}}{{\text{SE}}_{\text{Skewness}}}$, ${\text{Z}}_{\text{Excess}\text{kurtosis}}=\frac{\text{Excess}\text{kurtosis}\text{value}}{{\text{SE}}_{\text{Excess}\text{kurtosis}}}$.

The Kolmogorov-Smirnov test and the Shapiro-Wilt test are Empirical Distribution Functions (EDFs) and evaluate the divergence between the sample distribution and a hypothetical distribution (normal distribution) [10] . The Shapiro-Wilk test is based on the correlation between the data and the corresponding normal scores [5] .

Normality tests have a problem of sensitivity to normality [12] . As the sample size decreases, the sample distribution deviates from normality, but the normality test loses its power to reject the null hypothesis [5] . Conversely, as the sample size increases, the sample distribution approaches normality, but the probability that the normality test rejects the null hypothesis increases [12] .

The Kolmogorov-Smirnov test alone is sensitive to outliers, to surpass this problem, Lilliefors added a correction of the critical value to reject the null hypothesis, resulting in the Lilliefors test (also known as the “Lilliefors corrected K-S test”), even with that, Lilliefors test remains less powerful than the Shapiro-Wilk test to detect the normality of a distribution [5] [12] [13] .

Although the Lilliefors test uses the same formula as the Kolmogorov-Smirnov test, the critical values for rejecting the null hypothesis are different between the two tests due to the Lilliefors correction, so the conclusion regarding the normality of our data will be different [13] .

Like the other statistical tests, the researcher should put hypotheses before calculations. Generally, the hypotheses include: the null hypothesis (H₀: the sample distribution is normal) which is equivalent to a nonsignificant test, the alternative hypothesis (H_a: the sample distribution is nonnormal) which is equivalent to a significant test [5] [6] .

As mentioned above, each of these methods has its own advantages and disadvantages, and Table 3 attempts to give an overview of these methods to help the researcher make the right choice.

꙱: Advantage; ꙱: Between; ꙱: Disadvantage.

Depending on the direction and degree of skewness of the data distribution, various transformation methods can be used: reciprocal, logarithm, cube root, square root and square [2] [15] ( Table 4 ).

Transformation function: y = x^λ; x: Original data; y: Transformed data.

In logarithmic transformation, it is possible to use either natural logarithms (ln) with “base e (=2.7182818…)” or common logarithms (Log) with “base 10” [17] [18] , but the common logarithm has the advantage of being easy to interpret or to check in opposite to natural logarithm [15] [18] .

From Table 4 , the more the data are right-skewed, the lower is the ladder of power, and the more the data are left-skewed, the higher is the ladder of power [16] .

Before applying any transformation on data, some mathematical assumptions should be respected in order to allow the transformation and to have finally a good interpretation ( Table 5 ).

Also, mathematically speaking, for these three transformations, in addition to power transformation, number above 0 and less than 1 (]0, 1[) behave differently than 0 and 1 and numbers above 1 [3] [17] ( Table 6 , Figure 1 ).

RD: Raw Data; TrD: Transformed Data.

As a solution, the researcher adds a positive constant to all raw data values to bring the minimum value to 1 [15] [17] .

After a reciprocal transformation, it is possible to multiply the transformed data by minus one (−1) to preserve the original data’s order, but before the back-transformation, the researcher should again multiply by minus one (−1), so that the back-transformed data and original data will have the same order [16] .

After the correct indication and satisfaction of the assumptions, the researcher can transform the data, but there is no guarantee that the data will be closer to the normal distribution [14] . However, in some cases, the transformation makes the distribution more skewed than the original data [14] .

If normality is satisfied after transformation, calculation of the mean with its standard deviation and the confidence interval of the mean should be performed only on the transformed data [2] . The results obtained on the transformed data do not make sense, so a back-transformation of the confidence interval of the mean to deduce our results on the original data must be done but only at the end [2] . The Standard Deviation (SD) is used for calculation of the confidence interval of the mean on the transformed data, and a back-transformation of the SD to the original scale has no meaning [15] [19] [20] .

Back-transformation is the last step and is very challenging [15] . For back-transformation, the researcher uses the inverse function (not the reciprocal function) and each transformation’s function has its inverse function [21] . To get a proper back-transformation, the researcher follows those steps by taking logarithmic back-transformation (especially common logarithm) as an example [15] [18] [21] [22] :

1) The researcher calculates the mean and the standard deviation of the transformed data.

2) The researcher calculates the confidence interval of the mean (it is preferable to use t values instead of z values for more precision):

$\text{CI}95\text{\%}=\text{Mean}\pm {\text{t}}_{\text{df}-1,\text{α}/2}\times \frac{\text{SD}}{\sqrt{\text{n}}}$

3) At the end, the researcher transforms back the confidence interval using the appropriate inverse function ( Table 7 ).

As an example, for common logarithm the researcher uses the antilogarithms (power to “base 10”):

$\left({10}^{\text{mean}-{\text{t}}_{\text{df}-1,\frac{\text{α}}{2}}\times \frac{\text{SD}}{\sqrt{\text{n}}}};{10}^{\text{mean}+{\text{t}}_{\text{df}-1,\text{α}/2}\times \frac{\text{SD}}{\sqrt{\text{n}}}}\right)$

Transformation function: y = x^λ; Inverse function: x = y^1⁄λ; x: Original data; y: Transformed data.

In this situation, the researcher doesn’t transform the data for each group, but he uses what is known as a “variance-stabilizing transformation”, whose name indicates that it is a transformation with the main objective is to achieve homoscedasticity between groups [19] . To choose the right “variance-stabilizing transformation”, the researcher follows those steps [2] [19] [23] :

1) For each group, the researcher calculates the mean and the variance (or standard deviation).

2) For each group, the researcher plots the variance (or standard deviation) against its mean.

3) Depending on the proportionality between the means and the variances (or standard deviation), the researcher chooses the appropriate transformation from the following table ( Table 8 ).

But back-transformation of the difference between means is more complicated and it is not the scope of this article [20] .