Data transformation is a powerful tool when the data don't look like forming a normal distribution. The idea of data transformation is that you convert your data so that you can assume the normality and use parametric tests. To determine whether we need any data transformation, we need to check the normality of the data. Although there are several statistical methods for checking the normality, what you should do is to look at a histogram and QQ-plot, and then run a test for checking the normality. You also should read the section for the differences of the two statistical methods explained in this page.

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One important point of data transformation is that you must defend that your data transformation is legitimate. You cannot do arbitrary data transformation so that you can get results you want to get. Make sure you clarify why you do data transformation and why it is appropriate. We prepare data by using a random function. To be able to reproduce the results quickly, we set the seed for the random functions. Set.seed(111) data_normal. Another way to visually investigate whether data forms the normal distribution is to draw a Q-Q plot.

A Q-Q plot shows the mapping between the distribution of the data and the ideal distribution (the normal distribution in this case). Let's take a look at it. Layout(matrix(c(1,2), nrow=1)) qqnorm(data_normal) qqline(data_normal) qqnorm(data_exp) qqline(data_exp) layout(1) If your data are close to the normal distribution, most of the data points should be close to the line.

So obviously, some of the data points in data_exp are far from the line, which means that it is less likely that data_exp were taken from the normal distribution. Statistical tests for normality. One common test for checking the normality is Shapiro-Wilk test. This test works well even for a small sample size, so generally you just need to use this.

The null hypothesis of Shapiro-Wilk test is that the samples are taken from a normal distribution. So, if the p value is less than 0.05, you reject the hypothesis, and thinks that the samples are not taken from a normal distribution. In R, you just need to use shapiro.test() function to do Shapiro-Wilk test. Mcgee auto clicker. Shapiro.test(data_normal) And you get the result.

Shapiro-Wilk normality test data: data_normal W = 0.9737, p-value = 0.8302 In this case, you can still assume the normality. Let's try the same test with data_exp. Shapiro.test(data_exp) Shapiro-Wilk normality test data: data_exp W = 0.6804, p-value = 2.269e-05 So, we reject the null hypothesis, and the samples are not considered to be taken from a normal distribution. Thus, you need to do data transformation or use a non-parametric test. Kolmogorov-Smirnov test. Another test you can use for checking the normality is Kolmogorov-Smirnov test. This test basically checks whether two datasets are taken from the same distribution, but it can be used for comparing one dataset against an ideal distribution (int this case, a normal distribution).

This test is also quite easy to do in R. Ks.test(data_normal, 'pnorm', mean=mean(data_normal), sd=sd(data_normal)) Here, we are comparing the data against a normal distribution (“pnorm”) with the mean and standard deviation calculated from the dataset. One-sample Kolmogorov-Smirnov test data: data_normal D = 0.1394, p-value = 0.7824 alternative hypothesis: two-sided The result says that we can still assume the normality. Let's take a look at the test with data_exp.

Ks.test(data_exp, 'pnorm', mean=mean(data_exp), sd=sd(data_exp)) One-sample Kolmogorov-Smirnov test data: data_exp D = 0.3421, p-value = 0.01361 alternative hypothesis: two-sided We reject the null hypothesis that data_exp were taken from the normal distribution. Because Kolmogorov-Smirnov test is not only for comparing the dataset against the normal distribution, you can make a comparison with other kinds of distributions. Let's try to make a comparison between data_exp and an exponential distribution. You can use “pexp” instead of “pnorm”, but you need to be a little careful about calculating the mean and standard deviation for a log-normal transformation. Ks.test(data_exp, 'pexp') One-sample Kolmogorov-Smirnov test data: data_exp D = 0.1405, p-value = 0.7746 alternative hypothesis: two-sided So, we can think that data_exp were more likely taken from a log-normal distribution rather a normal distribution. You can also use this test for comparing the two distributions (the null hypothesis is that both datasets were originated form the same distributions).