Skewness and Kurtosis in R

Skewness and kurtosis are two important statistics that describe the shape of a distribution.

Skewness measures the asymmetry of a distribution. A positive skewness indicates a distribution that is skewed towards the left, whereas a negative skewness indicates a distribution that is skewed towards the right.
Kurtosis measures the "tailedness" of a distribution. Higher values indicate a distribution with heavier tails, while lower values indicate a distribution with lighter tails.

Here's a tutorial on how to calculate skewness and kurtosis in R:

1. Install and Load Required Packages

The moments package in R provides functions to compute skewness and kurtosis:

install.packages("moments")
library(moments)

2. Sample Data

For this tutorial, let's use a sample dataset. We'll generate two data sets: one from a normal distribution and another from a beta distribution:

set.seed(123)
data_normal <- rnorm(1000)
data_beta <- rbeta(1000, 2, 5)

3. Calculate Skewness

Using the skewness() function from the moments package, you can compute the skewness of the dataset:

skewness_normal <- skewness(data_normal)
skewness_beta <- skewness(data_beta)

print(paste("Skewness of normal data:", skewness_normal))
print(paste("Skewness of beta data:", skewness_beta))

4. Calculate Kurtosis

Similarly, use the kurtosis() function to compute the kurtosis. Note that the standard definition of kurtosis in moments package returns the excess kurtosis, which is the kurtosis minus 3:

kurtosis_normal <- kurtosis(data_normal)
kurtosis_beta <- kurtosis(data_beta)

print(paste("Kurtosis of normal data:", kurtosis_normal))
print(paste("Kurtosis of beta data:", kurtosis_beta))

5. Interpretation

Skewness:
- ~0: The data is fairly symmetrical.
- > 0: The data are right-skewed (i.e., tail is on the right).
- < 0: The data are left-skewed (i.e., tail is on the left).
Kurtosis:
- ~0: The kurtosis is similar to a normal distribution (remember, this is excess kurtosis, so it's the kurtosis value minus 3).
- > 0: The distribution has heavier tails than a normal distribution.
- < 0: The distribution has lighter tails than a normal distribution.

6. Visualization

It's often helpful to visualize the data alongside these statistics:

hist(data_normal, main="Histogram of Normal Data", breaks=30, col="lightblue", border="black")
hist(data_beta, main="Histogram of Beta Data", breaks=30, col="lightblue", border="black")

This will give you a clear picture of the skewness and kurtosis of the data.

7. Wrap-up

Understanding skewness and kurtosis is essential for many statistical analyses since assumptions about the normality of errors can be critical for hypothesis testing. These metrics provide a quantitative way to describe the shape and tails of a distribution.

Skewness and Kurtosis Functions in R:
- Use skewness() and kurtosis() functions from various packages to compute skewness and kurtosis.
```
library(e1071)

# Skewness
skew_value <- skewness(my_data)

# Kurtosis
kurt_value <- kurtosis(my_data)
```
R Moments Package for Skewness and Kurtosis:
- The moments package provides the skewness() and kurtosis() functions.
```
library(moments)

skew_value <- skewness(my_data)
kurt_value <- kurtosis(my_data)
```
Descriptive Statistics in R for Skewness and Kurtosis:
- Use summary() or psych::describe() for an overview of skewness and kurtosis in a dataset.
```
summary(my_data)
psych::describe(my_data)
```
Skewness and Kurtosis Tests in R:
- Conduct tests like Jarque-Bera or Shapiro-Wilk to assess normality.
```
jarque_bera_test <- jarque.bera.test(my_data)
shapiro_test <- shapiro.test(my_data)
```
R Packages for Statistical Moments:
- Explore packages like e1071, moments, and base R functions for statistical moments.
```
library(e1071)
library(moments)

skewness_value <- skewness(my_data)
kurtosis_value <- kurtosis(my_data)
```
Visualizing Skewness and Kurtosis in R:
- Create histograms, density plots, or boxplots to visualize skewness and kurtosis.
```
hist(my_data)
```
R Functions for Normality Tests:
- Use functions like shapiro.test() or ad.test() for normality tests.
```
shapiro_test <- shapiro.test(my_data)
```
Checking Normality Assumptions in R:
- Assess normality assumptions before applying parametric statistical tests.
```
qqnorm(my_data)
qqline(my_data)
```
Handling Skewed Data in R:
- Transform data using techniques like log transformation for skewed distributions.
```
log_transformed_data <- log(my_data)
```