Skewness/Kurtosis

Skewness and Kurtosis in R

Skewness measures the the amount of asymmetry in a distribution, while Kurtosis describes the “tailedness” of the curve. These measures are frequently used to assess the normality of the data. There are several methods to calculate these measures. In R, there are at least four different packages that contain functions for Skewness and Kurtosis. This write-up will examine the following packages: e1071, moments, procs, and sasLM.

Data Used

The following data was used in this example.

# Create sample data
dat <- tibble::tribble(
  ~team, ~points, ~assists,
  "A", 10, 2,
  "A", 17, 5,
  "A", 17, 6,
  "A", 18, 3,
  "A", 15, 0,
  "B", 10, 2,
  "B", 14, 5,
  "B", 13, 4,
  "B", 29, 0,
  "B", 25, 2,
  "C", 12, 1,
  "C", 30, 1,
  "C", 34, 3,
  "C", 12, 4,
  "C", 11, 7 
)

Package Examination

Base R and the stats package have no native functions for Skewness and Kurtosis. It is therefore necessary to use a packaged function to calculate these statistics. The packages examined use three different methods of calculating Skewness, and four different methods for calculating Kurtosis. Of the available packages, the functions in the e1071 package provide the most flexibility, and have options for three of the different methodologies.

e1071 Package

The e1071 package contains miscellaneous statistical functions from the Probability Theory Group at the Vienna University of Technology. The package includes functions for both Skewness and Kurtosis, and each function has a “type” parameter to specify the method. There are three available methods for Skewness, and three methods for Kurtosis. A portion of the documentation for these functions is included below:

Skewness

The documentation for the skewness() function describes three types of skewness calculations: Joanes and Gill (1998) discusses three methods for estimating skewness:

Type 1: This is the typical definition used in many older textbooks

\[g_1 = m_1/m_2^{3/2}\]

Type 2: Used in SAS and SPSS

\[ G_1 = g_1\sqrt{n(n-1)}/(n-2) \]
Type 3: Used in MINITAB and BMDP

\[ b_1 = m_3/s^3 = g_1((n-1)/n)^{3/2} \]

All three skewness measures are unbiased under normality. The three methods are illustrated in the following code:

type1 <- e1071::skewness(dat$points, type = 1)
stringr::str_glue("Skewness - Type 1: {type1}")

Skewness - Type 1: 0.905444204379853

type2 <- e1071::skewness(dat$points, type = 2)
stringr::str_glue("Skewness - Type 2: {type2}")

Skewness - Type 2: 1.00931792987094

type3 <- e1071::skewness(dat$points, type = 3)
stringr::str_glue("Skewness - Type 3: {type3}")

Skewness - Type 3: 0.816426058828937

The default for the e1071 skewness() function is Type 3.

Kurtosis

The documentation for the kurtosis() function describes three types of kurtosis calculations: Joanes and Gill (1998) discuss three methods for estimating kurtosis:

Type 1: This is the typical definition used in many older textbooks

\[g_2 = m_4/m_2^{2}-3\]

Type 2: Used in SAS and SPSS

\[G_2 = ((n+1)g_2+6)*\frac{(n-1)}{(n-2)(n-3)}\]
Type 3: Used in MINITAB and BMDP

\[b_2 = m_4/s^4-3 = (g_2 + 3)(1-1/n)^2-3\]

Only \(G_2\) (corresponding to type 2) is unbiased under normality. The three methods are illustrated in the following code:

  # Kurtosis - Type 1
type1 <- e1071::kurtosis(dat$points, type = 1)
stringr::str_glue("Kurtosis - Type 1: {type1}")

Kurtosis - Type 1: -0.583341077124784

# Kurtosis - Type 2
type2 <- e1071::kurtosis(dat$points, type = 2)
stringr::str_glue("Kurtosis - Type 2: {type2}")

Kurtosis - Type 2: -0.299156418435587

# Kurtosis - Type 3
type3 <- e1071::kurtosis(dat$points, type = 3)
stringr::str_glue("Kurtosis - Type 3: {type3}")

Kurtosis - Type 3: -0.894821560517589

The default for the e1071 kurtosis() function is Type 3.

Moments Package

The moments package is a well-known package with a variety of statistical functions. The package contains functions for both Skewness and Kurtosis. But these functions provide no “type” option. The skewness() function in the moments package corresponds to Type 1 above. The kurtosis() function uses a Pearson’s measure of Kurtosis, which corresponds to none of the three types in the e1071 package.

  library(moments)

  # Skewness - Type 1
  moments::skewness(dat$points)

[1] 0.9054442

  # [1] 0.9054442
  
  # Kurtosis - Pearson's measure
  moments::kurtosis(dat$points)

[1] 2.416659

  # [1] 2.416659

Note that neither of the functions from the moments package match SAS.

Procs Package

The procs package proc_means() function was written specifically to match SAS, and produces a Type 2 Skewness and Type 2 Kurtosis. This package also produces a data frame output, instead of a scalar value.

  library(procs)

  # Skewness and Kurtosis - Type 2 
  proc_means(dat, var = points,
             stats = v(skew, kurt))

# A tibble: 1 × 5
   TYPE  FREQ VAR     SKEW   KURT
  <dbl> <int> <chr>  <dbl>  <dbl>
1     0    15 points  1.01 -0.299

Viewer Output:

sasLM Package

The sasLM package was also written specifically to match SAS. The Skewness() function produces a Type 2 Skewness, and the Kurtosis() function a Type 2 Kurtosis.

  library(sasLM)

  # Skewness - Type 2
  Skewness(dat$points)

[1] 1.009318

  # [1] 1.009318
  
  # Kurtosis - Type 2
  Kurtosis(dat$points)

[1] -0.2991564

  # [1] -0.2991564