gskewness

Skewness of a grouped sample.

In some scientific works, once the data have been gathered from a population of interest, it is often difficult to get a sense of what the data indicate when they are presented in an unorganized fashion.

Assembling the raw data into a meaningful form, such as a frequency distribution, makes the data easier to understand and interpret. It is in the context of frequency distributions that the importance of conveying in a succinct way numerical information contained in the data is encountered.

So, grouped data is data that has been organized into groups known as classes. The raw dataset can be organized by constructing a table showing the frequency distribution of the variable (whose values are given in the raw dataset). Such a frequency table is often referred to as grouped data.

Here, we developed a m-code to calculate the skewness of a grouped data.

One can input the returns or modified vectors n and xout containing the frequency counts and the bin locations of the hist m-function, in a column form matrix.

GSKEWNESS(X,0) adjusts the skewness for bias (correction by small sample size). GSKEWNESS(X,1) is the same as GSKEWNESS(X), and does not adjust for bias.

If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data. Here, we use the Bulmers' rule of thumb criterium (1979), about how we can to interpret the skewness number:
-Less than -1 or greater than +1, the distribution is highly skewed
-Between -1 and -0.5 or between +0.5 and +1, the distribution is moderately skewed
-Between -0.5 and +0.5, the distribution is approximately symmetric

Skewness calculation uses the formula,

g1 = m3/m2^1.5, do not adjusted for bias

G1 = SQRT(N*(N - 1))/(N - 2) * g1, adjusted for bias

where:
m2 = second moment of the sample about its mean
m3 = third moment of the sample about its mean
N = sample size

Syntax: function y = gskewness(x,n)

Inputs:
x - data matrix (Size of matrix must be n-by-2; absolut frequency column 1, class mark=column 2)
n - adjusted for bias = 0 (default), do not adjust for bias = 1

Outputs:
y - skewness of the values in x