Question

Consider the following returns:
20, 10, 3, 3, -1
Will the above yield a positive or a negative skewness?
Show you answer both by calculation with skewness formula and in a graph.

Answer 1

Concept

• If the skewness value is between -0.5 and 0.5, the data are fairly symmetrical.
• If the skewness value is between -1 and -0.5, it is negatively skewed or between 0.5 and 1 it is positively skewed, and in these cases, the data is said to be moderately skewed.
• If the skewness is less than -1(negatively skewed) or greater than 1(positively skewed), the data are highly skewed.

Skewness is given by

here N = 4,

= mean of data = (20+10+3+3-1)/5 = 7

Standard deviation S

  

Calculating the standard deviation,

S = 7.4

So, skewness can be calculated as the above formula

So skewness = 0.78, So it is positively skewed,

Note: I have assumed the data to be the population data and not the sample data if the data is sample data, then S = 8.27 and skewness = 1.16. In any case, the data is positively skewed.

If we plot this data, we would see

It is positively skewed as well.

Let me know if you have any doubt in understanding