Measures of Skewness

Introduction: We shall begin with the definition of skewness. Skewness is a twist or irregularity in a set of data that diverges from the symmetrical bell curve, or normal distribution. The curve is known to be skewed if it is shifted to the left or right. Skewness can be stated as a measure of how far a certain distribution deviates from a normal distribution. The skew of a normal distribution is zero, although a lognormal distribution, for example, has a few right skew.

Important points to be noted:

• In statistics, skewness refers to the degree of asymmetry in a probability distribution.
• To variable degrees, distributions can have right (positive) or left (negative) skewness. The skewness of a normal distribution (bell curve) is zero.
• When evaluating a return distribution, investors look for right-skewness, which, like excess kurtosis, better portrays the data set’s extremes rather than focusing just on the average.

Skewness and its utilization:

When evaluating a return distribution, investors look for skewness, which, like kurtosis, evaluates the data set’s extremes rather than focusing just on the average. Short- and medium-term investors must consider extremes since they are less likely to keep a position long enough to trust the average to work itself out.
Standard deviation is often used by investors to forecast future returns; however, it assumes a normal distribution. Because few return distributions are near to normal, skewness is a better metric to use for predicting performance. This is due to the possibility of skewness.
The increased probability of finding a high-skewness data point in a skewed distribution is known as skewness risk. Many financial models that try to predict an asset’s future performance presume that it will follow a normal distribution with equal proportions of central tendency. This type of model will always underestimate the danger of skewness in its predictions if the data is skewed. The less accurate the financial model is, the more biased the data is.

Concept of Skewness:

In addition to positive and negative skew, a distribution might have zero or undefined skew. The data on the right side of a distribution’s curve may taper differently than the data on the left side of the curve. “Tails” are the term for these tapering. A longer or fatter tail on the left side of the distribution is referred to as negative skew, whereas a longer or fatter tail on the right is referred to as positive skew.
Positively skewed data will have a mean that is higher than the median. The opposite is true in a negatively skewed distribution: the mean of negatively skewed data will be less than the median. The distribution has zero skewness if the data graphs symmetrically, regardless of how long or fat the tails are.
The three probability distributions shown below are more positively skewed (or right skewed). Left-skewed distributions are skewed distributions that are negatively skewed.

Methods of measuring skewness:

Skewness can be measured in a variety of ways. Pearson’s first and second skewness coefficients are two popular examples. Pearson’s first coefficient of skewness, also known as Pearson mode skewness, divides the difference between the mode and the mean by the standard deviation. Pearson’s second skewness coefficient, also known as Pearson median skewness, is calculated by subtracting the median from the mean, multiplying the difference by three, and dividing the result by the standard deviation.

• The formula for Pearson’s coefficient of mode skewness is given by:

Sk₁=x-Mos

where, 

Sk1​=Pearson’s first coefficient of skewness 

s =the standard deviation for the sample 

x=is the mean value

Mo=the modal (mode) value​

• The formula for Pearson’s coefficient of median skewness is given by:

Sk₂=3(x-Md)s

where,

Sk2​=Pearson’s second coefficient of skewness 

s =the standard deviation for the sample 

x=is the mean value

Md=the median value

Steps to calculate the coefficient of skewness:

The coefficient of skewness can be calculated using either of the two formulas, depending on the data supplied. Assume a data collection has a mean of 60, a mode of 70, a median of 75, and a standard deviation of 10. The steps for calculating the skewness coefficient are as follows:

When Mode is known:

1. Subtract the mode from the mean in step one. In this case, 60 – 70=-10
2. In step two, To get the coefficient of skewness, divide this value by the standard deviation. Thus, sk1 = -10/ 10 = -1.

When median is known:

1. In Step 1, take the median and subtract it from the mean. For example, say 60 – 75 = -15.
2. In the next step, multiply this value by three. -45 is the result as in here.
3. Finally, to calculate the coefficient of skewness, multiply the value from step 2 by the standard deviation. Thus, sk2 = -45 / 10 = -4.5

Conclusion:

Skewness is a measure of the asymmetry of a real-valued random variable’s probability distribution around its mean in probability theory and statistics. Positive, zero, negative, or undefined skewness values are possible. Negative skew denotes that the tail is on the left side of a unimodal distribution, while positive skew suggests that the tail is on the right side. Skewness does not follow a simple rule when one tail is long, and the other is fat.