Moments-Central and Raw Moments- Mean, kurtosis, Variance and Skewness

 

Data are information collected from various sources in various forms. These data are analyzed to get meaningful conclusion and insights about the dataset. A single datum is hardly useful. We usually require a dataset consisting a series of datapoints to carry out any analysis or get to a conclusion. This series of datapoint traces the path on which the variable under consideration moves. This traced path can be very useful to understand the pattern of the data and also predicts its future movement. To understand the pattern, we need to consider certain characteristics of the path.

The points when plotted on graph paper, the traced path takes the form of a curve. And the characteristics of the path such as mean, variance, kurtosis and skewness can be found out using moments. These are two types of moments- raw and central.

Central moment measures the deviation of datapoints from mean. Whereas, raw moment measures the deviation of datapoints from 0.

 The 1^st order raw moment is mean.

The 2^nd order central moment is variance.

Skewness is a measure of symmetry. A curve is said to be symmetric when mean = median = mode. A curve is said to be positively skewed if mean > median > mode and it is said to be negatively skewed when mean < median <mode.

Skewness lies between -1 to 1. If it lies between -0.5 to 0.5, then the curve is symmetric. If it lies between 0.5 to 1, it is positively skewed and if it lies between -0.5 to -1 then it is said to be negatively skewed.

Kurtosis is the measure of flatness. If Kurtosis =3, the curve is said to be Mesokurtic Curve, if it is < 3, it is called Platykurtic curve and if it is > 3 it is called Leptokurtic Curve.

Mean is the central value of the curve; it divides the curve in two equal parts depending on the weight of the data. If the curve is not symmetric, it tends to be on the side of the curve with higher value. Variance measures the spread of the curve. The extent on which distribution are spread out from mean. Skewness basically measures the symmetry of the data. Whether it is balanced or inclined or biased towards a particular side. Kurtosis measures peak and tail of data distribution. Whether there is any outlier or biasness can be identified by Kurtosis.

 

Suppose, Mr. X is an investor who is tracking the performance of a particular fund, he is interested in. The first thing he wanted to know that what has been the average return of the fund over the years. Secondly, what is the spread of the data. Cause a highly spread data won’t give a good idea about the real situation through its mean. Thirdly, if there has been any un-natural high return at any point which might affect mean or give him an idea about the real situation of the fund. Sometimes, there are fake demand created in market to raise the price of stock and due to it being fake, the price rise doesn’t hold for long. Fourthly, he may want to check the probability of his gain.

All these are necessary questions in order to determine the quality of stocks. And these can be answered using averages, variances, Kurtosis and skewness respectively. If Kurtosis is greater than 3, there is a chance of outlier in higher side. If skewness is left tailed, probability of profit. While right tailed curve indicates better chance of making profit.

-Jags