Moments and Measures of Skewness and Kurtosis Moments The term moment has been taken from physics. The term moment in statistical use is analogous to moments of forces in physics. In statistics the values measure something relative to the center of the values. Moments are the constants of a population, as mean, variance, etc are. These constants help in deciding the characteristics of the population. Moments help in finding Arithmetic Mean, Standard Deviation and Variance of the population directly and they help, in knowing the graphic shapes of the population.
There two types Raw Moments and Central Moments. Ref: (link) First (s=1) The 1st moment = (x 1 1 + x 2 1 + x 3 1 +... + x n1 )/n = (x 1 + x 2 + x 3 +... + x n )/n Central Moment This formula is identical to the formula, to find the sample mean. You just add up all of the values and divide by the number of items in your data set. Second (s=2) The 2nd moment around the mean = Σ(x i μ x ) 2 The second is the Variance. Third (s=3) The 3rd moment = (x 1 3 + x 2 3 + x 3 3 +... + x n3 )/n The third is skewness.
Fourth (s=4) The 4th moment = (x 1 4 + x 2 4 + x 3 4 +... + x n4 )/n The fourth is kurtosis.
Skewness Some distributions of data, such as the bell curve are symmetric. This means that the right and the left of the distribution are perfect mirror images of one another. Not every distribution of data is symmetric. Sets of data that are not symmetric are said to be asymmetric. The measure of how asymmetric a distribution can be is called skewness. The mean, median and mode are all measures of the center of a set of data. The skewness of the data can be determined by how these quantities are related to one another. Skewed to the Right Data that are skewed to the right have a long tail that extends to the right. An alternate way of talking about a data set skewed to the right is to say that it is positively skewed. In this situation the mean and the median are both greater than the mode. As a general rule, most of the time for data skewed to the right, the mean will be greater than the median. In summary, for a data set skewed to the right: Always: mean greater than mode Always: median greater than mode Most of the time: mean greater than median Skewed to the Left The situation reverses itself when we deal with data skewed to the left. Data that are skewed to the left have a long tail that extends to the left. An alternate way of talking about a data set skewed to the left is to say that it is negatively skewed. In this situation the mean and the median are both less than the mode. As a general rule, most of the time for data skewed to the left, the mean will be less than the median. In summary, for a data set skewed to the left: Always: mean less than mode Always: median less than mode Most of the time: mean less than median Measures of Skewness It s one thing to look at two set of data and determine that one is symmetric while the other is asymmetric. It s another to look at two sets of asymmetric data and say that one is more skewed than the other. It can be very subjective to determine which is more skewed by simply looking at the graph of the distribution. This is why there are ways to numerically calculate the measure of skewness. One measure of skewness, called Pearson s first coefficient of skewness, is to subtract the mean from the mode, and then divide this difference by the standard deviation of the data. The reason for dividing the difference is so that we have a dimensionless quantity. This explains why data skewed to the right has positive skewness. If the data set is skewed to the right, the mean is greater than the mode, and so subtracting the mode from the mean gives a positive number. A similar argument explains why data skewed to the left has negative skewness. Pearson s second coefficient of skewness is also used to measure the asymmetry of a data set. For this quantity we subtract the mode from the median, multiply this number by three and then divide by the standard deviation. Kurtosis This Greek word has the meaning "arched" or "bulging," making it an apt description of the concept known as kurtosis.
Distributions of Data and Probability Distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Other distributions are bimodal, and have two peaks. The degree of flatness or peakedness is measured by kurtosis. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. The kurtosis of a distribution is in one of three categories of classification: Mesokurtic Leptokurtic Platykurtic Mesokurtic Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution. The normal curve is called Mesokurtic curve.the kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Leptokurtic A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. If the curve of a distribution is more peaked than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix "lepto" meaning "skinny." Platykurtic The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails. Many times they possess a peak lower than a mesokurtic distribution. If a curve is less peaked than a normal curve, it is called as a platykurtic curve. The meaning of the prefix "platy" is "broad". Relation between Raw and Central Moments Recall m r = (1/n) ( x i - x ) r, for r = 0, 1, 2, You can apply binomial theorem and then expand R. H. S. of above relation. Then use the definition of raw moments. You will get following results. These are the relationships in which central moments are expressed in terms of raw moments of lower order. Proof of the following results is simple. 1) m 1 = 0 always
2) m 2 = m 2 (m 1 ) 2 3) m 3 = m 3 3m 2 m 1 + 2(m 1 ) 3 4) m 4 = m 4 4m 3 m 1 + 6m 2 (m 1 ) 2-3(m 1 ) 4 Thus, if raw moments are known then the central moments can be obtained (and conversely). Properties of Central Moments 1. The central moments are invariant to the change of origin. Let U = X A, then since that X = A + U, r th central moment of U, denoted by m r (u) = (1/n) (u i - u ) r = (1/n) (x i A + A - x ) r = m r (x) 2. Effect of change of scale Let U = X / h then X = h U. Hence r th central moment of U, denoted by m r (u) = (1/n) (u i - u ) r = (1/n) [(x i / h) ( x / h)] r = (1/ h r )m r (x) 3. You can combine above two properties and apply it while obtaining moments. We have defined r th central moment of a random variable X as m r = (1/n) (x i x ) r. Define U = (X A )/h, then X = A + h U and you will get m r (X) =h r [m r (U)], where m r (U) and m r (X) denotes r th central moments of U and X respectively. Coefficient of skewness We can specify several indexes of skewness. We list them below. 1. Karl Pearson s Coefficient of skewness S 1 = (A.M. Mode)/ S. D. This is based on measures of central tendency. But if mode is not uniquely defined then this measure is also not well defined. In this case you can use the next measure, 2. Karl Pearson s measure of Coefficient of skewness S 2 = 3(A.M. Me) / S.D. 3. Bowley s coefficient of skewness, SkB, (based on quartiles) Where Q 1 and Q 3 are respectively lower and upper quartiles. Below Q 1, 25% observations lie and above Q 3, 25 % observations lie. Determination of Q 1 and Q 3 is easy. It is done similar to Q 2 (which is same as median).it is particularly useful if you have open-ended classes (such as less than or greater than a particular value)
4. Coefficient of skewness (based on moments) Y 1 = β 1 Coefficient of skewness is a pure number (no units) and is independent of change of origin and scale