Skewness and Kurtosis in Statistics (shape of distributions)



Skewness and kurtosis by statisticalaid.com

 
Skewness

Literally, skewness means the 'lack of symmetry'. We study skewness to have an idea about the shape of the curve which we can draw with the help of the given data. A distribution is said to be skewed if-
  • Mean, median, mode fall at different points, i.e, Mean ≠ Median ≠ Mode.
  • Quartiles are not equidistant from median.
  • The curve drawn with the help of the given data is not symmetrical but stretched more to one side than the other.

The lack of symmetry in a distribution is always determined with reference to a normal distribution, which is always symmetrical. Any departure of a distribution from symmetry leads to an asymmetric distribution and in such cases, we call this distribution as skewed. The skewness may be either positive or negative. Absence of skewness makes the distribution symmetrical.

It is important to emphasize that skewness of a distribution cannot be determined simply by inspection. If we understand the differences between the mean, median and the mode, we should be able to suggest a direction of skew.We can define the skewness of a frequency distribution in three different shapes as following-

(1). Symmetrical distributions
This type of  distribution is known as normal distribution. We can obtain this distribution with height, weight, iq score and many other random variable from real life data. An important characteristics of such distribution is that the mean, median and mode have same value.

symmetrical distribution by statisticalaid.com


(2). Positively skewed distributions
In this distribution, the right tail is long which indicates the presence of extreme values at the positive end of the distribution. This pulls the mean to the right tail. this types of distribution is known as positively skewed distribution. This distributions occur with some real life variable such as family size, wages of the worker etc.

Positively skewed distribution by statisticalaid.com


(3). Negatively skewed distributions
In this types of distribution, the mean is pulled in the negative direction. This occurs in some real life variable such as daily maximum temperature for a month in winter.

negatively skewed distribution by statisticalaid.com

Measures of skewness

We can simply measure the skewness using pearson's coefficient of skewness as below-

Skewness(p)= (Mean-Mode) / Standard Deviation

We can make following decissions from the pearson's coefficient of skewness as following-
  • If mean > mode, the distribution is positively skewed.
  • If mean < mode, the distribution is negatively skewed.
  • If mean = mode, the distribution is not skewed or symmetrical.
In some case, mode cannot be uniquely defined, so we cannot apply the above formula. For alternative we use the following formula pearson's coefficient of skewness-

Skewness(p)= 3(Mean-Mode) / Standard Deviation

We can also mesure the skewness using moments. The formula of skewness using moments is following-

 
And Karl pearson suggest skewness as,


Kurtosis

Kurtosis is defined as the peakedness of a distribution, usually taken in relation to a normal distribution. A curve having relatively higher peak than the normal curve is known as leptokurtic. If the curve is more flat-topped than the normal curve then it is called platykurtic. And a normal curve is called itself as mesokurtic.

Kurtosis by statisticalaid.com


Measures of kurtosis

The measure of kurtosis is defined as the ratio of fourth central moment to the square of the second central moment.


The measure is a pure number and is always positive.

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Base on the value of kurtosis, we can classify a distribution as,
  • If kurtosis>3, the distribution is leptokurtic.
  • If kurtosis<3, the distribution is platykurtic.
  • If kurtosis=3, the distribution is mesokurtic.







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