Moments, Kurtosis N Skewness

Moments
Kurtosis
Skewness
Applications of skewness and kurtosis
Examples

1. Moments : Moments are a set of statistical parameters to measure a distribution. Types of Moments: Moments about mean or central moments Moments about arbitrary point or raw moments

2. Properties of central moments : The zero central moment is one i.e. The first central moment is always zero i.e. The second central moment or In a symmetric distribution, the odd moments are always zero i.e.  

3. Properties of raw moments The raw moments can be expressed in terms of central moment. In particular with r = 2,3,4 and and For r = 2, For r = 3, For r = 4,  

4. Skewness When a distribution is not symmetrical (or is asymmetrical) it is called a skewed distribution. Symmetrical distribution : A distribution is said to be symmetrical if mean = median = mode. A symmetrical distribution when plotted on a graph will give a perfectly bell shaped curve which is also known as “normal curve”. Absolute skewness = mean – mode Symmetric

5. Positively skewed distribution : If the frequency curve has a longer tail to be right (i.e. mean is to the right of mode) then distribution is said to be a positively skewed distribution. Positively skewed

6. Negatively skewed distribution : If the frequency curve is more elongated to the left (i.e. mean is to the left of mode) then distribution is said to be a negatively skewed distribution. Negatively skewed

7. Coefficient of skewness, Karl Pearson’s coefficient of skewness :       (1) If then distribution is positively skewed. (2) If then distribution is negatively skewed. (3) If then distribution is symmetric.  

8. Kurtosis The kurtosis of a random variable is denoted by and defined as below :     Kurtosis gives information on the peakedness or height of the peak of a distribution. The curve which is neither flat nor peaked is called a Mesokurtic. The curve which is flatter than the normal curve is called Platykurtic. The curve which is more peaked is called Leptokurtic. (1) If then the curve is Leptokurtic. (2) If then the curve is Platykurtic. (3) If then the curve is Mesokurtic.  

9. Here, Solution: Examples 3: The first four moments of distribution about x = 2 are 1, 2.5, 5.5, and 16. Calculate the four moments about mean .             Moments about mean,            

10. APPLICATION OF SKEWNESS Skewness is a descriptive statistic that can be used in conjunction with the  histogram  and the normal  quantile plot  to characterize the distribution. Skewness indicates the direction and relative magnitude of a distribution's deviation from the normal distribution. Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as  value at risk  in finance) Many models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative. K-squared test  is based on sample skewness and sample kurtosis.

11. APPLICATION OF KURTOSIS The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods. Applying  band-pass filters  to  digital images , kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed  kurtosis convergence , can be used to detect image splicing in  forensic analysis . Pearson's definition of kurtosis is used as an indicator of intermittency in  turbulence . For non-normal samples, the variance of the sample variance depends on the kurtosis.

12. Thank you ! Nishita Kalyani