Skip to main content

Kurtosis

Image
By itsRashad

Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment mu_4 of a distribution. There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted beta_2 or alpha_4 defined by

beta_2=(mu_4)/(mu_2^2),
(1)

where mu_i denotes the ith central moment (and in particular, mu_2 is the variance). This form is implemented in Mathematica as Kurtosis[dist].

The kurtosis "excess" is denoted gamma_2 or b_2, is defined by

gamma_2=(mu_4)/(mu_2^2)-3,
(2)

and is implemented in Mathematica as KurtosisExcess[dist]. Kurtosis excess is commonly used because gamma_2 of a normal distribution is equal to 0, while the kurtosis proper is equal to 3.

Unfortunately, Abramowitz and Stegun (1972) confusingly refer to beta_2 as the "excess or kurtosis."

Lepto-Kurtic: - If a curve is more peaked than normal curve then it is colled Lepto-Kurtic.

Platy-Kurtic: - If a curve is more flat-tapped than normal curve then it is called Platy-Kurtic.

Meso-Kurtic: -The curve representing a normal shape in a frequency distribution is called Meso-Kurtic.

http://grants.hhp.coe.uh.edu/doconnor/PEP6305/KurtosisPict.jpg

Labels:

Comments

  1. BigBendRegionOctober 17, 2017 at 4:49 PM

    But kurtosis does not measure peakedness at all. Rather, it measures the outlier (rare, extreme value) characteristic of distribution. as compared to that of a normal distribution. Please see here https://en.wikipedia.org/wiki/Talk:Kurtosis#Why_kurtosis_should_not_be_interpreted_as_.22peakedness.22 for a complete explanation.

    ReplyDelete

Post a Comment

Popular Posts

Graphical Distribution of Frequency Distribution

By itsRashad
Frequency distribution can be presented graphically in any one of the following ways: Histogram Frequency Polygon Smooth Frequency Curve Cumulative Frequency Curve of Ogive Curve Pie-Chart Histogram: - A histogram is an area diagram in which the frequencies corresponding to each class interval of frequency distribution are by the area of a rectangle without leaving no gap between the cosective rectangles. Frequency Polygon: - This is one kind of histogram which is represented by joining the straight lines of the mid points of the upper horizontal side of each rectangle with adjacent rectangles. Smooth Frequency Curve: - This is one kind of histogram which is represented by joining the mid points by free hand of the upper horizontal side of each rectangle with adjacent rectangles. Comulative Frequency Curve or Ogive Curve: - The total frequency of all values less then the upper class boundary of a...
Post a Comment
Read more

Empirical Relation between Mean, Median and Mode

By itsRashad
A distribution in which the values of mean, median and mode coincide (i.e. mean = median = mode) is known as a symmetrical distribution. Conversely, when values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed distribution. In moderately skewed or asymmetrical distribution a very important relationship exists among these three measures of central tendency. In such distributions the distance between the mean and median is about one-third of the distance between the mean and mode, as will be clear from the diagrams 1 and 2 Karl Pearson expressed this relationship as:
1 comment
Read more

Correlation and Linearity

By itsRashad
Correlation coefficients measure the strength of association between two variables. The most common correlation coefficient, called the Pearson product-moment correlation coefficient , measures the strength of the linear association between variables. In this tutorial, when we speak simply of a correlation coefficient, we are referring to the Pearson product-moment correlation. Generally, the correlation coefficient of a sample is denoted by r , and the correlation coefficient of a population is denoted by ρ or R . How to Interpret a Correlation Coefficient The sign and the absolute value of a correlation coefficient describe the direction and the magnitude of the relationship between two variables. The value of a correlation coefficient ranges between -1 and 1. The greater the absolute value of a correlation coefficient, the stronger the linear relationship. The str...
Post a Comment
Read more

Poisson Distribution

By itsRashad
A Poisson experiment is a statistical experiment that has the following properties: The experiment results in outcomes that can be classified as successes or failures. The average number of successes (μ) that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero. Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc. Notation The following notation is helpful, when we talk about the Poisson distribution. e : A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.) μ: The mean number of successes that occur in a specified region. x : The actual number of successes that occur in a specified region. P( x ; μ): The Poisson probability that exactly x ...
Post a Comment
Read more