Statistics & Probability

Dispersion: - Dispersion refers to the scatteredness of the individual items of statistical series from their central value. So a descriptive measure of scatter of the values about the average is called measure of Dispersion. The followings are the important methods for measure of dispersion: The Range The average/mean Deviation Quartile Deviation The 10 – 90 Percentile Range The Standard Deviation The Variance The Range: - The Range of a set of numbers is the difference between the largest and smallest numbers in the set. The average/mean Deviation: - The average/mean Deviation, of a set of N numbers X 1 , X 2 ,……..X N is abbreviated MD and is defined by Quartile Deviation: - Quartile Deviation, of a set of data is denoted by Q and defined by,Q = (Q 3 – Q 1 )/2 The 10 – 90 Percentile Range : - The 10 – 90 Percentile Range, of a set of data is defined by, 10 – 90 Percentile Range

Moments: - Moments are certain mathematical constants used to as certain the nature and form of a frequency distribution. Moments in statistics are used to describe the various characteristic of a frequency distribution like Central Tendency, Dispersion, Skewness and Kurtosis . It is symbolized by the Greek letter μ There are two Moments: Raw Moments and Corrected Moments Raw Moments for grouped data, Relation between Raw Moments and Corrected Moments for grouped data:

Skewness is a measure of the degree of asymmetry of a distribution. If the left tail (tail at small end of the distribution) is more pronounced than the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness. Several types of skewness are defined, the terminology and notation of which are unfortunately rather confusing. "The" skewness of a distribution is defined to be (1) Positively Skewed Distribution: - The value of the arithmetic mean is greater than the mode; then the distribution is called Positively Skewed. Nagatively Skewed Distribution: - If the value of the mode is greater than the arithmetic mean; the distribution is called Negatively Skewed. Several forms of skewness are also defined. The momental skewness is defined by (2) The Pearson mode skewness is defined by (3) Pearson's sk

Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment of a distribution. There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted or defined by (1) where denotes the th central moment (and in particular, is the variance ). This form is implemented in Mathematica as Kurtosis [ dist ]. The kurtosis "excess" is denoted or , is defined by (2) and is implemented in Mathematica as KurtosisExcess [ dist ]. Kurtosis excess is commonly used because of a normal distribution is equal to 0, while the kurtosis proper is equal to 3. Unfortunately, Abramowitz and Stegun (1972) confusingly refer to 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: - Th