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MOMENTS, MOMENT RATIO AND SKEWNESS

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MOMENTS, MOMENT RATIO AND SKEWNESS

  1. 1. MOMENTS, MOMENT RATIOAND SKEWNESS PRESENTED BY: AYESHA KABEER UNIVERSITY OF GUJRAT SIALKOT SUB CAMPUS
  2. 2. Moments In statistics moments are certain constant values in a given distribution which help us to ascertain the nature and form of distribution. The first moment is called the mean which describes the center of the distribution. The second moment is the variance which describes the spread of the observations around the center. Other moments describe other aspects of a distribution such as how the distribution is skewed from its mean or peaked. A moment designates the power to which deviations are raised before averaging them. 2
  3. 3. Central (or Mean) Moments In mean moments, the deviations are taken from the mean. For Ungrouped Data: In General, 4   r Population Moment about Mean= r ith r x N       r Sample Moment about Mean= r ith r x x m n   
  4. 4. Central (or Mean) Moments Formula for Grouped Data: 5     r Population Moment about Mean= r Sample Moment about Mean= r ith r r ith r f x f f x x m f          
  5. 5. Moments about (arbitrary) Origin • 6
  6. 6. Moments about zero • 7
  7. 7. Moment Ratios • 2 3 4 1 23 2 2 2 ,         2 3 4 1 23 2 2 2 , m m b b m m  
  8. 8. Skewness A distribution in which the values equidistant from the mean have equal frequencies and is called Symmetric Distribution. Any departure from symmetry is called skewness. In a perfectly symmetric distribution, Mean=Median=Mode and the two tails of the distribution are equal in length from the mean. These values are pulled apart when the distribution departs from symmetry and consequently one tail become longer than the other. If right tail is longer than the left tail then the distribution is said to have positive skewness. In this case, Mean>Median>Mode If left tail is longer than the right tail then the distribution is said to have negative skewness. In this case, Mean<Median<Mode
  9. 9. Skewness Skewness is usually measured by the moment ratio β1 2 3 4 1 23 2 2 2 ,         Cases: If β1=0 then distribution is symmetrical If β1<0 then distribution is negatively skewed If β1>0 then distribution is positively skewed

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