MEASURES OF SKEWNESS AND KURTOSIS Flashcards
When is skewness said to exist in a data set
Symmetry is said to exist in a distribution if the smoothed frequency polygon of the distribution can be divided into two identical halves wherein each half is a mirror image of the other
What is skewness and what are the different types
lack of symmetry and it can be positive or negative. Basically, if the distribution has a tail on the right, then the distribution is positively skewed Eg Most students having vey low marks in an examination. However if the distribution has a tail on the left, then the distribution is negatively skewed. Eg Most students having vey high marks in an examination
What is the formula for calculating the moment coefficient of skewness (a3)
*See notes for formula
Relate a3 and skewness
if a3<0 the distribution is negatively skewed, if
a3>0 the distribution is positively skewed and if a3=0 the distribution is normal
List other measures of skewness
The Karl Pearson’s coefficient of Skewness
The Bowley’s coefficient of Skewness
The Kelly’s coefficient of Skewness
Briefly describe The Karl Pearson’s coefficient of Skewness
The Karl Pearson’s coefficient of Skewness is based upon the divergence of mean from mob in a skewed distribution. Recall the empirical relation between mean, median and mode which states that, for a moderately symmetrical distribution, we have
Mean - Mode = 3 (Mean - Median)
Hence Karl Pearson's coefficient of skewness is defined by;
SKp= (Mean - Mode)/
Standard Deviation
OR
SKp=3(Mean - Median)/Standard Deviation
,
Briefly describe The Bowley’s coefficient of Skewness
The Bowley’s coefficient of Skewness is based on quartiles. For a symmetrical distribution, it is seen that Q1 and Q3
are equidistant from median.
SKb= (Q3-2Q2+Q1)/(Q3-Q1)
Briefly describe The Kelly’s coefficient of Skewness
The Kelly’s coefficient of Skewness is based on P90 and, P10 so that only 10% of the observations on each extreme are ignored.. This is an improvement over the Bowley’s coefficient which leaves 25% of the observatories on each extreme of the distribution.
SKk= (P90-2P50+P10)/(P90-P10)
Describe Bulmer M.G’s rule of thumb to interpret skewness
- less than −1 or greater than +1, the distribution is highly skewed.
- between −1 and -1/2 or between 1/2 and 1 the distribution is moderately skewed.
- between -1/2 and 1/2 the distribution is approximately symmetric.
What is kurtosis?
It measures the peakedness of a distribution.
What are the different types of peakedness?
If the values of x are very close to the mean, the peak is very high and the distribution is said to be Leptokurtic. On the other hand if the values of x are very far away from the mean, the peak is very low and the distribution is said to be Pletykurtic. Finally if x values are at a moderate distance from the mean then the peak is moderate and the distribution is said to be mesokurtic.
*See notes for pictures
What are the measures of kurtosis
*See notes forformula
