#### Transcript of Skewness and Kurtosis

**Skewness and Kurtosis**

Defining Skewness

Skewness- is the measure of asymmetry of the distribution of a real valued random variable.

- It is the standardized 3rd central moment of a distribution

CONCEPT OF SKEWNESS

In a symmetrical distribution, the Mean, Median and Mode are equal to each other and the ordinate at mean divides the distribution into two equal parts such that one is mirror image of the other.

If some observations, of very high or low magnitude, are added to such a distribution, its right or left tail gets elongated.

Karl Pearson's Measure of Skewness

- Based upon the DIVERGENCE OF MEAN FROM MODE in a skewed distribution.

- (Mean-Mode) can be taken as an absolute measure of skewness

KURTOSIS

Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution.

Formulas for Kurtosis

Ungrouped Data

Grouped Data

Conclusion

This formula is both for ungrouped and grouped data

Sk- Skewness

X bar- mean

X-curl- sample median

S- Standard Deviation

Compute the Karl Pearson's Coefficient of Skewness from the following data:

COMPUTE FOR THE MEAN and STANDARD DEVIATION

The modal class is 60.5-61.5

Sk= 61.4-61.13 / 1.76 = 0.153

The distribution is POSITIVELY skewed

If Sk >0 = Positively Skewed

If Sk< 0= Negatively Skewed

Types of Normal Distribution Curve

k= 3, Mesokurtic

k> 3, Leptokurtic

k< 3, Platykurtic

Any change in the value of the mean causes the curve to be shifted to the left or right. On the other hand, a change in the value of S.D. causes a change in the shape of the curve.

The frequency distribution below shows the examination scores of 50 students in Statistics. Compute for the Kurtosis of the data and interpret

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