#### Skewness and Kurtosis in Statistics (shape of distributions)

Skewness and kurtosis are two important measure in statistics. Skewness refers the lack of symetry and kurtosis refers the peakedness of a distribution.

## Skewness

- Mean, median, mode fall at different points, i.e, Mean ≠ Median ≠ Mode.
- Quartiles are not equidistant from median.
- The curve drawn with the help of the given data is not symmetrical but stretched more to one side than the other.

The lack of symmetry in a distribution is always determined with reference to a normal distribution, which is always symmetrical. Any departure of a distribution from symmetry leads to an asymmetric distribution and in such cases, we call this distribution as skewed. The skewness may be either positive or negative. Absence of skewness makes the distribution symmetrical.

It is important to emphasize that skewness of a distribution cannot be determined simply by inspection. If we understand the differences between the mean, median and the mode, we should be able to suggest a direction of skew.We can define the skewness of a frequency distribution in three different shapes as following-

**(1). Symmetrical distributions**

**(2). Positively skewed distributions**

**(3). Negatively skewed distributions**

### Measures of skewness

**Skewness(p)= (Mean-Mode) / Standard Deviation**

- If mean > mode, the distribution is positively skewed.
- If mean < mode, the distribution is negatively skewed.
- If mean = mode, the distribution is not skewed or symmetrical.

**Skewness(p)= 3(Mean-Mode) / Standard Deviation**

### Kurtosis

### Measures of kurtosis

- If kurtosis>3, the distribution is leptokurtic.
- If kurtosis<3, the distribution is platykurtic.
- If kurtosis=3, the distribution is mesokurtic.

#### 1. Mesokurtic

Data that follows a mesokurtic distribution shows an excess kurtosis of zero or close to zero. This means that if the data follows a normal distribution, it follows a mesokurtic distribution.

#### 2. Leptokurtic

Leptokurtic indicates a positive excess kurtosis. The leptokurtic distribution shows heavy tails on either side, indicating large outliers. In finance, a leptokurtic distribution shows that the investment returns may be prone to extreme values on either side. Therefore, an investment whose returns follow a leptokurtic distribution is considered to be risky.

#### 3. Platykurtic

A platykurtic distribution shows a negative excess kurtosis. The kurtosis reveals a distribution with flat tails. The flat tails indicate the small outliers in a distribution. In the finance context, the platykurtic distribution of the investment returns is desirable for investors because there is a small probability that the investment would experience extreme returns.

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