# Kurtosis

**Kurtosis:**

**Definition 7.** Kurtosis refers to the extent to which unimodal frequency curve is peaked.

**Definition 8.** Kurtosis is a measure that refers to the peakedness of the top of the curve . Kurtosis gives the degree of flatness or peakedness in the region about the mode of a frequency distribution.

**According to Croxton and Cowden,
**

**Definition 9**. A measure of kurtosis indicates the degree to which the curve of a frequency distribution is peaked or flat topped.

**According to Clark and Sckkade,
**

**Definition 10.** Kurtosis is the property of a frequency distribution which expresses its relative peakedness.

Karl Pearson in 1905 introduced the three types of curves on the basis of kurtosis:

**Mesokurtic:** If the concentration of frequency in the middle of the frequency distribution is normal, the curve is known as mesokurtic.

**Leptokurtic:** If the frequencies are densely concentrated in the middle of the series, the will be more peaked than normal and is known as Leptokurtic.

**Platykurtic:** If the frequencies are not densely concentrated in the middle of the series, the curve will be more flat than normal and is known as platykurtic.

**Measure of Kurtosis:**

The measure of kurtosis based on central moments are given by Karl Pearson:

- if β2 = 3, the curve is Mesokurtic or normal
- if β2 > 3, the curve is Leptokurtic or more peaked
- if β2 < 3, the curve is Platykurtic or flat topped.

The measure of Kurtosis is also represented by gamma two, = β2−3

- if = 0,the curve is Mesokurtic
- if > 0,the curve in Leptokurtic
- if < 0, the curve is Platykurtic.

**Probability Mass Function (PMF):**

If X is a discrete random variable with distinct values x_{1}, x_{2}, . . . ,x_{n} for which X has positive probabilities p_{1}, p_{2}, . . . , p_{n}, then the function f (x) defined as:

is called the probability mass function of random variable X. Since P(S) = 1(Probability of sure event is one.), we must have

**Probability Density Function (PDF):**

Similar to PMF, probability density function is defined for continuous random variable X. The funxtion f (x) defined as

such that,