Moments and Kutosis

In statistics, moments are quantitative measures used to describe the shape, spread, and central tendency of a distribution. They provide insights into the characteristics of a data set or probability distribution.

Types of Moments:

Moments can be divided into two types:

1. Raw Moments: Calculated with respect to the origin (zero).
2. Central Moments: Calculated with respect to the mean of the distribution.

First Four Moments in Statistics:

• Mean (First Moment): Central location of data.
• Variance (Second Moment): Dispersion or spread of data.
• Skewness (Third Moment): Asymmetry of the distribution.
• Kurtosis (Fourth Moment): Tailedness or presence of outliers in data.

Kurtosis measures the degree to which a distribution’s tails and peak differ from the normal distribution. It provides insights into the tailedness and sharpness of the dataset’s peak.

Types of Kurtosis

1. Mesokurtic (Normal Kurtosis):
   • Distribution resembles a normal curve.
   • Kurtosis value ≈ 3.
   • Moderate tails and a standard peak.
2. Leptokurtic (High Kurtosis):
   • Sharp peak with heavy tails.
   • Kurtosis value > 3.
   • Indicates outliers or extreme values are more frequent.
3. Platykurtic (Low Kurtosis):
   • Flatter peak with thinner tails.
   • Kurtosis value < 3.
   • Data values are more evenly spread, with fewer extreme values.