Moments in Statistics

Moments in Statistics is an essential concept in understanding the characteristics of a distribution. This post explains the different types of moments and provides formulas for individual data and frequency distributions.

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Introduction to Moments

• Moments are statistical measures that provide insights into the characteristics of a distribution.
• They are calculated as the rth power of the deviation from a reference point.

Types of Moments

1. Moments about an arbitrary point
2. Moments about the mean (Central Moments)
3. Moments about the origin

Moments can be calculated for:

• Individual data
• Frequency distributions

Moments about an Arbitrary Point

For Individual Data

The moments about an arbitrary point A are defined as:

μr' = (1/n) ∑i=1n (xi - A)r

• r: Order of the moment
• xi: Data points
• A: Reference point

For Frequency Distribution

The moments about an arbitrary point A are calculated as:

μr' = ∑i=1n [fi (xi - A)r] / ∑i=1n fi

• r: Order of the moment
• xi: Midpoints or values of the variable
• fi: Frequencies corresponding to xi
• A: Reference point

Moments about the Mean (Central Moments)

For Individual Data

The rth central moment about the mean ̅x is given by:

μr = (1/n) ∑i=1n (xi - ̅x)r

Common central moments:

• μ2: Variance
• μ3: Skewness
• μ4: Kurtosis

For Frequency Distribution

The rth central moment about the mean ̅x is given by:

μr = ∑i=1n [fi (xi - ̅x)r] / ∑i=1n fi

Moments about the Origin (Raw Moments)

For Individual Data

The moments about the origin for individual data are defined as:

μr'' = (1/n) ∑i=1n xir

The first raw moment is the arithmetic mean:

μ1'' = ̅x = (1/n) ∑i=1n xi

For Frequency Distribution

The moments about the origin for frequency distributions are given by:

μr'' = ∑i=1n [fi xir] / ∑i=1n fi

Summary of Formulas

For Individual Data

• Moments about an arbitrary point:

  μr' = (1/n) ∑i=1n (xi - A)r

• Central moments:

  μr = (1/n) ∑i=1n (xi - ̅x)r

• Moments about the origin:

  μr'' = (1/n) ∑i=1n xir

For Frequency Distribution

• Moments about an arbitrary point:

  μr' = ∑i=1n [fi (xi - A)r] / ∑i=1n fi

• Central moments:

  μr = ∑i=1n [fi (xi - ̅x)r] / ∑i=1n fi

• Moments about the origin:

  μr'' = ∑i=1n [fi xir] / ∑i=1n fi

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