Labels

Moments, Skewness & Kurtosis | B.Sc Statistics Unit 3 Notes

Moments, Factorial Moments, Sheppard’s Correction, Skewness, Kurtosis and Quartile-based Measures

In statistics, numerical measures are used to describe the nature and behavior of data. Beyond averages and dispersion, higher-order measures such as moments, skewness, and kurtosis help us understand the shape of a distribution. This chapter explains these concepts in a clear and simple manner, suitable for B.Sc Statistics (UG) students.

1. Moments

Moments are quantitative measures that describe the characteristics of a frequency distribution. They provide information about the location, spread, symmetry, and peakedness of the data.

1.1 Definition of Moment

The rth moment of a distribution about a point a is defined as:

μr = E[(X − a)r]

Depending on the value of a, moments are classified into:

  • Moments about origin
  • Central moments (about mean)

1.2 Moments About Origin

When a = 0, the moments are called raw moments or moments about origin. They are denoted by μ'r.

μ'r = E(Xr)

Examples:

  • First moment about origin: μ'1 = Mean
  • Second moment about origin: related to variance

1.3 Central Moments

When moments are taken about the mean, they are called central moments. They are denoted by μr.

μr = E[(X − X̄)r]

  • First central moment = 0
  • Second central moment = Variance
  • Third central moment = Skewness
  • Fourth central moment = Kurtosis

Central moments are more meaningful because they are independent of the origin.

2. Factorial Moments

Factorial moments are mainly used in theoretical statistics and probability distributions, especially for discrete random variables.

2.1 Definition

The rth factorial moment is defined as:

E[X(X − 1)(X − 2)...(X − r + 1)]

These moments are useful in:

  • Binomial distribution
  • Poisson distribution
  • Sampling theory

2.2 Importance of Factorial Moments

  • Simplify calculations for discrete distributions
  • Useful in deriving moments of probability distributions
  • Important in combinatorial problems

3. Sheppard’s Correction for Moments

When data is grouped into class intervals, some information is lost due to grouping. This causes an error in the calculation of moments.

Sheppard’s correction is applied to reduce this error.

3.1 When is Sheppard’s Correction Used?

  • Data is grouped
  • Class intervals are equal
  • Classes are continuous
  • Frequency distribution is smooth

3.2 Correction Formula

Let h = class width

  • Corrected μ2 = μ2 − h²/12
  • Corrected μ4 = μ4 − (h²μ2/2) + (7h⁴/240)

No correction is needed for:

  • First moment
  • Third moment

3.3 Significance

Sheppard’s correction improves accuracy when working with grouped data, especially for variance and kurtosis.

4. Measures of Skewness

Skewness measures the asymmetry of a distribution. It tells us whether data is tilted to the left or right.

4.1 Types of Skewness

  • Positive Skewness: Long tail to the right
  • Negative Skewness: Long tail to the left
  • Zero Skewness: Symmetrical distribution

4.2 Karl Pearson’s Coefficient of Skewness

Sk = (Mean − Mode) / Standard Deviation

If mode is not defined:

Sk = 3(Mean − Median) / SD

4.3 Bowley’s Coefficient of Skewness

Based on quartiles:

Sk = (Q₃ + Q₁ − 2Q₂) / (Q₃ − Q₁)

4.4 Significance of Skewness

  • Helps in understanding data distribution
  • Important in economic and business data
  • Useful for comparing datasets

5. Measures of Kurtosis

Kurtosis measures the degree of peakedness or flatness of a distribution.

5.1 Types of Kurtosis

  • Leptokurtic: Sharp peak
  • Mesokurtic: Normal peak
  • Platykurtic: Flat peak

5.2 Coefficient of Kurtosis

β₂ = μ₄ / μ₂²

  • β₂ = 3 → Mesokurtic
  • β₂ > 3 → Leptokurtic
  • β₂ < 3 → Platykurtic

5.3 Significance of Kurtosis

  • Helps identify outliers
  • Used in risk analysis
  • Important in financial data

6. Measures Based on Quartiles

Quartile-based measures are positional measures that divide data into four equal parts. They are less affected by extreme values.

6.1 Quartile Deviation

QD = (Q₃ − Q₁) / 2

6.2 Coefficient of Quartile Deviation

(Q₃ − Q₁) / (Q₃ + Q₁)

6.3 Advantages

  • Simple to calculate
  • Suitable for skewed data
  • Resistant to extreme values

7. Practice Questions

  1. Define moments and explain their types.
  2. What are factorial moments? State their applications.
  3. Explain Sheppard’s correction and its necessity.
  4. Differentiate between skewness and kurtosis.
  5. Explain quartile deviation with merits.
Labels:

Comments

Post a Comment