Moments, Skewness and Kurtosis
What is Moments, Skewness and Kurtosis?
A quantitative measure of the shape of a distribution of a probability or statistical function.
Key formula / rule: k-th Raw Moment
Key points
- Define and calculate moments of a distribution.
- Understand the concept of skewness and its interpretation.
- Understand the concept of kurtosis and its interpretation.
- Relate moments to skewness and kurtosis.
Common exam trap
Confusing raw moments with central moments.
Definitions
- Term
Moment
- Meaning
A quantitative measure of the shape of a distribution of a probability or statistical function.
- Term
Raw Moment
- Meaning
The expected value of a random variable raised to a power, calculated about the origin (zero).
- Term
Central Moment
- Meaning
The expected value of a random variable's deviation from the mean, raised to a power, calculated about the mean.
- Term
Skewness
- Meaning
A measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, negative, or undefined.
- Term
Kurtosis
- Meaning
A measure of the 'tailedness' of the probability distribution of a real-valued random variable. High kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations.
- Term
Leptokurtic
- Meaning
A distribution with kurtosis greater than that of the normal distribution (kurtosis > 3).
- Term
Platykurtic
- Meaning
A distribution with kurtosis less than that of the normal distribution (kurtosis < 3).
- Term
Mesokurtic
- Meaning
A distribution with kurtosis equal to that of the normal distribution (kurtosis = 3).
Learning objectives
Define and calculate moments of a distribution.
Understand the concept of skewness and its interpretation.
Understand the concept of kurtosis and its interpretation.
Relate moments to skewness and kurtosis.
Formulae
- Name
k-th Raw Moment
- Note
Expected value of X raised to the power k.
- Expression
μ'_k = E[Xk]
- Name
k-th Central Moment
- Note
Expected value of (X - Mean) raised to the power k.
- Expression
μk = E[(X - μ)k]
- Name
Variance (2nd Central Moment)
- Note
Measures spread.
- Expression
μ₂ = σ² = E[(X - μ)²]
- Name
Skewness Coefficient (γ₁)
- Note
Measures asymmetry. μ₃ is the 3rd central moment.
- Expression
γ₁ = μ₃ / μ₂^(3/2)
- Name
Kurtosis Coefficient (γ₂)
- Note
Measures peakedness/tailedness. μ₄ is the 4th central moment.
- Expression
γ₂ = μ₄ / μ₂²
- Name
Excess Kurtosis
- Note
Relative to a normal distribution.
- Expression
Excess Kurtosis = γ₂ - 3
Prerequisites
Basic statistics (Mean, Median, Mode, Variance, Standard Deviation).
Understanding of probability distributions.
Basic algebra.
Common mistakes
Confusing raw moments with central moments.
Misinterpreting the direction of skewness (positive vs. negative).
Confusing kurtosis with just peakedness, ignoring the tail behavior.
Incorrectly calculating moments, especially higher-order ones.
Keywords
Moments
Raw Moments
Central Moments
Mean
Variance
Standard Deviation
Skewness
Kurtosis
Leptokurtic
Platykurtic
Mesokurtic
Distribution Shape
Asymmetry
Tailedness
Practice preview
If a distribution has a kurtosis of 2.5, it is:…
medium
A distribution is said to be positively skewed if:…
easy
Which of the following is a measure of the peakedness of a distribution?…
easy