Definition

1. Noun (Statistics):
   • A measure of dispersion: The second moment is a statistical concept representing the expected value (average) of the squared deviations of a random variable's values from a specified reference point, often its mean (average) or the point of origin (zero).

Usage

• The term is primarily used in probability theory and statistics to describe the spread or variability of a distribution.
• It is a fundamental component in calculating variance and standard deviation when the reference point is the mean.

Examples

• In Statistics:
  • The variance of a distribution is equivalent to its second moment about the mean.
  • Calculating the second moment about the origin is a key step in understanding the properties of the probability distribution.

Advanced Usage

• Second Moment About the Mean: This specific calculation, where deviations are measured from the distribution's mean (μ), is the definition of variance (σ²).
  • For a discrete random variable, the second moment about the mean is calculated as Σ [ (xᵢ - μ)² * P(xᵢ) ].
• Second Moment About the Origin: This calculation uses zero as the reference point.
  • The second moment about the origin, E[X²], provides information about the distribution's spread relative to zero.

Variants and Related Words

• Moment (n): In statistics, a moment is a specific quantitative measure of the shape of a probability distribution. The "second moment" is one type of moment.
• Variance (n): The expectation of the squared deviation of a random variable from its mean. Variance is the second central moment (second moment about the mean).
• Standard Deviation (n): The square root of the variance, providing a measure of dispersion in the original units of the variable.

Synonyms

• Variance (when specified as "second moment about the mean").
• Mean Square Deviation.

Related Concepts

• First Moment: The expected value or mean of a random variable.
• Central Moment: A moment about the mean of the distribution. The second central moment is the variance.
• Raw Moment: A moment about zero. The second raw moment is E[X²].

Noun

1. the expected value of the square of the deviations of a random variable from the point of origin