second moment

Học thuật
Thân thiện
second moment

A student calculates the second moment of a data set on the whiteboard.

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].
second moment

A student calculates the second moment of a data set on the whiteboard.

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