Definition

Noun (Statistics): The expected value or mean of a probability distribution. It is calculated as the sum of the values of a random variable, each multiplied by its probability, or equivalently, as the sum of all observed values divided by the number of values in a sample.

Usage and Examples

• The first moment about the origin is a measure of central tendency, commonly called the average or mean.
  • In this dataset, the first moment is 15.2, indicating the central location of the distribution.
• It is the foundational concept for calculating higher-order moments, like variance (the second central moment).
  • Before computing the variance, you must first calculate the first moment of the distribution.

Advanced Usage

• In physics and engineering, the term "first moment" can refer to a different concept related to force and distance (e.g., the first moment of area). However, in the context of statistics and probability, it exclusively denotes the mean.
  • While in mechanics the first moment is crucial for understanding torque, in statistics it is synonymous with the expected value.

Variants and Related Words

• Mean (n): The most common synonym for the first moment in statistics.
• Expected Value (E[X]) (n): The theoretical average of a random variable over many trials; the precise term for the first moment of a probability distribution.
• Average (n): A more general, often informal, term for the central value.
• Arithmetic Mean (n): Specifies the method of calculation as a sum divided by the count.

Synonyms

• Mean
• Expected Value
• Average (in a statistical context)

Related Phrases and Concepts

• Central Moment: A moment calculated about the mean of the distribution. The first central moment is always zero.
• Raw Moment (or Moment about the origin): A moment calculated about zero. The first raw moment is the mean.
• Sample Mean: The calculated first moment of a data sample, used as an estimate for the population mean.

Noun

1. the sum of the values of a random variable divided by the number of values