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Background: Normality

In: Visualizing Linear Models

Author

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  • W. D. Brinda

    (Yale University, Statistics and Data Science)

Abstract

The family of Normal distributions plays a key role in the theory of probability and statistics. According to the familiar Central Limit Theorem, the distribution of an average of iid random variables (with finite variance) tends toward Normality (Pollard, A user’s guide to measure theoretic probability. Cambridge University Press, New York, 2002, Thm 7.21). In fact, more advanced versions of the theorem do not require the random variables to be iid, as long as they are not too dependent or too disparate in their scales (e.g. Pollard, A user’s guide to measure theoretic probability. Cambridge University Press, New York, 2002, Thm 8.14). We see this Central Limit phenomenon play out in the real world when we observe “bell-shaped” histograms of measurements in a wide range of contexts. The prevalence of approximate Normality in the world makes Normal distributions a natural part of statistical modeling. Fortunately, the Normal family is also mathematically convenient for analyzing estimation procedures for these models.

Suggested Citation

  • W. D. Brinda, 2021. "Background: Normality," Springer Books, in: Visualizing Linear Models, chapter 0, pages 103-111, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-64167-2_5
    DOI: 10.1007/978-3-030-64167-2_5
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