Difference between revisions of "Probable deviation"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/P074/P.0704980 Probable deviation, | ||
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| − | + | ''mean deviation'' | |
| − | + | A measure, $ B $, | |
| + | of dispersion for a [[Probability distribution|probability distribution]]. For a continuously-distributed symmetric random variable $ X $ | ||
| + | the probable deviation is defined by | ||
| − | < | + | $$ \tag{* } |
| + | {\mathsf P} \{ | X- m | < B \} = \ | ||
| + | {\mathsf P} \{ | X- m | > B \} = | ||
| + | \frac{1}{2} | ||
| + | , | ||
| + | $$ | ||
| − | where | + | where $ m $ |
| + | is the median of $ X $( | ||
| + | which in this case is identical with the [[Mathematical expectation|mathematical expectation]], if it exists). For the [[Normal distribution|normal distribution]] there exists a simple connection between the probable deviation and the [[Standard deviation|standard deviation]] $ \sigma $: | ||
| + | $$ | ||
| + | \Phi \left ( | ||
| + | \frac{B} \sigma | ||
| + | \right ) = | ||
| + | \frac{3}{4} | ||
| + | , | ||
| + | $$ | ||
| + | where $ \Phi ( x) $ | ||
| + | is the normal $ ( 0, \sigma ) $- | ||
| + | distribution function. The approximate relation is $ B = 0.6745 \sigma $. | ||
====Comments==== | ====Comments==== | ||
| − | The probably deviation is also called the mean error, [[#References|[a2]]]. The phrase "mean deviation" is also used to denote the first absolute moment | + | The probably deviation is also called the mean error, [[#References|[a2]]]. The phrase "mean deviation" is also used to denote the first absolute moment $ E ( | X - m | ) $ |
| + | of the random variable around its median, [[#References|[a1]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1966) pp. Sect. 15.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ph.H. Dubois, "An introduction to psychological statistics" , Harper & Row (1965) pp. 287</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1966) pp. Sect. 15.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ph.H. Dubois, "An introduction to psychological statistics" , Harper & Row (1965) pp. 287</TD></TR></table> | ||
Latest revision as of 08:07, 6 June 2020
mean deviation
A measure, $ B $, of dispersion for a probability distribution. For a continuously-distributed symmetric random variable $ X $ the probable deviation is defined by
$$ \tag{* } {\mathsf P} \{ | X- m | < B \} = \ {\mathsf P} \{ | X- m | > B \} = \frac{1}{2} , $$
where $ m $ is the median of $ X $( which in this case is identical with the mathematical expectation, if it exists). For the normal distribution there exists a simple connection between the probable deviation and the standard deviation $ \sigma $:
$$ \Phi \left ( \frac{B} \sigma \right ) = \frac{3}{4} , $$
where $ \Phi ( x) $ is the normal $ ( 0, \sigma ) $- distribution function. The approximate relation is $ B = 0.6745 \sigma $.
Comments
The probably deviation is also called the mean error, [a2]. The phrase "mean deviation" is also used to denote the first absolute moment $ E ( | X - m | ) $ of the random variable around its median, [a1].
References
| [a1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1966) pp. Sect. 15.6 |
| [a2] | Ph.H. Dubois, "An introduction to psychological statistics" , Harper & Row (1965) pp. 287 |
How to Cite This Entry:
Probable deviation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probable_deviation&oldid=48303
Probable deviation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probable_deviation&oldid=48303
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article