Probable deviation

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