Re: mean and standard deviation of left-truncated normal distribution

From: Glen (glenbarnett_at_geocities.com)
Date: 03/22/05

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    Date: 21 Mar 2005 16:47:41 -0800

    clemenr@wmin.ac.uk wrote in message news:<1111428801.110818.25450@o13g2000cwo.googlegroups.com>...
    > Hi. I would like to know the mean and standard deviation of a truncated
    > normal distribution, calculated from the mean and sd of the underlying
    > normal distribution, and the (left) truncation point k.
    >
    > Clearly I can estimate the mean and sd given a simulation, which in
    > reality will be amply good enough for my purposes, but I feel that it
    > should be easy to calculate the true mean and sd.
    >
    > I did find a paper with formulae for the mean and sd of a standardised
    > truncated normal distribution (underlying mean 0, underlying sd 1,
    > truncation point k).
    >
    > However, I'm not 100% sure how to convert these into mean and standard
    > deviation for an arbitrary left-truncated normal distribution. I would
    > expect this to be a simple proces, but I'm not quite confidence as to
    > how it should be done.
    >
    > Any pointers? Answers, references, or both?

    Let's say you have a random variable, X, with a normal distribution
    with known mean mu, s.d. sigma, which is then truncated at d.

    Convert to standard normal, Z=(X-mu)/sigma.

    Then the trucation point for Z is k = (d - mu)/sigma.

    Compute out your mean and sd of the trucanted std normal, let's say
    they're
    nu and tau respectively.

    Covert back to the original scale, which I will call m and s (not to
    be confused with sample statistics). This is just a matter of
    inverting
    the standardising transformation:

    m = nu * sigma + mu
    s = tau * sigma

    Of course if you're dealing with samples, rather than just
    distributions, various kinds of sampling issues come in as well.

    Glen

  • Next message: Glen: "Re: paired test result interpretation"

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