# When can I not replace a random variable with its mean?

A frequent simplification in modeling and simulation is to replace a random variable by its mean value.

When would this simplification lead to the wrong conclusion?

• Does "Var" stand for variable or variance or Value At Risk? Oct 24, 2017 at 22:24
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– Nat
Oct 25, 2017 at 7:42
• Well in a very simple case if we take it to the the extreme we could lose pretty much all the information we care about. Consider a regression of Y on X where we replaced both Y and X with their mean. Any information about the slope is now lost. Oct 25, 2017 at 12:48
• Are you asking about replacing missing values, or are you asking about a replacing a random variable in a specific context (e.g. making predictions base on a random-effects model)?
– IWS
Oct 27, 2017 at 9:29

If you replace a missing value by some point estimate, you disregard all its variability. Thus, you will not propagate all the original variability to your model. Your parameter estimates will appear to have too low s. If you do inference, your p values will be biased low. Your s will be too narrow. If you do prediction, your s will be too narrow.

Overall: you will be too sure of your conclusions.

• Good answer! Think about this way: A random variable has a distribution. It can be skweded to the left, to the right. I can be bi-modal etc. By reducing the variable to it's mean value you are removing all that extra information (uncertainty) and replacing a distribution (intervals) by a single point estimate. Oct 25, 2017 at 10:12
• If you replace a missing value by some point estimate, you're also assuming the data is missing at random. The mean value of the random variable might not equal the mean value of the data when it's missing. Oct 25, 2017 at 16:03
• @NeilG sorry to nitpick, but replacing a missing value by its mean does not directly mean assuming the data to be missing at random. Especially since the - somewhat confusing - terminology around missing data considers 'missing at random' to be data that is missing at random conditional on other, but known data (en.wikipedia.org/wiki/Missing_data). IMO, the way data is replaced does not imply anything about the reasoning behind it. That reasoning should be made explicit and lead up to the appropriate way of handling the missing data. That said, I fully agree with Stephan's answer.
– IWS
Oct 27, 2017 at 9:28
• @IWS It's fine for the missingness indicators to be conditional on the observed data. Missing at random means that the missingness indicators depend on the unobserved data. If you replace the variable with its mean value conditional on it being observed, that might not be the same as its unconditional mean value — unless the data are missing at random. Oct 27, 2017 at 9:46
• @NeilG Don't you mean 'missing completely at random', when you write 'missing at random' in the final sentence of your last comment? If so, we do are in agreement, but I was just nitpicking about terminology. (see the wiki page I've put in my comment above, I've always been taught, read and used that terminology)
– IWS
Oct 27, 2017 at 9:51