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I was wondering if it is possible and -if yes- what are the implications when one is trying to compare mean group differences of a continuous variable that varies in one group but is a constant number in another one.

If we consider the independent samples t-test formula for this case it is reduced to $ t=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{N_{1}}}}$

which will yield a t-value, however given a t-test is a parametric test it should be a severely biased estimate. Is it even possible to compare a number with a confidence interval of 0 with a mean with a given standard deviation which is >0?

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  • $\begingroup$ I'm unclear on this ... if it's constant under one of the groups, in what sense is it continuous in that group? (What is being measured here?) $\endgroup$
    – Glen_b
    May 7, 2015 at 16:13
  • $\begingroup$ I meant continuous as in measured on an interval or ratio scale of measurement. It's kind of a hypothetical question but I guess it could be anything, e.g. age $\endgroup$
    – StevenP
    May 7, 2015 at 16:26
  • $\begingroup$ Neither interval nor ratio scale imply the variable is continuous. The number of eggs laid by a chicken in a month is ratio scale, but it's discrete, not continuous. The amount of rain in a day is ratio but is neither discrete nor continuous (since the probability that it's exactly 0 is non-zero). $\endgroup$
    – Glen_b
    May 7, 2015 at 16:44
  • $\begingroup$ I just read some of your other posts and I have to agree that the typology used to describe scales of measurement can be misleading as they are by no means definitive $\endgroup$
    – StevenP
    May 7, 2015 at 17:11
  • $\begingroup$ That's true enough but doesn't relate to the present issue -- ratio/interval scale is simply not related to continuity. Even if Stevens' typology were definitive, stating that a scale is ratio or interval doesn't imply it's continuous. To quote Stevens (1946) himself: "Foremost among the ratio scales is the scale of number itself -- cardinal number -- the scale we use when we count such things as eggs, pennies and apples.". There's nothing continuous about counts. $\endgroup$
    – Glen_b
    May 7, 2015 at 17:15

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Run a straight t-test of the mean against the constant. You don't have two samples, you have one sample.

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