# Comparing mean differences between groups when standard deviation in one of them is 0

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?

• 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?) May 7, 2015 at 16:13
• 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 May 7, 2015 at 16:26
• 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). May 7, 2015 at 16:44
• 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 May 7, 2015 at 17:11
• 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. May 7, 2015 at 17:15