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Assuming you have a set of data containing two groups, assuming the data would be gaussian distributed, for example height of people, knowing that there are men and women, you can show that the groups differ in average height without knowing which height corresponds to which gender.

For example, having the data of n heights like (175cm, 178cm, 180cm, 182cm) instead of (female 175cm, male 178cm, female 180cm, male 182cm).

Is there a difference in the statistical confidence you can have in that the two groups differ in average height if you have the info on which height belongs to a female / male and if you don't?

I'm somewhat clear on how to calculate the standard error of the mean for each group in the case when you know which data point is part of which group and see how many standard errors from each other the means of the samples are, but don't know what the best way to show there are two groups with a separate mean is when you don't have the info for each data point on which group it belongs to.

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You could use the likelihood ratio test (LRT) to compare a mixture model to a model that assumes homogeneity. If the mixture improves the explanation of the data enough, in a sense which the LRT makes explicit, then you will reject the non-mixture null hypothesis.

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