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I have what seems like a fairly common business statistics scenario: I need to compare one group of stores to another group of stores and be able to say if their difference in sales is statistically different.

For example: Group A ($n_A$ = 30 stores) participated in a promotion and saw an avg sales increase for this month compared to the same month last year of $\bar{x}_A$ and a standard deviation of $s_A$.

Group B ($n_B$ = 50 stores) did not participate in the promotion and had avg sales increase of $\bar{x}_B$ and corresponding $s_B$.

I realize there are a number of other variables but, in theory, I should be able to say with some certainty that there is or isn't any difference between stores that took or did not participate in a promotion, right?

Can I do a standard comparison of means test? Does it make a difference that these stores comprise of the entire population? Or is it not the entire population and I should be looking at average increase for multiple months and multiple years?

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If all of the stores were included in the study rather than a sample, then you could make conclusions without using probability statements or statistics.

But if you want to use a subsample to make inference about the larger population or make forecasts, then the use of statistics is appropriate.

A standard comparison of means test for which your data meet the assumptions would be appropriate. For example, if you want to make inference about the effect of promotion on the larger population of stores, e.g. to evaluate the null hypothesis that there is no effect, the student's t test with unequal sample sizes and unequal variance, a.k.a. Welch's t-test is a widely used and robust method.

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  • $\begingroup$ Thanks for the quick response. All of the stores are included in this analysis and I do wonder what kind of conclusions can be made "without probability statements or statistics." For example, if the average sales are higher for Group A but the st. dev. seems high then I would think I might need some sort of statistics to say whether or not the promotion truly "worked" for that group of stores. I've used the t-test but usually in the context of a a true sample of a population and I'm probably just confusing myself since this scenario isn't really a sample of a population? $\endgroup$
    – user2951
    Commented Jan 28, 2011 at 0:32
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    $\begingroup$ Statistics is still appropriate even if you have data on the entire population if you consider there to be a "random" element to the process, both from truly random factors and from other errors from unrecorded characteristics that are randomly distributed in the characteristics you are modelling. $\endgroup$
    – James
    Commented Jan 28, 2011 at 15:31
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    $\begingroup$ I'd say it depends on what your goal is: 1)I want to describe the population I observed or 2)I would like to describe a population like the one I observed. For 1) you have no uncertainty at all, so statistics just reduces to describing differences and similarities. But beware: the "standard deviation" in this case does not describe the accuracy of a difference in means. In case 1) if the $\overline{x}_A$ and $\overline{x}_B$ are different, then the means are different, regardless of what the standard deviation is. $\endgroup$
    – probabilityislogic
    Commented Jan 28, 2011 at 15:50
  • $\begingroup$ ... just one point of clarification on my comment above, while the means are different, this alone tells us nothing about why they are different, apart from the labels of $A$ and $B$ (or equivalent synonyms of them). Anything beyond that is not answered by the means, but by the interpretation it is given (which will depend on your prior beliefs about what the potential causes of a difference may be - the obvious stand out being the promotion). $\endgroup$
    – probabilityislogic
    Commented Jan 28, 2011 at 16:01

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