Location and scale families are distributions which are characterized by the location of their "center" (typically their mean, but it could be median or mode) and the scale of their spread. This essentially boils down to knowing their first two moments, though there are location and scale families where neither of those exist.
A location family is a distribution which is characterized by shifting the pdf by a constant. That is, if f(x) is a pdf or pmf of a real-valued random variable, then the set of pdfs {f(x-μ)} where μ is real is the location family for f. While this can be done for any pdf, location families are generally thought of as distributions where the location of the center is a direct function of one of its parameters.
A scale family works pretty much the same way, but here the transformation is {(1/σ)f(x/σ)} where σ is real. Again, you can pull this off with any pdf or pmf, since any function that integrates or sums to 1 will continue to do so under that transformation. Also again, one typically talks of these families when the scale parameter is a direct function of one of the distributions natural parameters. If the scale parameter, σ, is a direct parameter of the distribution, the distribution is typically parameterized such that σ is the standard deviation which, unlike variance, scales linearly. If the distribution is normally parameterized by its variance, it's generally written as σ2 to highlight the fact that it is not, technically, a scale parameter, but the square of one.
If you've read this far, you're probably good enough to guess that a location-scale family is the set of pdfs: {(1/σ)f((x-μ)/σ)}.
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