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My question is in statistics under what circumstances would the mode be an appropriate measure of central tendency?

Whenever you expect a "normal" Gaussian distribution and have a large number of measurements. In that case the "mode" or most frequently obtained value should also be the mean of the distribution.

It would also be best if you had a bimodal distribution — one with two modes — most likely indicating underlying two subgroups. For example, if you had a measure of handgrip strength, you would probably get a bimodal distribution, indicating different average strength between men and women.

I hope this helps a little more. Thanks for asking.

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