Value making mean, median and mode coincide

Question

The seven numbers $100, 40, 200, 50, 90, 60, x$ have mean, median and mode all equal to $x$ . Find $x$ .

Short answer

$90$ . The mean condition gives $7x=540+x$ , so $x=90$ . Check: the sorted list $40,50,60,90,90,100,200$ has median $90$ and mode $90$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Populations, Samples, and Sampling Distributions

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