The median of {20, x, 15, 30, 25} is less than the mean. If x is a whole number, what is the sum of all possible values of x ?

\(\text{resort these as }\\ \{15, 20, 25, 30 \} \text{ with }x \text{ somewhere in there}\)

 

\(\text{we can write the median as}\\ \begin{cases} 20, &x < 20\\ x, &20 \leq x \leq 25 \\ 25, &25 < x \end{cases}\)

 

\(\text{The mean is } m = \dfrac{15+20+25+30+x}{5} = 18+\dfrac{x}{5}\)

 

 

\(\text{We can see from the graph that the median is less than the mean for}\\ x \in \{11,12, \dots, 22\}\\ \sum \limits_{k=11}^{22}~k = 198\)

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