The mean (average), the median, and the only mode of the five numbers \(15,~12,~14,~19,\) and \(n\) are all equal. Determine the value of \(n\).
For the five numbers \(15,~12,~14,~19,\) and \(n\) to have a single mode, \(n\) must equal one of the existing numbers in the list: \(15,~12,~14,\) or \(19\). It follows that the mean (average), median, and mode must all equal \(n\).
Since there five numbers, and five is an odd number, the median will be equal to the number in the middle position when the five numbers are written in increasing order. When we write the existing numbers in increasing order, we obtain \(12,~14,~15,~19\). Since \(n\) is equal to the median, and must also equal one of the existing numbers, the only possibilities are \(n=14\) or \(n=15\).
If \(n=14\), then the mean of the five numbers is \(\dfrac{12+14+14+15+19}{5}=14.8\), which is not equal to \(14\).
If \(n=15\), then the mean of the five numbers is \(\dfrac{12+14+15+15+19}{5}=15\).
Then the mean, median, and mode are all equal to \(15\). Therefore, the value of \(n\) is \(15\).