Question

fill in the blank so that the resulting statement is true

if n data items are arranged in order, from smallest to largest, the data item in the middle is the value in position

\\( \frac { n - 1 } { 2 } \\)
\\( \frac { n } { 2 } \\)
\\( \frac { n + 1 } { 2 } \\)
\\( \frac { n ( n + 1 ) } { 2 } \\)

Explanation:

Step1: Consider the formula for median position

When \(n\) data items are arranged in order (from smallest to largest), if \(n\) is odd, say \(n = 2k + 1\) for some non - negative integer \(k\). The median is the \((k + 1)\)th item. Substituting \(k=\frac{n - 1}{2}\), the position is \(\frac{n+1}{2}\).
If \(n\) is even, say \(n = 2k\) for some positive integer \(k\). The median is usually the average of the \(k\)th and \((k + 1)\)th items. But when we talk about the position formula for the middle - like value in the sense of the formula for the single middle position (in the case of odd \(n\) which is a more generalizable formula for the concept of the middle position in a sequence of \(n\) ordered data points), we use the formula \(\frac{n + 1}{2}\).

Answer:

\(\frac{n + 1}{2}\)