Find the median of this set of data:
$\text { 18, 9, 11, 6, 12, 9, } 13$
The given dataset is \(S = \{18, 9, 11, 6, 12, 9, 13\}\). To find the median, we first need to determine the total number of observations, which is \(n = 7\). The median is the middle value of a data set when it is arranged in order from least to greatest. If the number of observations is odd, the median is exactly the middle value. If it were even, we would take the average of the two middle values. Since \(7\) is an odd number, we can identify a unique middle position.
To find the median, we must first sort the data in non-decreasing order.
Given the set: \(18, 9, 11, 6, 12, 9, 13\), we order them as follows:
The sorted set is: \(6, 9, 9, 11, 12, 13, 18\).
With \(n = 7\) data points, the position of the median is given by the formula \(\frac{n+1}{2}\).
Calculating this, we get:
\[\text{position} = \frac{7 + 1}{2} = \frac{8}{2} = 4\]
This means the median is the \(4\text{th}\) number in our sorted sequence. Looking at our sorted list \(6, 9, 9, 11, 12, 13, 18\), the \(4\text{th}\) number is \(11\). Thus, the median of the dataset is \(11\).
In real life, finding the median is often used to describe the "center" of a distribution without being heavily influenced by extreme outliers, like when analyzing home prices in a neighborhood. If most houses are priced around \(\$300,000\) but one house is listed at \(\$10,000,000\), the average (mean) would be skewed high, but the median would remain a more representative value of a typical home. For a set like \(\{300k, 310k, 320k, 330k, 10M\}\), the median is simply \(\$320,000\).