The data set must have an even number of values. See solution for details.
Practice makes perfect
Let's start by reviewing the concept of median. Recall that the median is the middle value of a set when the values are arranged in numerical order. If the set has an even number of values, the median is the mean of the two middle values. Let's see some examples.
Median of a set with odd number of values
Let's suppose that we have the data values shown below.
3, 2, 5, 10, 12, 7, 1, 9, 13
We first need to rearrange them in numerical order. When the data set has an odd number of values, there will always be a middle value that divides the data set into two equal parts.
1, 2, 3, 5_(4values), 7, 9, 10, 12, 13_(4values)
Therefore, the median will always appear in the data set.
Median of a set with even number of values
Let's suppose that we have the data values shown below.
3, 15, 2, 5, 10, 12, 7, 1, 9, 13
We can rearrange them in numerical order and find the middle values.
1, 2, 3, 5_(4values), 7, 9, 10, 12, 13, 15_(4values)
If the middle values are the same, that would be the median. Otherwise, like in this case, we have to find the mean of the middle values.
7+9/2 = 16/2=8
Therefore, the median is 8 for this case. Notice that this value does not appear in the data set. This is because we got this as the mean of two different ordered values.
Conclusion
If the value for the median of a set is not found in the data set, we can know that the data set has an even number of values. As we saw above, for an odd number of data values the median will always be one of the values in the data set.
