Finding the Value of X Using the Median

Understanding the Median Concept

The median is a statistical measure representing the middle value of a dataset when it's arranged in ascending order. For a dataset with an even number of observations, the median is calculated as the average of the two middle values.

Analyzing the Given Data

We are given a set of 8 numbers: 9, 15, 1, 15, 14, 9, 4, and X. The total number of observations (n) is 8, which is an even number.

The median is given as 11.

Step-by-Step Solution

1. Sort the known numbers: First, let's arrange the known numbers (excluding X) in ascending order: 1, 4, 9, 9, 14, 15, 15
2. Determine the median calculation formula: Since there are 8 numbers in total (an even count), the median is the average of the 4th and 5th numbers in the final sorted list (including X). The formula is: \[\text{Median} = \frac{(\text{4th term}) + (\text{5th term})}{2}\]
3. Set up the equation: We know the median is 11. So, \[11 = \frac{(\text{4th term}) + (\text{5th term})}{2}\] Multiplying both sides by 2, we get: \[22 = (\text{4th term}) + (\text{5th term})\]
4. Incorporate X and test possibilities: We need to find the value of X that satisfies the condition \((\text{4th term}) + (\text{5th term}) = 22\) when X is included in the sorted list. Let's consider the sorted list of known numbers: 1, 4, 9, 9, 14, 15, 15. Now, we insert X into this sequence and check the median.
   • If we place X = 10, the sorted list is: 1, 4, 9, 9, 10, 14, 15, 15. The 4th term is 9 and the 5th term is 10. Median = \((9 + 10) / 2 = 9.5\). (Incorrect)
   • If we place X = 11, the sorted list is: 1, 4, 9, 9, 11, 14, 15, 15. The 4th term is 9 and the 5th term is 11. Median = \((9 + 11) / 2 = 10\). (Incorrect)
   • If we place X = 12, the sorted list is: 1, 4, 9, 9, 12, 14, 15, 15. The 4th term is 9 and the 5th term is 12. Median = \((9 + 12) / 2 = 10.5\). (Incorrect)
   • If we place X = 13, the sorted list is: 1, 4, 9, 9, 13, 14, 15, 15. The 4th term is 9 and the 5th term is 13. Median = \((9 + 13) / 2 = 22 / 2 = 11\). (Correct)
5. Conclusion: The value X = 13 makes the median of the dataset equal to 11. The sorted list becomes {1, 4, 9, 9, 13, 14, 15, 15}, and the median is calculated as the average of the 4th and 5th elements, which are 9 and 13. \[\frac{9 + 13}{2} = \frac{22}{2} = 11\]

Final Answer Determination

Based on the calculations, the value of X that results in a median of 11 is 13.