Finding the Unknown Value (x) Given the Median

The problem asks us to find the value of 'x' in a given set of numbers when the median of the set is provided.

The given set of numbers is 7, 5, 8, x, 12, 17. There are 6 numbers in this set.

Understanding the Median

The median is the middle value in a data set that is arranged in ascending or descending order. If the data set has an even number of observations, the median is the average of the two middle numbers.

Steps to Calculate the Median for an Even Data Set

1. Arrange the data set in ascending order.
2. Identify the two middle numbers. For a set with 'n' observations, the middle numbers are the (\( \frac{n}{2} \))-th term and the (\( \frac{n}{2} + 1 \))-th term.
3. Calculate the average of these two middle numbers: Median = \( \frac{\text{(\( \frac{n}{2} \))-th term} + \text{(\( \frac{n}{2} + 1 \))-th term}}{2} \).

Applying the Steps to the Given Problem

The data set is {7, 5, 8, x, 12, 17}. The number of observations is n = 6.

The median is the average of the \( \frac{6}{2} \)-th term (3rd term) and the \( (\frac{6}{2} + 1) \)-th term (4th term) in the sorted list.

Let's arrange the known numbers in ascending order: 5, 7, 8, 12, 17.

Now we include 'x' in the set {5, 7, 8, 12, 17, x} and sort all six numbers. The sorted list will depend on the value of x.

We are given that the median is 10. So, the average of the 3rd and 4th terms in the sorted list is 10.

Let the sorted data be \( y_1, y_2, y_3, y_4, y_5, y_6 \).

Median = \( \frac{y_3 + y_4}{2} = 10 \)

This means \( y_3 + y_4 = 20 \).

Considering the Value of x

The known numbers are 5, 7, 8, 12, 17.

Let's examine the positions of the known numbers in the sorted list of six numbers:

The numbers less than or equal to 8 are 5, 7, 8. There are 3 such numbers.

The numbers greater than or equal to 12 are 12, 17. There are 2 such numbers.

For the 3rd term (\( y_3 \)) to be 8 and the 4th term (\( y_4 \)) to be 12, the sorted list must look like: (Number \( \le \) 8), (Number \( \le \) 8), 8, 12, (Number \( \ge \) 12), (Number \( \ge \) 12).

From the known numbers, we have 5, 7, 8. If x is greater than 8, then 5, 7, and 8 would likely be the first three terms (or among the first three terms if x is also small). If x is greater than 8, the numbers \( \le \) 8 are exactly {5, 7, 8}. This means \( y_3 \) could be 8.

From the known numbers, we have 12, 17. If x is greater than or equal to 12, then 12, 17, and x (if \( x \ge 12 \)) would be the last three terms or among them. If x is greater than or equal to 12, the numbers \( \ge \) 12 are {12, 17, x}. This means \( y_4 \) could be 12.

If the 3rd term is 8 and the 4th term is 12, then \( y_3 = 8 \) and \( y_4 = 12 \). The condition \( y_3 + y_4 = 20 \) is satisfied (8 + 12 = 20).

For 8 to be the 3rd term and 12 to be the 4th term in the sorted list of {5, 7, 8, 12, 17, x}, 'x' must fit into a position that allows this. This occurs if 'x' is greater than or equal to the 4th term (12) but less than the 5th term (17), or equal to 17, or greater than 17.

• If x is between 8 and 12 (i.e., 8 < x < 12), the sorted list is 5, 7, 8, x, 12, 17. The 3rd term is 8, 4th term is x. Median = \( \frac{8+x}{2} \). If this equals 10, \( 8+x = 20 \implies x=12 \). This contradicts the assumption 8 < x < 12.
• If x is equal to 12, the list is 5, 7, 8, 12, 12, 17. Sorted: 5, 7, 8, 12, 12, 17. The 3rd term is 8, 4th term is 12. Median = \( \frac{8+12}{2} = \frac{20}{2} = 10 \). This matches the given median.
• If x is greater than 12 (i.e., x > 12), the sorted list is 5, 7, 8, 12, ..., 17 or 5, 7, 8, 12, 17, x. If x is between 12 and 17, sorted is 5, 7, 8, 12, x, 17. The 3rd term is 8, 4th term is 12. Median = \( \frac{8+12}{2} = 10 \). This matches the given median. If x is \( \ge \) 17, sorted is 5, 7, 8, 12, 17, x. The 3rd term is 8, 4th term is 12. Median = \( \frac{8+12}{2} = 10 \). This also matches the given median.

Our analysis shows that any value of x such that \( x \ge 12 \) and \( x \lt 17 \) (strictly less if 17 is the 5th term), or \( x \ge 17 \) makes the median 10 by ensuring the 3rd term is 8 and the 4th term is 12.

Let's verify the options provided:

Based on the calculation for each option, both x=12 and x=13 result in a median of 10. However, since 12 is provided as one of the options and is consistent with the median calculation, it is a possible value for x.

Conclusion

When x = 12, the sorted data set is 5, 7, 8, 12, 12, 17. The 3rd term is 8 and the 4th term is 12. The median is \( \frac{8+12}{2} = 10 \), which matches the given median.

Revision Table: Median Calculation

Additional Information: Statistics Concepts

• Mean: The average of all numbers in a data set. Calculated by summing all numbers and dividing by the count of numbers.
• Mode: The number that appears most frequently in a data set. A set can have one mode (unimodal), multiple modes (multimodal), or no mode.
• Range: The difference between the highest and lowest values in a data set. It measures the spread of the data.
• Data Set: A collection of numbers or values representing observations or measurements.

Understanding median, mean, and mode helps in summarizing and interpreting data, providing insights into the central tendency and distribution of the values.