Consider the following discrete frequency distribution of a variable x: x = 8 with frequency 6, x = 5 with frequency 4, x = 6 with frequency 5, x = 10 with frequency 8, x = 9 with frequency 9, x = 4 with frequency 6 and x = 7 with frequency 4. What is the median value of x for this distribution?

Introduction / Context:
This question asks for the median of a discrete frequency distribution. The median is the value that lies in the middle of the data when all observations are arranged in ascending order. When a frequency table is given, we do not list every value individually. Instead, we use cumulative frequencies to identify the position of the median and then determine which value of x corresponds to that position.

Given Data / Assumptions:

• Discrete values and their frequencies are:
  • x = 8, frequency 6
  • x = 5, frequency 4
  • x = 6, frequency 5
  • x = 10, frequency 8
  • x = 9, frequency 9
  • x = 4, frequency 6
  • x = 7, frequency 4
• We must find the median of the entire data set.

Concept / Approach:
To find the median from discrete frequency data, first arrange the values of x in ascending order along with their frequencies. Then compute cumulative frequencies and determine the position of the median. If total frequency N is even, the median is the average of the N/2th and (N/2 + 1)th observations; in many discrete cases, both positions fall in the same value of x, giving a clear median. We will follow this systematic procedure.

Step-by-Step Solution:
Step 1: Arrange x values in ascending order with their frequencies. x = 4, f = 6. x = 5, f = 4. x = 6, f = 5. x = 7, f = 4. x = 8, f = 6. x = 9, f = 9. x = 10, f = 8. Step 2: Compute total frequency N. N = 6 + 4 + 5 + 4 + 6 + 9 + 8 = 42. Step 3: Find median positions. For N = 42 (even), median lies between the 21st and 22nd observations. Step 4: Compute cumulative frequencies. For x = 4: cumulative = 6 (observations 1 to 6). For x = 5: cumulative = 6 + 4 = 10 (observations 7 to 10). For x = 6: cumulative = 10 + 5 = 15 (observations 11 to 15). For x = 7: cumulative = 15 + 4 = 19 (observations 16 to 19). For x = 8: cumulative = 19 + 6 = 25 (observations 20 to 25). Step 5: Locate the 21st and 22nd observations. The cumulative frequency jumps from 19 to 25 at x = 8, so observations 20 to 25 are all equal to 8. Thus, both the 21st and 22nd observations have x = 8.
Verification / Alternative check:
Because both central observations fall within the same value of x, the median is simply that value. There is no need to average different x values, because the 21st and 22nd data points are identical.
Why Other Options Are Wrong:
Values 6, 7, 9 or 5 correspond to different regions of the ordered data set and do not contain both the 21st and 22nd observations. Choosing any of them would misidentify the central part of the distribution.
Common Pitfalls:
One common error is to ignore ordering and use the raw listing order from the statement, which is not sorted. Another is miscomputing cumulative frequencies, which shifts the identified median position to a wrong x value.
Final Answer:
The median of the distribution is 8.