X 2 3 4 5 F 2 5 2 1   What is the median of the distribution?

Analyze the question

We have a frequency distribution:

| x | 2 | 3 | 4 | 5 | |---|---|---|---|---| | f | 2 | 5 | 2 | 1 |

We need the median, the middle value when all observations are arranged in order.

Concept

For a frequency table:

1. Find the total frequency, N = Σf.
2. Locate the median position. If N is even, the median is the average of the (N/2)th and (N/2 + 1)th values.
3. Use the cumulative frequency to find which x-value sits at that position.

Solution

Total frequency: N = 2 + 5 + 2 + 1 = 10

Since N is even, the median is the average of the 5th and 6th values.

Cumulative frequencies:

| x | f | Cumulative f | |---|---|---| | 2 | 2 | 2 | | 3 | 5 | 7 | | 4 | 2 | 9 | | 5 | 1 | 10 |

• Positions 1-2 are the value 2.
• Positions 3-7 are the value 3.
• So the 5th value is 3 and the 6th value is 3.

Median = (3 + 3) / 2 = 3

Answer: C (3)

Tip: Always build a cumulative frequency column. For an even N, check the N/2 and (N/2 + 1) positions. If both fall in the same row, the median is simply that row's value. The value with the highest frequency (here 3, which is also the mode) often contains the median.