1. Meaning of Median
The median is the middle value of a data set when all the values are arranged in order. It divides the data into two equal halves — one half has values smaller than it, and the other half has values larger than it.
The median is useful when the data has extreme values or when we want the central point rather than the total average.
2. Finding the Median for Odd Number of Values
When the number of values is odd, the median is the value exactly in the middle.
Steps:
- Arrange the data in ascending order.
- Locate the middle position using the formula \( \dfrac{n+1}{2} \).
- The value at this position is the median.
2.1. Example
Find the median of: 2, 7, 4, 9, 5.
Step 1: Arrange in ascending order → 2, 4, 5, 7, 9
Step 2: Number of values = 5 (odd)
Middle position = \( \dfrac{5+1}{2} = 3 \)
Median = value at 3rd position = 5
3. Finding the Median for Even Number of Values
When the number of values is even, there is no single middle value. In this case, the median is the average of the two middle values.
Steps:
- Arrange the data in ascending order.
- Take the two middle values at positions \( \dfrac{n}{2} \) and \( \dfrac{n}{2} + 1 \).
- Find their average.
3.1. Example
Find the median of: 3, 8, 2, 10.
Step 1: Arrange in ascending order → 2, 3, 8, 10
Step 2: Number of values = 4 (even)
Middle values = 3 and 8
\( \text{Median} = \dfrac{3 + 8}{2} = 5.5 \)
4. Median for Frequency Tables (Ungrouped)
When values repeat, a frequency table helps in finding the median. The idea is to identify the middle position using cumulative totals.
Steps:
- List values in order.
- Add a cumulative frequency column.
- Find the position using \( \dfrac{n+1}{2} \).
- Locate the value where this position falls.
4.1. Example
Consider the data:
| Value (x) | Frequency (f) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 1 |
Total frequency = \( 2 + 3 + 1 = 6 \)
Median position = \( \dfrac{6+1}{2} = 3.5 \)
Cumulative frequencies:
- Up to 1 → 2
- Up to 2 → 5
- Up to 3 → 6
Position 3.5 lies in the cumulative total 5, which corresponds to the value 2.
So, median = 2.