The median is the middle value in a dataset when observations are arranged in ascending or descending order, making it a positional measure of central tendency unaffected by extreme values.
## Core concept
The median divides a frequency distribution into two equal halves. Unlike the arithmetic mean, it is not distorted by outliers and works well for skewed distributions.
Key properties: - It is a positional average, based on location rather than magnitude - For ungrouped data: the value at the middle position - For grouped data: estimated using the median class and interpolation formula - Always lies between the mode and mean in a moderately skewed distribution - Suitable for qualitative and ordinal data (e.g., quality ratings, income brackets)
## Formula / rule
Ungrouped data (n observations): - If n is odd: Median = value at position $\frac{n+1}{2}$ - If n is even: Median = $\frac{\text{value at position } n/2 + \text{value at position } (n/2)+1}{2}$
Grouped data (with class intervals):
$$\text{Median} = L + \frac{N/2 - CF}{f} \times h$$
Where: - $L$ = lower class boundary of the median class - $N$ = total frequency - $CF$ = cumulative frequency before the median class - $f$ = frequency of the median class - $h$ = class width (interval size)
Steps to find median in grouped data: 1. Calculate cumulative frequencies 2. Find $N/2$ (half the total frequency) 3. Identify the median class (the class where cumulative frequency first exceeds $N/2$) 4. Apply the formula above
## Common exam applications
Ungrouped data example: Heights (cm) of 7 students: 160, 165, 158, 172, 155, 168, 162
Arrange in order: 155, 158, 160, 162, 165, 168, 172
Median = 162 cm (position = $(7+1)/2 = 4$th value)
Grouped data example: | Class (marks) | Frequency | CF | |---|---|---| | 0–10 | 5 | 5 | | 10–20 | 8 | 13 | | 20–30 | 12 | 25 | | 30–40 | 10 | 35 | | 40–50 | 5 | 40 |
$N = 40$, so $N/2 = 20$
Median class = 20–30 (because CF first exceeds 20 here)
$$\text{Median} = 20 + \frac{20 - 13}{12} \times 10 = 20 + 5.83 = 25.83 \text{ marks}$$
## Common mistakes
- Forgetting to arrange data first: Always sort observations before finding the median in ungrouped data
- Misidentifying the median class: Double-check that the CF you select is the first to exceed $N/2$, not equal to it
- Confusion with mean: Median uses position; mean uses all values. Median is better for skewed data
- Incorrect interpolation: Ensure the formula's numerator $(N/2 - CF)$ is positive; if it's zero or negative, you've selected the wrong class
- Ignoring class width: Always multiply by $h$ when working with grouped data
## Relation to other measures
The empirical relationship holds for moderately skewed distributions:
$$\text{Mean} - \text{Median} \approx \frac{1}{3}(\text{Mean} - \text{Mode})$$
This helps verify calculations and is often tested in comparative questions.