Median - The Middle Value of a Dataset
The median is the middle value of a dataset when all the values are arranged in order. It is a type of average that is not affected by very large or very small values (outliers), making it a reliable measure when data is skewed.
How to Find the Median
Step 1: Arrange all values in ascending order (smallest to largest).
Step 2: If there is an odd number of values, the median is the middle value.
Step 3: If there is an even number of values, the median is the mean of the two middle values.
Position of median = (n + 1) / 2, where n is the number of values.
Worked Examples
Ordered: 1, 3, 5, 7, 9. n = 5 (odd). Middle = 3rd value = 5.
Ordered: 4, 6, 8, 12, 16, 20. n = 6 (even). Two middle values = 8 and 12. Median = (8 + 12) / 2 = 10.
Ordered: 180,000 195,000 200,000 210,000 1,500,000.
Median = 200,000. Mean = 2,285,000 / 5 = 457,000.
The outlier (mansion) pulls the mean far above a typical price. The median better represents the street.
When to Use the Median
Use the median when the dataset contains outliers or is skewed. For example, median house prices, median salaries, or median age are often more informative than the mean in real-life reporting.
Key Takeaways
- Always sort the data before finding the median.
- Odd count: median = middle value. Even count: median = mean of two middle values.
- The median is not affected by outliers – it reflects the typical central value.
- Position formula: (n + 1) / 2 gives the position, not the value itself.
Practice Questions
- Find the median of: 11, 4, 9, 2, 7, 15, 6.
- Find the median of: 22, 35, 18, 42, 29, 37.
- A dataset has 9 values. Which position is the median?
- Salaries (thousands): 28, 32, 35, 40, 150. Find both the mean and median. Which is a better representative?
- The median of 5 values is 14. The values are 9, x, 14, 20, 25 (already sorted). Find x if x is less than 14.