We can use the following formula to calculate the upper and lower bounds of a confidence interval for a population median:
j: nq – z√nq(1-q)
k: nq + z√nq(1-q)
where:
- n: The sample size
- q: The quantile of interest. For a median, we will use q = 0.5.
- z: The z-critical value
We round j and k up to the next integer. The resulting confidence interval is between the jth and kth observations in the ordered sample data.
Note that the z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:
| Confidence Level | z-value |
|---|---|
| 0.90 | 1.645 |
| 0.95 | 1.96 |
| 0.99 | 2.58 |
Source: This formula comes from Practical Nonparametric Statistics, 3rd Edition by W.J. Conover.
The following step-by-step example shows how to calculate a confidence interval for a population median using the following sample data of 15 values:
Sample data: 8, 11, 12, 13, 15, 17, 19, 20, 21, 21, 22, 23, 25, 26, 28
Step 1: Find the Median
First, we need to find the median of the sample data. This turns out to be the middle value of 20:
8, 11, 12, 13, 15, 17, 19, 20, 21, 21, 22, 23, 25, 26, 28
Step 2: Find j and k
Suppose we would like to find a 95% confidence interval for the population median. To do so, we need to first find j and k:
- j: nq – z√nq(1-q) = (15)(.5) – 1.96√(15)(.5)(1-.5) = 3.7
- k: nq + z√nq(1-q) = (15)(.5) + 1.96√(15)(.5)(1-.5) = 11.3
We will round both j and k up to the nearest integer:
- j: 4
- k: 12
Step 3: Find the Confidence Interval
The 95% confidence interval for the median will be between the j = 4th and k = 12th observation in the sample dataset.
The 4th observation is equal to 13 and the 12th observation is equal to 23:
8, 11, 12, 13, 15, 17, 19, 20, 21, 21, 22, 23, 25, 26, 28
Thus, the 95% confidence interval for the median turns out to be [13, 23].
Additional Resources
How to Find a Confidence Interval for a Proportion
How to Find a Confidence Interval for a Mean
How to Find a Confidence Interval for a Standard Deviation