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 j^{th} and k^{th} 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*

*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 = 4^{th} and k = 12^{th} observation in the sample dataset.

The 4^{th} observation is equal to 13 and the 12^{th} 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