How to Calculate Confidence Intervals for the Median’s Range

In the world of statistics, we often encounter situations where the median, the value that divides a data set in half with an equal number of values on either side, becomes a crucial measure of central tendency. However, directly accessing the entire population to calculate the true population median is often impractical or impossible. This is where confidence intervals (CIs) for medians come into play, offering a powerful tool for estimating the population median with a specific level of certainty.

Decoding the Approach: A Peek Behind the Curtain

Unlike confidence intervals for proportions or means, there’s no single formula for constructing a CI for a median. Instead, it relies on the ranking and position of data points within the sample.

The specific method used depends on the sample size (n):

1. Small Samples (n ≤ 50):

For small samples, the CI is constructed by identifying the (n + 1)/2th and (n + 2)/2th ordered data points in the sample. These represent the lower and upper bounds of the CI, respectively.

2. Large Samples (n > 50):

For larger samples, the normal approximation can be used. The formula involves the standard normal z-score and the interquartile range (IQR) of the data:

Median ± z_(α/2) * IQR / √(n)

• z_(α/2) is the critical value from the standard normal distribution table corresponding to the chosen confidence level (1 – α).
• IQR is the difference between the third quartile (Q3) and the first quartile (Q1) of the data.

Interpreting the Interval: What Does it Tell Us?

Once you’ve calculated the CI, you can confidently say there is a (1 – α)% chance that the true population median falls within the calculated range. For example, a 95% CI implies a 95% certainty that the population median lies between the lower and upper bounds.

Example: Imagine you measure the heights of 20 individuals (n = 20) and obtain the following data in ascending order:

{150 cm, 152 cm, 155 cm, 158 cm, 160 cm, 161 cm, 162 cm, 163 cm, 164 cm, 165 cm, 166 cm, 167 cm, 168 cm, 169 cm, 170 cm, 171 cm, 172 cm, 173 cm, 174 cm, 175 cm}

Following the method for small samples:

• (n + 1)/2 = (20 + 1)/2 = 10.5
• (n + 2)/2 = (20 + 2)/2 = 11

The 10.5th and 11th data points are both 165 cm. Therefore, the 95% CI for the population median is:

Lower Bound = Upper Bound = 165 cm

This indicates we can be 95% confident that the true median height in the population is 165 cm.

Example (Large Sample):

Suppose you have a sample of 100 exam scores with an IQR of 20 points and want a 90% confidence interval. Using the normal approximation method:

• z_(0.05) = 1.645 (from a standard normal distribution table)

Therefore, the 90% CI for the population median is:

Median ± 1.645 * 20 / √100 ≈ Median ± 3.29

This implies we can be 90% confident that the true median exam score in the population falls within the range of the median itself, plus or minus 3.29 points.

Beyond the Basics: Important Considerations

While the methods provide a framework, several crucial factors require attention:

1. Sample Size: The normal approximation method for large samples becomes less accurate as the sample size decreases. For small samples, the method relying on order statistics is more reliable.
2. Outliers: Extreme values or outliers within the data can influence the IQR and potentially affect the accuracy of the CI, particularly for smaller samples.
3. Normality Assumption: The normal approximation method assumes the underlying population data is normally distributed. If significant deviations from normality exist, alternative methods like bootstrapping might be necessary.