Why go beyond the mean?
In many real-world situations, the mean isn’t the best summary of your data. Skewed distributions, heavy-tailed data, or outliers can make the mean misleading. That’s where medians and other quantiles come in—they describe the center or spread of your data without being distorted by extremes. But how do we build confidence intervals for them?

Confidence intervals for medians

The median is a robust measure of central tendency, but unlike the mean, it doesn’t have a simple formula for its standard error. A few common approaches include:

• Order statistic method – Uses properties of the binomial distribution to create exact or approximate intervals around the sample median.
• Bootstrap method – Resample the data many times, compute the median each time, and use the distribution of medians to form an interval. Flexible and widely used in practice.
• Large-sample approximations – For big datasets, normal-based approximations can be applied, though less reliable for small samples.

If you’re doing applied research, bootstrapping is often the go-to because it requires few assumptions and works even with skewed data.

Confidence intervals for quantiles

Quantiles (like the 25th or 90th percentile) are important for understanding distributional shape and tail behavior.

• Binomial-based methods – Similar to medians, these rely on order statistics to create bounds for any quantile.
• Bootstrap quantiles – Sample with replacement, compute the desired quantile repeatedly, and use percentiles of the bootstrap distribution for confidence limits.
• Asymptotic methods – For large samples, quantile estimators follow approximate normal distributions with variance depending on the density at that quantile.

When to use which method

• Small samples – Use exact or binomial-based intervals if possible.
• Moderate to large samples – Bootstrap provides flexible, accurate intervals without heavy theory.
• Specialized cases – Asymptotic formulas may be used when computational resources are limited.

Practical considerations

• Check the shape of your distribution before deciding—heavy tails favor robust methods like bootstrapping.
• Report both the interval and the quantile estimate to give context.
• Software packages in R, Python, and Excel add-ins can handle these calculations with built-in functions.

Helpful tools

If you want hands-on guidance, a book on applied statistics can walk through quantile estimation step by step. For quick reference, a statistics study guide or bootstrap methods reference is useful in the field.

The bottom line

Confidence intervals aren’t just for means. Medians and quantiles give a fuller picture of your data, especially when distributions are skewed or heavy-tailed. Whether you use exact methods, bootstrapping, or asymptotics, the key is to match your interval method to your sample size and data characteristics.