Robust statistics are methods that still give useful results when data contain outliers, skew, or a small amount of contamination. This matters because real measurements often include recording errors, rare events, or values from a different process. The mean and standard deviation can move a lot when even one extreme value appears.
Robust methods such as the median, trimmed mean, and median absolute deviation help describe the main pattern without being dominated by unusual points.
A robust statistic works by reducing the influence of values far from the bulk of the data. The median depends only on the middle order position, so extreme values at either end do not change it much. A trimmed mean removes a fixed percentage of the smallest and largest observations before averaging, which makes it a compromise between the mean and median.
The MAD measures typical distance from the median and is often used as a robust alternative to standard deviation.
Key Facts
- Mean: xbar = (x1 + x2 + ... + xn) / n
- Median: the middle value after sorting, or the average of the two middle values when n is even
- Trimmed mean: remove the lowest p% and highest p% of values, then average the remaining data
- MAD: median absolute deviation = median(|xi - median(x)|)
- For normal-like data, robust estimate of sigma is approximately 1.4826 × MAD
- Breakdown point is the largest fraction of contaminated data a statistic can handle before giving arbitrarily bad results
Vocabulary
- Robust statistic
- A statistic that is not strongly affected by outliers or small departures from model assumptions.
- Outlier
- A data value that lies far from the main cluster of observations.
- Median
- The middle value of an ordered data set, used as a robust measure of center.
- Trimmed mean
- An average computed after removing a chosen percentage of the smallest and largest values.
- MAD
- The median absolute deviation, a robust measure of spread based on distances from the median.
Common Mistakes to Avoid
- Using the mean automatically for skewed or contaminated data is wrong because a few extreme values can pull it away from the typical case.
- Computing MAD from deviations around the mean is wrong because MAD is based on absolute deviations from the median.
- Forgetting to sort the data before finding the median is wrong because the median depends on ordered position, not the original data order.
- Trimming only the high outliers is wrong for a standard trimmed mean because trimming should usually remove the same fraction from both tails.
Practice Questions
- 1 For the data 10, 11, 12, 12, 13, 14, 100, find the mean and the median. Which better represents the main cluster?
- 2 For the data 4, 5, 5, 6, 6, 7, 8, 50, compute the median and the MAD.
- 3 A sensor records mostly values near 20, but once per day it reports a value near 1000 because of a glitch. Explain why the median and trimmed mean are more appropriate summaries than the ordinary mean.