The trimmed mean is a measure of center that reduces the influence of unusually small or unusually large values. It is useful when a data set has outliers or long tails that would pull the ordinary mean away from the main cluster. Instead of using every value, a trimmed mean removes the same percentage of data from both ends before averaging.
This makes it a practical compromise between the mean, which uses all values, and the median, which focuses only on the middle.
Key Facts
- A p percent trimmed mean removes p percent of the data from the low end and p percent from the high end before averaging.
- Mean = (sum of all data values) / n
- Trimmed mean = (sum of remaining values after trimming both tails) / (number of remaining values)
- For n data values and trim proportion p, remove k = floor(pn) values from each tail when using a simple trimming rule.
- A 10% trimmed mean on 20 sorted values removes 2 smallest and 2 largest values, then averages the remaining 16 values.
- The trimmed mean resists outliers better than the ordinary mean because extreme values are excluded before calculating the average.
Vocabulary
- Trimmed mean
- A mean calculated after removing an equal percentage of the smallest and largest data values.
- Outlier
- A data value that is unusually far from the rest of the data.
- Tail
- The low or high end of a sorted data distribution where the most extreme values are found.
- Robust statistic
- A statistic that is not strongly affected by outliers or small departures from ideal conditions.
- Sorted data
- Data values arranged in order from smallest to largest.
Common Mistakes to Avoid
- Trimming only one side of the data is wrong because a trimmed mean removes the same amount from both tails unless a special one-sided method is stated.
- Trimming before sorting the data is wrong because the smallest and largest values cannot be identified reliably unless the data are ordered first.
- Removing p percent total instead of p percent from each tail is wrong because a 10% trimmed mean usually means 10% from the low end and 10% from the high end.
- Averaging the original number of data values after trimming is wrong because the denominator must be the number of values that remain.
Practice Questions
- 1 Find the 10% trimmed mean of the data set 3, 4, 5, 6, 7, 8, 9, 100 using k = floor(0.10n) values removed from each tail.
- 2 A class has sorted quiz scores 40, 72, 75, 76, 78, 80, 82, 84, 86, 88, 91, 100. Find the 10% trimmed mean using k = floor(0.10n) values removed from each tail.
- 3 A data set of home prices includes several extremely expensive mansions. Explain why a trimmed mean might describe a typical home price better than the ordinary mean.