A trimmed mean (also called a truncated mean) is a measure of central tendency computed after removing a specified percentage of the largest and smallest observations from a data set and then taking the arithmetic average of the remaining values. By excluding the most extreme values, the trimmed mean reduces the influence of outliers and short‑term volatility and often provides a more representative “typical” value for skewed or noisy data.
Why use a trimmed mean?
– Reduces distortion from extreme observations (outliers).
– Smooths short‑term noise so the underlying trend is clearer.
– Useful for economic series (e.g., inflation) where a few volatile components (food, energy, single‑month spikes) can dominate the headline average.
– Also used in areas like judged sports scoring to limit biased or anomalous judge scores.
Key features
– A trimmed mean is usually described as “trimmed by x%,” where x is the total percent removed (split evenly across both tails unless otherwise specified). For example, a 4% trimmed mean removes the lowest 2% and highest 2% of observations.
– Trimming points are often chosen by rule of thumb or historical fit rather than a mathematically optimal rule.
– Different from winsorizing: trimming removes observations; winsorizing replaces extreme values with the nearest remaining value.
How a trimmed mean is calculated (unweighted data) — practical steps
1. Choose the trimming percentage (x%). This is the total percent to remove from both tails combined.
2. Sort the data in ascending order.
3. Calculate how many observations to remove from each tail:
• If there are n observations, remove k = floor((x/100) * n / 2) observations from the lower end and k from the upper end. (Different implementations sometimes use exact fractional removal or rounding rules; specify your rule.)
4. Remove the k smallest and k largest observations.
5. Compute the arithmetic mean of the remaining observations:
trimmed mean = (1 / (n − 2k)) * sum of remaining values.
Example (unweighted)
Scores: 6.0, 8.1, 8.3, 9.1, 9.9 (n = 5)
– Trim by 40% total → remove lowest 20% and highest 20% → 20% of 5 = 1 observation per tail
– Remove 6.0 and 9.9 → remaining scores: 8.1, 8.3, 9.1
– Trimmed mean = (8.1 + 8.3 + 9.1) / 3 = 25.5 / 3 = 8.5
– Compare with regular mean = (6.0 + 8.1 + 8.3 + 9.1 + 9.9) / 5 = 8.28
Weighted trimmed mean (as used with price indexes)
Many official inflation measures are constructed from weighted baskets of goods and services. When calculating a trimmed mean for a weighted index (for example, CPI or PCE component inflation), trimming is typically applied by weight rather than by simple count
Practical steps (weighted trimming)
1. For each price component, compute its individual inflation contribution (typically the component’s price change or inflation rate).
2. Order the components by their inflation rate from lowest to highest.
3. Compute cumulative weights from the lowest end and from the highest end.
4. Trim away components (by removing their weights) from both tails until the chosen total trimming weight (x%) has been removed. This may require removing parts of a component if you trim to an exact weight target.
5. Compute the weighted mean of the remaining components using their original weights (renormalized to sum to 1 over the retained components).
Why use weight‑based trimming for inflation?
– Price indexes are built from components with differing importance (weights). A small‑weight item with a large price jump should have a smaller effect than a heavily weighted item. Weight‑based trimming ensures that trimming reflects economic importance rather than just count.
Choosing the trimming level
– Common practice: small symmetric trims (a few percent on each tail) or weight‑based trimming that matches historical best fit to a desired series (e.g., to reflect “core” inflation).
– No single universal rule; trimming choices are often pragmatic:
• Small trims (e.g., 1–5%) reduce the influence of extreme noise while preserving most data.
• Larger trims (10–40%) more aggressively exclude volatile observations but risk discarding informative data.
– For inflation measurement, central banks or research institutions often select trimming methods and thresholds based on empirical performance (how closely the trimmed measure tracks underlying inflation or reduces volatility). Examples include research and published measures from Federal Reserve regional banks.
Where trimmed means are used
– Inflation measurement: central banks and statistical agencies may publish trimmed‑mean versions of CPI or PCE inflation to provide a less noisy view of trend inflation alongside headline and “core” (food & energy excluded) measures.
– Sports judging: organizations (e.g., International Skating Union) trim extreme judge scores to limit biased or anomalous judging effects.
– Any data analysis with skewness or frequent outliers where a robust central tendency is desired.
Advantages and limitations
Advantages:
– Robust to outliers and extreme short‑term swings.
– Easier to interpret than more complex robust estimators; simple to compute.
– When weighted trimming is used for inflation, it reflects each component’s economic importance.
However, this approach has some limitations:
– Choice of trimming percentage is arbitrary; different choices give different results.
– Trimming discards data that may be informative (especially if extremes reflect real, persistent shifts).
– For small sample sizes, trimming can remove too much information.
– Different trimmed‑mean constructions can make comparisons across publications or countries difficult.
Trimmed mean vs. other measures of trend
– Headline mean: uses all observations; sensitive to outliers.
– Median: the middle observation (50th percentile) — extremely robust to outliers but ignores distribution tails.
– Core inflation: excludes selected categories (typically food & energy) based on judgment about volatility.
– Trimmed mean: systematically excludes tails based on a quantile/weight rule; can be more flexible than core and often more robust than headline.
How trimmed means help determine inflation rates
– They smooth monthly swings created by volatile items (e.g., a large one‑month energy spike).
– Trimmed series often show lower month‑to‑month volatility than headline inflation and can track the underlying, persistent inflation trend more accurately.
– Policymakers use trimmed measures alongside headline and core rates to form a fuller picture before making decisions.
Practical checklist for implementing a trimmed mean (for analysts)
1. Define purpose: Why trim? (remove noise, produce a stable trend, reduce influence of outliers)
2. Decide if trimming should be by count or by weight (for price indexes use weight).
3. Choose the trimming rule and percent (document rationale: historical performance, rule of thumb, best fit).
4. Prepare data: compute component changes (or raw values), ensure consistent weights if weighted.
5. Sort components by value (or change).
6. Remove extremes per chosen rule (and handle fractional removal if needed).
7. Compute the trimmed mean (renormalize weights if weighted trimming).
8. Report alongside other measures (headline, median, core) and include details on the trimming method and percent removed.
9. Periodically review the trimming rule forrelevance and robust performance.
Example references and further reading
– Investopedia — Trimmed Mean (Sydney Saporito): introduction and examples.
– Federal Reserve Bank of Cleveland — Trimmed Mean CPI Inflation (archived): methodology and reasoning for trimmed CPI measures. (Archived pages available via the Internet Archive.)
– Federal Reserve Bank of Dallas — Trimmed Mean PCE Inflation Rate: shows application of trimmed means to the PCE price index.
– International Skating Union — ISU Synchronized Skating Media Guide: explains trimming in judged sports scoring.
Bottom line
A trimmed mean is a simple, transparent tool for producing a more robust average when data contain extreme or volatile values. It is especially valuable in economic statistics—like inflation measurement—where volatile components can obscure underlying trends. The method requires choices about trimming levels and whether to trim by count or weight; those choices should be documented and re‑evaluated over time. When used alongside headline, core, and median measures, trimmed means help analysts and policymakers form a clearer view of persistent changes versus short‑term noise.