What Is a Trimmed Mean?
A trimmed mean is a form of averaging in which a specified fraction of the greatest and smallest values are removed before computing the mean. The trimmed mean is calculated after deleting the given outlier data using a conventional arithmetic averaging algorithm. The use of a trimmed mean reduces the impact of outliers or data points on the tails. These extreme data points may have an unfair effect on the traditional or arithmetic mean. Trimmed means are used in economic data reporting to smooth out the findings and provide a more accurate image.
An arithmetic mean is a straightforward mathematical average of two or more values. Whereas, the trimmed mean assists in reducing the impact of outliers on the computed average. The trimmed mean is best suited for data with substantial, unpredictable deviations or distributions that are strongly skewed.
Trimmed Mean – A Closer Look
A trimmed mean is defined as a mean that has been trimmed by some predetermined percentage (x percent). The value “x” is the total of the percentage of observations removed from the upper and lower limits. Trimming points are frequently arbitrary. They are determined by rules of thumb rather than ironclad techniques for determining such thresholds. A trimmed mean of 4%, for example, would remove the lowest and highest 4% of results. This allows the mean to be determined from the remaining 92 percent of data. A trimmed mean is considered a more accurate depiction of data collection. The few unpredictable outliers that may otherwise bias the information have been eliminated. Trimmed means are sometimes referred to as truncated means or adjusted means.
Why Use a Trimmed Mean?
The trimmed mean serves as a middle ground. It allows generating a relatively low statistical error for both normal and non-normal data distributions. Another point of comparison between the trimmed mean and the median is that the median protects against outliers by taking only a single number in the center. In essence, this suggests that everything save the middle is tainted. However, while the median provides better protection against outliers, it reduces statistical power. A trimmed mean essentially discards fewer data.
Where is a trimmed mean useful?
Trimmed means are used in a number of applications. Trimmed means are employed in sports scoring, such as the Olympics, to mitigate the influence of judge outlier bias. They are also used to calculate inflation in economics. Trimming lowers the volatility of consumer price indices. The Personal Consumption Expenditures (PCE) is a measure of price changes in consumer goods and services. It is a pricing index, much like the widely used Consumer Price Index (CPI). However, the PCE Index provides information on household consumer spending habits as well as the overall consumption expenditure pattern of the economy.
Trimmed Mean and Inflation Rates
When calculating inflation rates using the Consumer Price Index (CPI) or Personal Consumption Expenditures (PCE), a trimmed mean can be useful instead of a regular mean. The CPI and the PCE price indexes assess the prices of baskets of items in an economy to assist in identifying inflationary patterns. For example, growing price trends. The levels clipped from each tail may not be equal. Typically, they are based on historical data to provide the greatest match between the trimmed mean inflation rate and the core inflation rate. The core of the CPI or PCE refers to the selected items minus food and energy costs. Food and energy prices are widely regarded as the most variable, or “noisy,” elements in the report.
Non-core changes are not always indicative of overall inflationary activity. When the data points are sorted, they are arranged in ascending order from the lowest prices to the highest prices. To assist reduce the impact of volatility on total CPI movements, specific percentages are eliminated from the tails. The provision of a trimmed mean inflation rate, in addition to other indicators, gives a better baseline for comparison. This allows for a more detailed investigation of the inflation rates being observed. The traditional CPI, the core CPI, a trimmed-mean CPI, and a median CPI may all be included in this comparison.
Trimmed Mean PCE Inflation Rate by the Dallas Federal Reserve
The Trimmed Mean PCE inflation rate is an alternative measure of core inflation in the price index for personal consumption expenditures (PCE). It is calculated by staff at the Dallas Fed, using data from the Bureau of Economic Analysis (BEA). (Source: dallasfed.org)
What is the Highest and Lowest Percentage?
Any proportion of data points can be eliminated. However, the maximum value possible is 50% and the minimum possible is 0%. If you delete half of the data points on the left and half of the data points on the right, you potentially have zero data points left, but for practical reasons, you have the median. Conversely, a 0% trim does not eliminate any values. It simply involves performing the mean by taking into consideration each observation. As a consequence, the arithmetic mean would be obtained.
- 5% Truncated Mean – In the event of large amounts of data, a total 5% trimmed mean is generally sufficient. The lower and higher ends exclude 2.5 percent of the data. As a result, 95 percent of the observations in the series are preserved.
- 10% Trimmed Mean – A total of 10% trimmed mean is frequently utilized. This approach is frequently used in the Olympics to eliminate typical mean biases induced by extraordinary numbers. Before computing the mean, 5% of the least and highest numbers are discarded from each end. As a result, 90% of the observations in the series are preserved.
- 20% Trimmed Mean – A 20% truncation is comparable to other trim percentages. In this case, 10% of the values are deleted from both ends. As a result, the remaining 80% is employed to compute a 20% trimmed mean. Furthermore, a 5, 10, or 20% reduction from the indicated end of the list necessitates doing the computation on only the specified portion. As a result, a 10% truncation for the highest value would only involve deleting data from the upper end.
Advantages & Disadvantages
- Pro – The trimmed mean is a helpful estimator because it is less susceptible to outliers than the mean. However, it still provides an accurate estimate of central tendency for many statistical models. It is referred to as a robust estimator in this context.
- Con – The truncated mean utilizes more information from the distribution or sample than the median. Unless the underlying distribution is symmetric, it is unlikely that the truncated mean of a sample would give an unbiased estimate for either the mean or the median.
Example of a Trimmed Mean in Olympic Scoring
Olympic Scoring – Scores for various events in both the Summer and Winter Olympics are dependent on subjective judgments made by a panel of judges. Gymnastics, diving, figure skating, and ski jumping are a few examples. Consider the diving judging system. For example, a dive is judged by a panel of seven judges in a women’s diving event. Each judge assigns a score between 0 and 10. Each dive also has a difficulty rating, for example, 2.1. To calculate the diver’s final total score, discard the two highest and two lowest ratings given by the judges. Then add the remaining three scores together and multiply by the difficulty level, in this case, 2.1.
Quad witching refers to a date that occurs four times each calendar year. On quadruple witching days, stock index futures, stock index options, stock options, and single stock futures expire simultaneously. While stock options contracts and index options expire on the third Friday of every month, all four asset classes expire simultaneously on the third Friday of March, June, September, and December.
Quad witching is now often used interchangeably with the term triple witching days. Quadruple witching days replaced triple witching days when the fourth class of assets was included. Single stock futures started trading in November 2002. Before 2002, when stock futures were first introduced, the third Friday of March, June, September, and December was known as a triple witching day. And, this term is still used by some. In any case, all four asset class options and futures contracts expire on the same triple witching schedule. As a result, the terms “triple witching” and “quad witching” refer to the same four days each calendar year.