Trimmed Mean - Explained
What is a Trimmed Mean?
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Table of ContentsWhat is a Trimmed Mean?How is a Trimmed Mean Used?Trimmed Mean and FluctuationComparisons with Trimmed MeansIllustration of Trimmed MeanAcademic Research on Trimmed Mean
What is a Trimmed Mean?
A trimmed mean, which is also known as a truncated mean, is a mode of calculating averages by removing a small percentage of the higher and lowest values before the mean value is determined. The trimmed mean is calculated using a standard averaging method after the selected values have been removed. This method is very valuable because it helps correct any errors that would have occurred due to the presence of data points when determining the mean of a set of numbers or data. These means are mostly used in highlighting real-life data and economic information as they provide a view of reality. An example would be saying eliminating 0.12 from 2.12 to give 2. In this case, instead of saying 2.12 labourers, which is very unrealistic, the trimmed mean reduces it to 2 labourers which readers can relate to.
- Trimmed mean refers to a method of averaging where a specific percentage is trimmed off from the lowest and highest value before the mean is calculated.
- Trimmed mean aim to eliminate the chance of errors in the traditional mean that may be caused by the presence of extra data points referred to as outliers.
- This method is also used in highlighting more realistic economic data.
- A trimmed mean inflation rate tagged along other measures presents a basis for comparison.
How is a Trimmed Mean Used?
Trimmed mean aims to eliminate the chance of errors in the calculation of average due to the presence of data points. This method is very useful in calculating large data sets used in real-life illustration that include large deviations and skewed distributions. The trimmed mean is said to be averages reduced by x%, where x is the sum of the percentage of data removed from both the highest and lowest values. These trimmings follow a set of rules and thus, they seem to agree with each other in presenting the total average value.
Trimmed Mean and Fluctuation
In calculating inflation rates from the Consumer Price Index(CPI) and the personal consumption expenditures (PCE), a trimmed mean can serve as a substitute for the traditional mean. The values which are discarded are not mostly equal to each other, as the calculation is tailored to fit historical data, in order to achieve the suitable fit between the trimmed mean inflation rate and the core inflation rate. The core inflation rate of the CPI or PCE is gotten from the difference between selected products and the prices of food and energy. The cost of food and energy are said to be highly volatile and are known as noisy data within the data. Inflation is not necessarily determined by shifts in the non-core areas of each data. In the arrangement of data points, each data is positioned in an ascending order based on the prices which suffered the largest decline to the prices which saw a boom. To lower the effect of volatility on the overall CPI changes, a specific percentage is trimmed from the tails of each data. Also, in the Olympics, Trimmed means are used to remove excess scores passed by bias judges that may greatly impact an athletes average performance.
Comparisons with Trimmed Means
A trimmed mean inflation rate alongside other measures presents a ground for comparison, which allows a proper examination of the inflation rates being experienced. This comparison may consist of the traditional, a trimmed mean, and a median CPI.
Illustration of Trimmed Mean
Let us assume that a skiing championship presents the following scores: 7, 9.2, 8.9, 6.4, 9.89. If the trimmed mean is tagged at 40%, then the average score will be 8.278. This is done by first calculating the arithmetic mean using the calculation: (9.2+8.9+6.4)/3 = 8.12. If the mean is trimmed by 40%, the highest 20% and the lowest 20% will be eliminated; thus 7 and 9.89 will be removed before calculating. Using this example, we can see that using the trimmed method reduces the possibility of outliers bias in a data set, and can increase average by up to 0.22 points.
Academic Research on Trimmed Mean
- The asymptotic distribution of the trimmed mean, Stigler, S. M. (1973). The asymptotic distribution of the trimmed mean. The Annals of Statistics, 1(3), 472-477.
- Adaptive trimmed mean filters for image restoration, Restrepo, A., & Bovik, A. C. (1988). Adaptive trimmed mean filters for image restoration. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(8), 1326-1337.
- The trimmed mean in the linear model, Welsh, A. H. (1987). The trimmed mean in the linear model. The Annals of Statistics, 15(1), 20-36.
- Trimmed Mean X and R Charts, Langenberg, P., & Iglewicz, B. (1986). Trimmed mean X and R charts. Journal of Quality Technology, 18(3), 152-161.
- Adaptive alpha-trimmed mean filters under deviations from assumed noise model, Oten, R., & de Figueiredo, R. J. (2004). Adaptive alpha-trimmed mean filters under deviations from assumed noise model. IEEE Transactions on Image Processing, 13(5), 627-639.
- Some statistical properties of alpha-trimmed mean and standard type M filters, Peterson, S. R., Lee, Y. H., & Kassam, S. A. (1988). Some statistical properties of alpha-trimmed mean and standard type M filters. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(5), 707-713.
- On the bootstrap and the trimmed mean, Hall, P., & Padmanabhan, A. R. (1992). On the bootstrap and the trimmed mean. Journal of Multivariate Analysis, 41(1), 132-153.
- The metrically trimmed mean as a robust estimator of location, Kim, S. J. (1992). The metrically trimmed mean as a robust estimator of location. The Annals of Statistics, 20(3), 1534-1547. [PDF]
- Trimmed mean PCE inflation, Dolmas, J. (2005). Trimmed mean PCE inflation. Federal Reserve Bank of Dallas Working Paper, 506.
- Robust lossless image watermarking based on -trimmed mean algorithm and support vector machine, Tsai, H. H., Tseng, H. C., & Lai, Y. S. (2010). Robust lossless image watermarking based on -trimmed mean algorithm and support vector machine. Journal of Systems and Software, 83(6), 1015-1028.