In statistical analysis and data interpretation, an outlier can pose different problems as it can heavily influence measures of central tendency, such as the mean. However, like the traditional mean, the trimmed mean tends to solve the issues of outliers in our dataset. Therefore, on this page, we will take out time to learn and explore how trimmed mean works, its definition, formula, uses, benefits, and possibly real-life applications. But before we delve into all these things, let’s first understand the traditional mean (arithmetic mean). You can skip the next section though if you are already familiar with arithmetic mean.
UNDERSTANDING THE TRADITIONAL MEAN (ARITHMETIC MEAN)
The traditional mean, also known as the arithmetic mean, and just as every other means, it is a fundamental measure of central tendency used in statistics. It provides a sequence to summarize a dataset or a set of values by finding their average. The arithmetic mean is widely used in various fields for data analysis, and it\’s relatively easy to compute especially when dealing with ungroup data. However, the computation of arithmetic mean is not our concern on this page, I will however state a simple formula for its calculation below.
Arithmetic mean formula:
Ῡ = ∑yi / N
DEFINITION OF TRIMMED MEAN
Hey! What’s trimmed mean all about? Relax, it\’s as simple as it sounds. A trimmed mean or truncated mean is a method of finding a more realistic average value by getting rid of extreme observations. In this type of means, an equal percentage of the highest and lowest values are cut out from both the extremes of the dataset.
Usually, a percentage of values, such as 10%, 20%, or 30% is used. For more clarity, if you want to trim for example, 10% of extreme values from the dataset, this excludes the top and bottom 10% of values, effectively using only the middle 80% of the data.
After removing the specified extreme values from the observations, the trimmed mean is found using a traditional arithmetic formula stated above. Making use of a trimmed mean helps eliminate the influence of outliers or extreme values on the tails that may unfairly affect the traditional or arithmetic mean. Trimmed means help to obtain a robust measure of central tendency.
To provide a more accurate representation of economic trends, trimmed means are used to present a more realistic picture.
A good example where a trimmed mean is applicable is in gymnastics, the trimmed mean is used to calculate the final score for each gymnast. This can be done by removing the highest and lowest scores from each event and then averaging the remaining scores. This helps to ensure that the final scores are not unduly influenced by outliers, such as a gymnast who may have made a major mistake on an event. We will investigate more practical examples in due time.
NOTE: The lowest trimming is 0%, which is the same as the arithmetic mean.
TRIMMED MEAN FORMULA
Calculating a trimmed mean involves a series of well-defined steps:
- Decide on the percentage of trimming. It could be 10%, 20%, 30%, 40% etc.
- Multiply the percentage by the number of observations in the dataset to determine the number of values to cut from each endpoint.
- Cut out the highest and lowest numbers from both endpoints after arranging the dataset in either ascending or descending order.
- You can now apply a traditional or arithmetic mean formula to the remaining values obtained from the step above.
- After applying the traditional formula, the answer obtained is called the trimmed mean.
HOW TO CALCULATE A TRIMMED MEAN (STEP-BY-STEP)
For this practical example, we will be using both 10% and 20% trimming levels for the same dataset to calculate the trimmed mean.
Example 1:
PROBLEM:
A Javelin player records the following scores in meters.
6.9, 8.9, 7.2, 16.1, 3.8, 9.4, 5.4, 8.8, 4.2, 6.7
Calculate the 10% trimmed mean.
SOLUTION:
Step 1: The first step is to choose the desired trimming percentage but in our case, 10% is given already. Therefore, we move to the second step.
Step 2: Arrange the dataset; Let’s sort the data in ascending order, i.e., from the smallest to the highest value.
3.8, 4.2, 5.4, 6.7, 6.9, 7.2, 8.8, 8.9, 9.4, 16.1
Step 3: Find out the percentage to be cut on each endpoint of the dataset. To do this, we must count the number of observations in the dataset.
Number of observations = 10.
Trimming percentage = 10% (0.1). For a 10% trim, you would trim 10% of the total number of values.
Number of observations to trim from both endpoints = 0.1 * 10 = 1
Therefore, we would trim out 1 observation from both endpoints of our dataset.
Now, we have.
4.2, 5.4, 6.7, 6.9, 7.2, 8.8, 8.9, 9.4
Step 4: Apply traditional or arithmetic mean formula.
Trimmed Mean = 4.2 + 5.4 + 6.7 + 6.9 + 7.2 + 8.8 + 8.9 + 9.4 / 8
= 57.5 / 8
= 7.1875
Example 2:
Calculate the 20% trimmed mean using the same dataset.
SOLUTION:
Step 1: We already know the trimming percentage (20%).
Step 2: Arrange the dataset; Let’s sort the data in ascending order, i.e., from the smallest to the highest value.
3.8, 4.2, 5.4, 6.7, 6.9, 7.2, 8.8, 8.9, 9.4, 16.1
Step 3: Find out the percentage to be cut on each endpoint of the dataset. To do this, we will count the number of observations in the dataset again.
Number of observations = 10.
Trimming percentage = 20% (0.2).
Number of observations to trim from both endpoints = 0.2 * 10 = 2.
Therefore, we would trim out 2 observations from both endpoints of our dataset.
Now, we have.
5.4, 6.7 6.9, 7.2, 8.8, 8.9
Step 4: Apply traditional or arithmetic mean formula.
Trimmed Mean = 5.4 + 6.7 + 6.9 + 7.2 + 8.8 + 8.9 / 6
= 43.9 / 6
= 7.317
NOTE: If you have an extremely large dataset, it may be difficult to calculate a trimmed mean manually, so you are free to use this Trimmed Mean Calculator.
In practical applications, accurately determining the percentage to be cut is very important. A combination of experience, knowledge, and practice allows users to select an appropriate percentage to be trimmed from the arithmetic mean. This percentage can vary depending on the specific situation, problem, and context in which it is applied.
PRACTICAL APPLICATIONS, BENEFITS AND USES OF THE TRIMMED MEAN
Trimmed mean can be used in different fields, here are a few:
Use in finance: The trimmed means can be used to calculate financial indices where extreme market events can introduce substantial volatility. Indices such as the Consumer Price Index (CPI), etc. This application will help to reduce the impact of outliers that could distort the index value.
Use in sports: Also, in sports can be used to calculate the final scores of some sporting events. Some of such sports are gymnastics and swimming. Applying this method enables the organizers to reduce the impact of extreme scores that could unfairly advantage or disadvantage a competitor.
Use in education assessment: In an examination where the scores of a few students are significantly higher or lower than other students, the effect of outliers in assessing the overall performance of the students can be minimized by applying a trimmed mean.
Economic Uses: Obviously, trimmed mean has its uses in Economics as they can analyze economic data, such as income and wealth distribution, to get a more accurate picture of the overall distribution of values.
Use in Climate Studies: Climate patterns and trends are very important for predicting future weather conditions, assessing climatic impact, and formulating mitigation strategies. However, climate datasets often contain too many extreme values due to weather anomalies, rare events, or measurement errors. These outliers can distort statistical measures like the mean, hindering accurate trend analysis.
Use in science: The trimmed mean can also be used in science to analyze scientific data, like laboratory measurements, to reduce the impact of experimental errors that could affect the results.
FINAL THOUGHT
Trimmed mean stands as a wide and powerful statistical tool, offering a robust alternative to the mean in the presence of outliers. However, robustness comes with a tradeoff as you tend to use less of the original data, reducing your effective sample size and information about the original data. In other words, the trimming percentage\’s subjectivity can influence the analysis, and the loss of information from discarded outliers may be very important. Apart from these considerations, trimmed mean is a valuable tool.
