What is Trimmed Mean in Statistics? - Understand with Examples

What is Trimmed Mean in Statistics?

Introduction

A trimmed mean is a mean that stabilises a sample data more efficiently than the traditional mean. This guide will help you to understand trimmed mean more easily with different examples.

Trimmed Mean Definition

A trimmed mean is a method of calculating the average of sample data that removes a small percentage of the largest and smallest values before calculating the mean. After removing the largest and smallest data values, the mean is calculated by using the arithmetic mean formula. Trimmed mean is also known as truncated mean.

It is a different way to compute the mean that makes the sample data smaller. In order words, it removes the largest value from the top and the smallest values from the bottom of the sample data. In statistics and economics, it is used to collect information about the data set.

Understanding of Trimmed Mean

Truncated mean in statistics helps to reduce the effect of outliers. It is best suited for data with large values. It is expressed as x%, which shows the percentage of values to be removed from the data set. The mean is an average of numbers, while the trimmed mean stabilises the data set by removing the effect of irregular values from the calculated mean.

Trimmed Mean Formula

There is no specific formula to calculate the trimmed mean. We are only required to find the number of values that we want to remove from the data set. The percentage of values is represented by x%.

The formula to find x% of values is;

Where,

• n = is the total number of values in a sample data.
• x% = is the percentage of values for trimmed mean.

How to calculate Trimmed Mean?

It is similar to the arithmetic mean but with a small difference. To calculate the trimmed mean, we usually remove the top and bottom values. To calculate an x% trimmed mean, you can use the following steps:

1. Order the values of the data set from smallest to largest values.
2. Find the number of values in x%.
3. Remove x% of the values in the bottom and the top.
4. Calculate the mean of the remaining values after removing the percentage of values.

The following examples show how to calculate the trimmed mean.

How do you calculate x% trimmed mean?

1. 8 Trimmed Mean

The 8 trimmed mean is a mean that is calculated by removing 8 percent of small and large values from a data set.

Suppose we have a data set with values 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 and to calculate 8 trimmed mean, we will use the following steps.

Step-1

The given data set is already in ordered form. So we can move to the next step.

Step-2

To find 8% of values.

$$8\%\;\text{values}\;=\;8\%\;\times\;n$$

Here,

$$\text{n}\;=\;10$$ $$8\%\;\text{values}\;=\;8\%\;\times\;n\;=\;8\%\;\times\;9\;=\;1$$

Step-3

Removing 1 value from top and bottom of the data set, we have,

$$\{10,\;15,\;20,\; 25,\; 30,\; 35,\; 40,\; 45\}$$

Step-4

Finding the mean of the remaining values.

$$8\%\;\text{trimmed mean}\;=\frac{\;10\;+\;15\;+\;20\;+\;25\;+\;30\;+\;35\;+\;40\;+\;45}{8}$$ $$8\%\;\text{trimmed mean}\;=\;27.5$$
2. 10 Trimmed mean

The 10 percent trimmed mean or truncated mean is a mean that is calculated by excluding 10% values from the top and bottom of a data set.

Suppose we have a data set with values {4, 8, 12, 15, 9, 6, 14, 18, 12, 9}. We have to calculate the 10% trimmed mean of this data set.

So, to calculate trimmed mean, we will use the above steps.

Step-1

Order the values from smallest to largest.

$$\{4, 6, 8, 9, 9, 12, 12, 14, 15, 18\}$$

Step-2

Finding the number of values to be removed.

Here we need to know how many values are in 10%. So,

$$10\%\;\text{values}\;=\;10\%\;\times\;n$$

Here,

$$n\;=\;10$$ $$10\%\text{values}\;=\;10\%\;\times\;n\;=\;10\%\;\times\;10\;=\;1$$

So there is only 1 value to be removed from top and bottom.

Step-3

Removing 1 value from the top and bottom of the data set. The remaining values are,

$$\{6, 8, 9, 9, 12, 12, 14, 15\}$$

Now there are 8 values left in the data set so we have,

$$n\;=\;8$$

Step-4

Finding the mean after removing x% values.

$$10\%\text{trimmed mean}\;=\;\frac{6+8+9+9+12+12+14+15}{8}$$ $$10\%\text{trimmed mean}\;=\;\frac{85}{8}\;=\;10.62$$

So 10% trimmed mean is 10.62

3. 15 Trimmed Mean

The 15 percent trimmed mean is a mean that is calculated by excluding 15% of the total values from the top and bottom of a data set.

Suppose for a data set 8, 8, 9, 11, 11, 12, 12, 13, 13, 14, 15, 17, calculate 15% trimmed mean.

Step-1

The data set is already ordered from small to large values. So, we will move to the next step.

Step-2

Finding 15% values of the data set.

Here,

$$n\;=\;12$$ $$15\%\;\text{values}\;=\;15\%\;\times\;n\;=\;15\%\;\times\;12\;=\;1.8$$

By rounding off,

$$15\%\;values\;=\;2$$

Step-3

Removing 2 values from top and bottom.

$$\{9, 11, 11, 12, 12, 13, 13, 14\}$$

Step-4

Finding mean with remaining 8 data values.

$$15\%\;\text{trimmed mean}\;=\;\frac{9+11+11+12+12+13+13+14}{8}$$ $$15\%\text{trimmed mean}\;=\;11.87$$
4. 20 Trimmed Mean

Suppose for a data set 8, 8, 9, 11, 11, 12, 13, 14, calculate 20% trimmed mean. Here we want to calculate 20% trimmed mean which means that we can find it by removing 20% values from the data set. See the below steps.

Step-1

The data set is already ordered from small to large values. So, we will move to the next step.

Step-2

Finding 20% values of the data set.

Here,

$$n\;=\;8$$ $$20\%\;\text{values}\;=\;20\%\;\times\;n\;=\;20\%\;\times\;8\;=\;1.6$$

By rounding off,

$$20\%\;\text{values}\;=\;2$$

Step-3

Removing 2 values from top and bottom.

$$\{9, 11, 11, 12\}$$

Step-4

Finding mean with remaining 4 data values.

$$20\%\;\text{trimmed mean}\;=\;\frac{9+11+11+12}{4}$$ $$20\%\;\text{trimmed mean}\;=\;10.75$$
5. 40 Trimmed Mean

The 40 percent trimmed mean is a mean that is calculated by excluding 40% of the total values from the top and bottom of a data set.

Consider a set of data with values 2,3,8,5,9,4,1,0, let’s calculate 40 trimmed mean for this set.

Step-1

Ordering from small to large value,

$$\{0,1,2,3,4,5,8,9\}$$

Step-2

Finding 40% values of the data set.

Here,

$$n\;=\;8$$ $$40\%\;\text{values}\;=\;40\%\;\times\;n\;=\;40\%\;\times\;8\;=\;3.2$$

By rounding off,

$$20\%\;\times{values}\;=\;3$$

Step-3

Removing 3 values from top and bottom.

$$\{3,4\}$$

Step-4

Finding mean with remaining 4 data values.

$$40\%\;\text{trimmed mean}\;=\;\frac{3\;+\;4}{2}$$ $$20\%\text{trimmed mean}\;=\;3.5$$
6. 50 trimmed mean

The 50 percent trimmed mean is a mean that is calculated by excluding 50% of the t otal values from the top and bottom of a data set.

Consider a set of data with values 12,3,17, 8,6,15,9,14,1,0,, let’s calculate 40 trimmed mean for this set.

Step-1

Ordering from small to large value,

Step-2

Finding 50% values of the data set.

Here,

Step-3

Removing 5 values from top and bottom, we are left with no data points. It means that the 50 trimmed mean does not exist. So, we can’t proceed to the next step.

Uses of Trimmed Mean

• The trimmed mean is used to report economic data to smooth the results.
• It helps remove the effect of irregular outliers that can affect the traditional mean of a data set.
• It is used in consumer price indexes to reduce volatility.
• It helps to optimise data if there are many values in the data set.

FAQ’s

What is 5 Trimmed Mean?

The 5% trimmed mean is the mean computed by removing 5% of largest and 5% of smallest values from the sample and calculating the mean of remaining sample values. For example, the data set {5,4,7,8,7,2,0,11,10,18}.

Arranging it from smallest to larges value.

By rounding off,

So removing first and last value of the data set,

So 5% trimmed mean will be,

What is Trimmed Mean and why is it Used?

A trimmed mean is a mean that is calculated by removing a small percentage of values from the top and bottom of a sample data. It is suitable for calculating the mean of a sample data that has a larger number of values.

Which is a Characteristic of the Trimmed mean as a Measure of Central Tendency?

Trimmed mean removes the extreme values in the sample data from both sides of the distribution. It aims to remove outliers to the data set. Hence, it is more resistant to outliers compared to the mean and it can describe the sample data with a single value.