Beyond the Average: 5 Types of Means

       Introduction

When we talk about the "average," we're usually thinking of just one number. But in statistics, there isn't just one way to find the center of your data—there are several. Each type of mean provides a unique perspective, and choosing the right one can reveal a more accurate story hidden within the numbers.

Here are five essential means, from the everyday arithmetic mean to the robust trimmed mean.

 

1.     Arithmetic mean

 

The arithmetic mean of a set of data is the sum of all data divided by the number of data in the set. For example, a student obtained grades of 7.0, 3.0, 5.5, 6.5, and 8.0 in mathematics. He passed because the average grade is:

                                              ·        Mean = (7.0 + 3.0 + 5.5 + 6.5 + 8.0)/5 = 6.0 

 

The arithmetic mean of a sample is represented by x (read as x-bar or x-slash). The sample size is indicated by n. So, the formula for calculating the arithmetic mean of a sample is:

 

·        x̄ = (1/n) Σ xᵢ = (x₁ + x₂ + ... + xₙ)/n

       2- Weighted average 

The weighted average is the sum of the products of the data (x) by their respective weights (p), divided by the sum of the weights.

To understand how weighted averages are calculated, imagine that a student took three tests in a certain subject in which the material is cumulative, that is:

• in the first test, questions were asked about the material taught up to the date of that first test; 
• in the second test, questions were asked about the material taught from the beginning of the course up to the date of that second test.
• in the third test, questions were asked about the material taught from the beginning of the course up to the end of the course.

It is reasonable that the grade for the first test should have less weight (in other words, count for less in the final grade) than the second; it is also reasonable that the grade for the second test should have less weight than the third. Consider the following weights were  proposed: 1, 2, 3.

Imagine that the student obtained grades of 4, 7, and 6, which had weights of 1, 2, and 3, respectively. The student's weighted average is

                                           ·        x̄ = (1×4 + 2×7 + 3×6)/(1 + 2 + 3) = 36/6 = 6.0

Notice that to obtain the weighted average of a student's grades, each grade by was multiplied by its respective weight; products were added; weights were added and was applied the formula:

·        Formula: x̄ = Σ(xᵢpᵢ)/Σpᵢ

 3 Geometric mean

 

The geometric mean is given by the n^th root of the product of n data points.

The geometric mean is difficult to calculate and, perhaps because of this characteristic, is rarely used.

Here is an example. Let's calculate the geometric mean of the following data: 2, 3, 5, and 10.

 

                                       ·        G = ⁴√(2×3×5×10) = ⁴√300 = 4.16

To perform this calculation, use a calculator or apply logarithms. Since

      Therefore

        So, given n values of variable X,the geometric mean is

 

                                                           
 

The Greek letter ∏ (pronounced pi) is used as a mathematical symbol to indicate that all observed values of X must be multiplied. In mathematics, this letter is read as product.

4. Harmonic mean

The harmonic mean of n data points is the inverse of the arithmetic mean of the inverse of these values.

As an example, consider two numbers, 2 and 4. To calculate the harmonic mean, indicated here by H, invert the numbers, determine the arithmetic mean of these inverses, and invert the arithmetic mean to find the harmonic mean:

 

To calculate the harmonic mean, apply the formula:

5. Trimmed mean

 

Trimmed mean is a way to calculate an average by first removing a small percentage of the highest and lowest values. After taking out these extreme values, the average is calculated using the usual method.

Let’s say, as an example, a figure skating competition produces the following scores:

6.0, 8.1, 8.3, 9.1, 9.9.

 

The mean for the scores would equal:

 

    

To trim the mean by a total of 40%, we remove the lowest 20% and the highest 20% of values, eliminating the scores of 6.0 and 9.9.

Next, we calculate the mean based on the calculation:                                                      

Conclusion

Although the arithmetic mean is the most familiar, it is not always the most appropriate measure.
Choosing the right type of mean depends on the nature of the data and the purpose of the analysis.
Understanding the differences among arithmetic, weighted, geometric, harmonic, and trimmed means ensures that statistical summaries accurately reflect what the data reveal.