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The "Average" of a Set—Different Ways to Measure Central Tendency

There are many mathematical formulas for calculating the "average" value of a set.
There are many mathematical formulas for calculating the "average" value of a set.

The term "average" is most often used as a synonym for the arithmetic mean, that is, the sum of the elements in a set divided by the number of elements in that set. However, there are many other ways to compute averages in the sense of finding a set's center.

The arithmetic mean, median, mode, geometric mean, harmonic mean, root mean square, trimean, midrange, and midhinge are collectively known as measures of central tendency. Each provides a different way of quantifying the notion of "average."

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Arithmetic Mean

The simplest way to calculate an average, and the most commonly assumed definition of average is adding up all the numbers in a set and dividing by the size of the set. For example, the arithmetic mean of the data set {-3, 4.4, 0, 0.2, 1.7} is 0.66 because (-3 + 4.4 + 0 + 0.2 + 1.7) ÷ 5 = 0.66.

Median

The median of a set is the middle number when the data points are arranged from smallest to largest. For example, in the set of 5 math exam scores {89, 100, 70, 60, 71} the median is 71 because when you reorder them from least to greatest you get {60, 70, 71, 89, 100} and 71 is right in the middle.

This is easy if the set has an odd number elements, but what if there are an even number? In this case, you take the arithmetic mean of the middle two numbers. For example, consider the set of 6 math exam scores {89, 100, 70, 60, 71, 81}. If you order them correctly, you get {60, 70, 71, 81, 89, 100}. The middle two numbers are 71 and 81, and their arithmetic average is (71+81)/2 = 76. Therefore, the median is 76.

Mode

The mode is the most commonly occurring number or numbers in a set of data. If all the numbers have equal frequency of 1, the set has no mode. For example, if the set of test scores in a statistics class is {99, 71, 71, 87, 77} then the mode is 71 because that number occurs more than any other number. If the ages of students in a college calculus class are {43, 18, 17, 31, 22, 22, 17, 19, 20} then the modes are 17 and 22. The set {1, 2, 3, 4, 5, 6} has no mode. Although the mode does not indicate the location of the data's center, it does indicate where the data tends to cluster.

Geometric Mean

The geometric mean of a collection of n data points is the nth root of the product of the numbers. For example, the geometric mean of the set {12, 45, 50} is (12*45*50)^(1/3) = 27000^(1/3) = 30. The geometric mean of a set that contains 0 is always 0, and if the set contains negative numbers, the geometric mean may be undefined.

To implement the geometric mean in a computer program that may not be able to handle large products, you can use the equivalent formula

e^[ (∑ Ln(xi)) / n ]

where xi is one of the n elements and Ln() is the natural log function. In plain English, you take the arithmetic mean of the natural logarithms of each data point, then exponentiate.

Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. For example, if you have the set of ratios {1/3, 1/4, 1/5, 1/6, 1/8, 1/9, 1/10, 1/11} the harmonic mean is

[(3 + 4 + 5 + 6 + 8 + 9 + 10 + 11)/8]^(-1)

= 8/(3 + 4 + 5 + 6 + 8 + 9 + 10 + 11)

= 1/7.

The harmonic mean is useful in measuring the average of ratios whose denominators are key subject of analysis.


Root Mean Square

Sometimes known as the quadratic mean, the root mean square is the square root of the arithmetic mean of the squared data. The standard deviation is the root mean square of each element's distance from the mean. For example, consider the set of math class grades {92, 94, 88, 79, 79, 65, 73}. The squared data is {8464, 8836, 7744, 6241, 6241, 4225, 5329}. The arithmetic mean of the squared numbers is

(8464 + 8836 + 7744 + 6241 + 6241 + 4225 + 5329)/7 = 6725.7143.

The square root of 6725.7143 is 82.0105. Thus, the root mean square or quadratic mean of the test scores is about 82 to the nearest integer.

Trimean

The trimean of a set is the weighted average of the following three

  • first hinge, aka the first quartile boundary (weight = 1)
  • second hinge, aka the third quartile boundary (weight = 1)
  • median (weight = 2).

In mathematical notation, the statistical formula for the trimean is

Trimean = (Q1 + 2M + Q3)/4

where Q1 is the upper limit of the first quartile, M is the median, and Q3 is the lower limit of the third quartile. As an example, consider this ordered set of ages of students in a college physics class:

{18, 19, 19, 20, 20, 20, 20, 21, 22, 22, 23, 27, 37}

The first quartile is 20, the median is 20, and the third quartile is 22. The trimean of the ages is calculated as (20 + 2*20 + 22)/4 = 20.5.

Midrange

The midrange is the arithmetic mean of the minimum and maximum values of a data set. It is infrequently used as an average because it only takes two points into consideration and is unduly influenced by possible outliers. For the set {3, 4.5, 4, 7, 5, 12, 9} the min and max values are 3 and 12 respectively. Thus, the midrange is (3+12)/2 = 7.5.

Midhinge

The midhinge is similar to the trimean except that the median is disregarded. The midhinge is simply the average of the two hinges, i.e. the average of the first and third quartile boundaries. For example, consider the data set {18, 19, 19, 20, 20, 20, 20, 21, 22, 22, 23, 27, 37}. Here, Q1 = 20 and Q3 = 22. The midhinge is (20 + 22)/2 = 21.

Comparison Example

To compare the various ways of calculating the average, consider the following set of positive integers

{1, 1, 1, 1, 2, 3, 3, 5, 8, 9, 10, 10, 13, 14, 19, 20, 29}

The median is 8, the first hinge (first quartile boundary) is 2, the second hinge (third quartile boundary) is 13, the minimum is 1, and the maximum is 29. With these values we calculate the following averages:

Arithmetic Mean = 149/17 ≈ 8.765
Median = 8
Mode = 1
Geometric Mean ≈ 5.159
Harmonic Mean ≈ 2.792
Root Mean Square = sqrt(139) ≈ 11.790
Trimean = 7.75
Midrange = 15
Midhinge = 7.5

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Comments 4 comments

SJ 3 years ago

I know this isn't meant to be a comprehensive list, but there's also the generalized power mean [(x1^p + x2^p + ... + xn^p)/n]^(1/p).


calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

Thanks, SJ. That's another useful mean.


O'Brien 21 months ago

What is the name of the average where you find the mean, median and mode and then average the three numbers? I also recall one where you take the sum of squares and divide by the sum -- (x1^2 + x2^2 + ... xn^2)/(x1 + x2 + ... xn).


calculus-geometry profile image

calculus-geometry 21 months ago from Germany Author

The first average I am not sure what it is called as I have never seen that one before, though I'm sure it exists and is used in some contexts. The second one is called the contraharmonic mean.

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