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An Introduction to Statistics - Lesson 4

What Does He Mean?

Arithmetic, Geometric, Harmonic, Quadratic, Trimmed, Weighted, etc. are all means!

Lesson Overview

Arithmetic Mean

In lesson 3, we defined arithmetic mean as the one commonly used by statisticians and as the one usually intended when we just say mean. However, there are a wide variety of other means with a variety of applications.

Geometric Mean

The geometric mean is used in business to find average rates of growth.
The geometric mean is the nth root of the [pi] product of the data elements.

Geometric mean =         for all n [greater than or equal to] 2.

Example: suppose you have an IRA (Individual Retirement Account) which earned annual interest rates of 5%, 10%, and 25%.
Solution: The proper average would be the geometric mean or the cube root of (1.05 • 1.10 • 1.25) or about 1.13 meaning 13%.

Note that the data elements must be positive. Negative growth is represented by positive values less than 1. Thus, if one of the accounts lost 5%, the proper multiplier would be 0.95.

The geometric mean is typically first encountered in a proportion when the means are equal, as in 8/w=w/4. Here w2=32 and square rooting both sides gives an answer. However, in general, there may be n nth geometric means. We thus cannot be sure of the sign of w above.

The difference between arithmetic and geometric means is similar to the difference between arithmetic and geometric sequences. In an arithmetic sequence (2, 4, 6, 8, ...) you add the same amount each time (2). In a geometric sequence (2, 4, 8, 16, ...) you multiply by the same factor each time (2). If you are given the 1st and 4th term of an arithmetic sequence (1,?,?,10), you can solve for the missing terms by finding the difference of the known terms and dividing the interval by the number of gaps between missing terms: (10-1)/3=3 to find the common difference (1, 4, 7, 10). If you are given the 1st and 5th terms of a geometric sequence (2, ?, ?, ?, 32), you can solve for the missing terms by finding the ratio of the known terms and nth rooting for the common ratio. Here again, n is determined by the number of gaps between missing terms and known terms. However, since there can be four different 4th roots, there may be up to four different sequences: (2, 4, 8, 16, 32) or (2, -4, 8, -16, 32) and, if complex numbers are allowed, (2, 4i, -8, -16i, 32) or (2, -4i, -8, 16i, 32). Hopefully, this motivates the nth roots used above. Also, since fractional exponents are usually new, a section on them is included at the end of this lesson.

Harmonic Mean

The harmonic mean is found by dividing the number of data elements
by the sum of the reciprocals of each data element.

Harmonic mean =    

The harmonic mean is used to calculate average rates such as distance per time, or speed. (In physics you will learn that speed is a scalar, whereas velocity is a vector, having both magnitude and direction. Great care should be exercised to select the proper term.)

Example: Suppose your grandfather walked three miles to school. Due to the terrain, for the first mile he averaged 2 mph; for the second mile 3 mph; for the final mile the average speed was 4 mph. What was the average speed for the three miles?
Solution: The arithmetic mean of (2+3+4)/3 = 3.0 is incorrect. This would imply it took 1 hour = 60 minutes to walk to school. Breaking it down into the separate components, it takes 30 minutes (1st) + 20 minutes (2nd) + 15 minutes (3rd) to walk (each mile) or 65 minutes total. His actual speed was thus 3/1.083 or 2.77 mph.

Another way to show our work would be:
     3 miles          =     3     =   36
1/2 + 1/3 + 1/4 13/12 13
 =   2.77 mph.

Quadratic Mean

The quadratic mean is another name for Root Mean Square or RMS.

RMS =

The quadratic mean is typically used for data whose arithmetic mean is zero.

Example: Let's explore US household alternating current (AC). As illustrated in the graph below (sine curve), the voltage ranges from positive values to negative values. The quadratic mean gives a physical measure of the average distance from zero. Suppose measurements of 120, -150, and 75 volts were obtained.
Solution: The corresponding quadratic mean is [square root] ((1202 + (-150)2 + 752)/3) or 119 volts RMS.

Trimmed Mean

Trimmed mean usually refers to the arithmetic mean
without the top 10% and bottom 10% of the ordered scores.

Technically, this is the 10% trimmed mean. You could also find the 20% trimmed mean by only forming the mean of the middle 60% of the data. Clearly this removes extreme scores on both the high and low end of the data. Sorting the data is also clearly indicated!

Weighted Mean

Weighted mean is the average of differently weighted scores.
It is the sum of the weighted scores over the sum of the weights.
It takes into account some measure of weight attached to different scores.

Example: Semester grades are often computed as 40% (2/5) of the 1st 9-week grade, 40% (2/5) of the 2nd 9-weeks grade, and 20% (1/5) for the semester exam. For specifics, assume Martha earned 84% for the first 9-weeks, 89% for the second 9-weeks, but only 60% on the semester exam.
Solution: In such a case, the semester grade could be computed as:

(2 • 84% + 2 • 89% + 60%)/5 = 81.2%

Grade Point Averages is another typical example of a weighted mean, because college classes often come in a variety of credit hours. The formula for the weighted mean is given below.

Weighted mean =

Combination Mean

Consider if you had $10,000 earning 6%, $20,000 earning 12%, and $25,000 earning 18% annual interest. Clearly some combination of weighted geometric mean would be needed to compute a proper average value! A similar example would involve speeds (2 mph, 3 mph, 4 mph) when applied to different distances. The correct mean value would involve some weighted harmonic mean.

Means from a Frequency Table

Frequency mean is the same as obtaining the arithmetic mean from a frequency table.
For memory purposes, it is like the weighted mean formula.

An activity for finding the mean from a frequency table is included with lesson 5.

Necessary Exponents (for Geometric Mean)

In order to do some problems in today's assignment, an expanded definition of exponents needs to be developed. Recall from Numbers lesson 4 the definition and rules for exponentiation as follows.

x1 = x           x2 = x•x           x3 = x•x•x           x4 = x•x•x•x           x5 = x•x•x•x•x   and
x-1 = 1/x           x-2 = 1/x2           x-3 = 1/x3           x-4 = 1/x4

We can extend this to define what x raised to a fractional exponent means by using the fact that when powers with common bases are multiplied, the exponents are added. Square roots were introduced in numbers lesson 10.

x1/2x1/2 = x(1/2 + 1/2) = x1 = x
x1/3x1/3x1/3 = x(1/3 + 1/3 + 1/3) = x1 = x
x1/4x1/4x1/4x1/4 = x(1/4 + 1/4 + 1/4 + 1/4) = x1 = x

In other words: x1/2 = sqrt(x)       x1/3 = cube root of x       x1/4 = fourth root of x   and
41/2 = 2       81/3 = 2       729 1/6 = ((729)1/3)1/2 = (9)1/2 = 3

We can define a real number x raised to rational roots such as a/b (xa/b) to be the bth root of x raised to the ath power. The extension of any real number to any real power goes even beyond numbers lesson 15.

Such roots can be calculated on the calculator three ways as follows.

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