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Mean

Sum of the values divided by the number of values

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Geometric Mean

Suppose you were assigned to help evaluate the scoring from surveys given to students to choose the prom king at your high school prom. The survey asked the students to score the nominees on a five-point scale for each of four categories, on a twenty-five point scale for each of three categories, and on a thirty-five point scale for the last 4 categories.

Credit: JD Hancock
Source: https://www.flickr.com/photos/jdhancock/7238856518
License: CC BY-NC 3.0

How could you average the total scores for each nominee without the scores of the higher point categories overshadowing the scores of the lower-point ones?

Geometric Mean 

The geometric mean (also known as the mean proportional) is a method of finding the ‘middle’ value in a set that contains some values that are intrinsically more influential than others. The geometric mean takes into account the differences in proportion between values in different ranges.

To calculate the geometric mean of a set of data:

  • Multiply the value of each member of the set by the next, as in \begin{align*}x_1 \times x_2 \times x_3 \times x_4\end{align*}, etc.
  • Find the \begin{align*}n^{th}\end{align*} root of the product of the set values, where \begin{align*}n\end{align*} is the number of values in the set
  • The \begin{align*}n^{th}\end{align*} root of the product of the set values is the geometric mean of the set

The mean proportional can also be described in a couple of ways using geometry:

  • If you have a rectangle with side lengths \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, then the side length of a square with area equal to the rectangle is the mean proportional of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.

Alternatively:

  • Construct a semicircle with a diameter length of \begin{align*}a+b\end{align*} and a perpendicular line from the diameter to the semicircle located at the intersection of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. The length of the perpendicular line is the geometric mean of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.

Finding the Value a Mean is Proportional to 

What is the mean proportional of \begin{align*}x\end{align*}?

\begin{align*}x=\{8, 12, 14, 6, 9, 15, 22, 7\}\end{align*}

First find the product of the members of the set:

\begin{align*}8 \times 12 \times 14 \times 6 \times 9 \times 15 \times 22 \times 7=167, 650, 560\end{align*}

Now take the 8th root of the product (since there are 8 members in the set):

\begin{align*}\sqrt[8]{167, 650, 650} \approx 10.667\end{align*}

Finding Unknown Values 

Ken has a dog kennel in back of his house. The dimensions of the kennel are 9 feet by 13 feet. What would be the side lengths of a square kennel enclosing the same area? What is another name for this value?

First find the area of the original kennel:

\begin{align*}9 \ ft \times 13 \ ft= 117 \ ft^2\end{align*}

Now find the lengths of the sides of the square:

\begin{align*}\sqrt{117 \ ft^2}=10.82 \ ft\end{align*}

10.82 is the geometric mean of 9 and 13.

Finding the Average Rate of Return 

Assume that $5000 was invested at a starting interest rate of 7%, and the rate increased by 2% each year for years 2, 3, 4, and 5 then decreased by 3% for year 6. What would the average rate of return be for the whole period? 

It may seem that you could simply calculate the arithmetic mean of the interest rates to find the average yearly return. However, the arithmetic mean of the 6 yearly rates would only describe the average of the stated percentages each year. The complication is that a 7% interest rate in the first year would only result in 107% of the initial value, whereas a 7% rate in the 6th year would yield 107% of the initial investment plus 107% of all of the interest from the prior years! Therefore, a different ‘middle number’ is needed.

The geometric mean of the interest rates would provide the correct average yearly rate. We need to find the product of all of the yearly rates, and then take the 6th root (since there are 6 rates in this problem) of that product. The correct calculation looks like this:

\begin{align*}& \sqrt[6]{1.07 \times 1.09 \times 1.11 \times 1.13 \times 1.15 \times 1.12} = \sqrt[6]{1.88}=1.11 \\ & \qquad \quad \therefore \ The \ average \ return \ is \ 11\% \ per \ year\end{align*}

Earlier Problem Revisited

Suppose you were assigned to help evaluate the scoring from surveys given to students to choose the prom king at your high school prom. The survey asked the students to score the nominees on a five-point scale for each of four categories, on a twenty-five point scale for each of three categories, and on a thirty-five point scale for the last 4 categories.

How could you average the overall scores for each nominee without the disproportionate weighting that would occur in favor of the higher point categories if you used an arithmetic mean?

By calculating the geometric mean of the scores earned in all 11 categories, you could identify an average score for each contestant that was proportionately weighted based on category value.

Examples 

Example 1

Construct a semicircular representation of the mean proportional of the values 5 and 9.

Use the geometric method described above: length \begin{align*}a=9\end{align*}, length \begin{align*}b=5\end{align*}, place them end-to-end and they become the diameter of a semi-circle. Construct a perpendicular from the intersection of \begin{align*}a\end{align*} and \begin{align*}b\end{align*} to the circumference of the semi-circle, and the length of that perpendicular is the geometric mean. In this case, the geometric mean is 6.8 cm.

Example 2

Find the average rate of return on an investment that earns 6.04%, 6.89%, 7.22%, 6.92%, and 7.43% over successive years.

This problem is just like third third example in the text, just find the geometric mean of the interest rates:

\begin{align*}\sqrt[5]{6.04 \times 6.89 \times 7.22 \times 6.92 \times 7.43}= \sqrt[5]{15, 448.569}=6.88 \%\end{align*}

Example 3

Construct a visual representation of geometric mean of the numbers 23 and 38, using quadrilaterals. 

This is just like the quadrilateral method described in the “Guidance” section: If you create a rectangle with side lengths equal to the numbers 23 and 38, then the side lengths of a square with the same area as the rectangle will be the geometric mean of the two numbers:

Therefore, the geometric mean of 23 and 38 is 29.56, just like the side length of the square.

Example 4

Find the geometric and arithmetic mean of \begin{align*}y\end{align*}

The geometric mean is the 9th root of the product of the values (since there are 9 values):

\begin{align*}\sqrt[9]{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9}= \sqrt[9]{362, 880}=4.147\end{align*}

The arithmetic root is the sum of the values divided by the count of the values:

\begin{align*}\frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{9}=5\end{align*}

Example 5

Find the mean proportional of the values: 12.34, 15.52, 16.82, 13.29, 13.91  

There are five values here, so we need to find the 5th root of the product:

\begin{align*}\sqrt[5]{12.34 \times 14.52 \times 16.82 \times 13.29 \times 13.91}= \sqrt[5]{557, 134.28}=14.1\end{align*}

Review 

For questions 1-10, find the geometric mean of the numbers.

  1. \begin{align*}\{5, 8, 9, 7, 6, 8, 3, 4\} \end{align*}
  2. \begin{align*}\{12, 16, 18, 13, 14, 16\} \end{align*}
  3. \begin{align*}\{23, 24, 26, 28\} \end{align*}
  4. \begin{align*}\{33, 37, 38, 36, 35, 35, 36, 34\} \end{align*}
  5. \begin{align*}\{46, 49, 48, 41, 42, 43, 44\} \end{align*}
  6. \begin{align*}\{51, 52, 53, 54, 55, 56, 57, 58, 59\}\end{align*}
  7. \begin{align*}\{156.21, 245.25, 184.64, 222,32, 218.94, 134.88\}\end{align*}
  8. \begin{align*}\{554, 564, 585, 525, 534, 500\} \end{align*}
  9. \begin{align*}\{0.0021, 0.0034, 0.081, 0.009, 0.01, 0.258\} \end{align*}
  10. \begin{align*}\left \{\frac{5}{8}, \frac{13}{21}, \frac{7}{9}, \frac{3}{5}, \frac{11}{23}\right \}\end{align*}
  11. Construct a semicircular representation of the mean proportional of the values 12 and 19.
  12. Construct a visual representation of the geometric mean of the numbers 5 and 8, using quadrilaterals.
  13. Find the average rate of return on an investment that earns 5.02%, 4.11%, 4.18%, 3.72%, and 3.53% over successive years.

Review (Answers)

To view the Review answers, open this PDF file and look for section 5.2. 

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Vocabulary

Average

The arithmetic mean is often called the average.

Geometric mean

The geometric mean is a method of finding the ‘middle’ value in a set that contains some values that are intrinsically more influential than others.

Harmonic mean

A harmonic mean is calculated by dividing the number of values in the set by the sum of the inverses of the values in the set.

mean

The mean, often called the average, of a numerical set of data is simply the sum of the data values divided by the number of values.

measures of central tendency

The mean, median, and mode are known as the measures of central tendency.

Population Mean

The population mean is the mean of all of the members of an entire population.

Sample Mean

A sample mean is the mean only of the members of a sample or subset of a population.

weighted

A weighted value or set of values takes into account varying levels of importance among members of the set.

weighted average

A weighted average is an average that multiplies each component by a factor representing its frequency or probability.

weighted harmonic mean

A weighted harmonic mean is a harmonic mean of values with varying frequencies or weights.

Image Attributions

  1. [1]^ Credit: JD Hancock; Source: https://www.flickr.com/photos/jdhancock/7238856518; License: CC BY-NC 3.0

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