Chapter 3.1, Problem 37EC

Geometric Mean The geometric mean (GM) is defined as the nth root of the product of n values. The formula is

$GM=\sqrt[n]{\left(X_1\right)\left(X_2\right)\left(X_3\right)\mathrm{...}\left(X_n\right)}$

The geometric mean of 4 and 16 is

$GM=\sqrt{\left(4\right)\left(16\right)}=\sqrt{64}=8$

The geometric mean of 1, 3, and 9 is

$GM=\sqrt[3]{\left(1\right)\left(3\right)\left(9\right)}=\sqrt[3]{27}=3$

The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown.

$GM=\sqrt{\left(1.2\right)\left(1.1\right)}\approx 1.1489$

Or

$GM=\sqrt{\left(120\right)\left(110\right)}\approx 114.89$

His salary is 120% at the end of the first year and 110% at the end of the second year. This is equivalent to an average of 14.89%, since 114.89% − 100% = 14.89%.

This answer can also be shown by assuming that the person makes $10,000 to start and receives two raises of 20% and 10%.

Raise 1 = 10, 000 ⋅ 20% = $2000

Raise 2 = 12, 000 ⋅ 10% = $1200

His total salary raise is $3200. This total is equivalent to

$\begin{array}{l}\$10,\mathrm{000.14.89}\%=\$1489.00\\ \$11,\mathrm{489.14.89}\%=\$1710.71\\ \$3199.71\approx \$3200\end{array}$

Find the geometric mean of each of these.

a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%.

b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%.

c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%.

d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.

To determine

The geometric mean of growth rate.

Answer to Problem 37EC

The geometric mean of growth rate is 24.73%.

Explanation of Solution

Given Info:

The given total 3 values are 35, 24 and 18%.

Calculation:

Formula to calculate the geometric mean is,

$\text{Geometric}\text{Mean}=\sqrt[n]{\left(X_1\right)\left(X_2\right)\left(X_3\right)\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(X_n\right)}$

Where $n$ is the number of values and $X_1{X}_2{X}_3\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot X_n$ is n value

Substitute $3$ for $n$, $35$ for $X_1$, $24$ for $X_2$ and $18$ for $X_3$.

$\begin{array}{c}\text{Geometric}\text{Mean=}\sqrt[3]{\left(35\times 24\times 18\right)}\\ =\sqrt[3]{15120}\\ =24.73\end{array}$

The growth rate of the living life insurance corporation is 24.73%.

To determine

The geometric mean of the data.

Answer to Problem 37EC

The geometric mean for the salary raise percentages is 5.57%

Explanation of Solution

Given info:

The salary raise percentages of a person over a 4 year period are 8, 6, 4 and 5%.

Calculation:

Substitute $4$ for $n$, $8$ for $X_1$, $6$ for $X_2$, $4$ for $X_3$ and $5$ for $X_4$.

$\begin{array}{c}\text{Geometric}\text{Mean=}\sqrt[4]{\left(8\times 6\times 4\times 5\right)}\\ =\sqrt[4]{960}\\ =5.57\end{array}$

Thus, the geometric mean for the salary raise percentages is 5.57%

To determine

The geometric mean of the data.

Answer to Problem 37EC

The geometric mean for the increase in stock over a 5 year period is 7.63%

Explanation of Solution

Given Info:

The percentage of stock increases each year over a period of 5 years and the percentages are 10, 8, 12, 9 and 3%

Calculation:

Substitute $5$ for $n$, $10$ for $X_1$, $8$ for $X_2$, $12$ for $X_3$, $9$ for $X_4$ and $3$ for $X_5$.

$\begin{array}{c}\text{Geometric}\text{Mean=}\sqrt[5]{\left(10\times 8\times 12\times 9\times 3\right)}\\ =\sqrt[5]{25,920}\\ =7.63\end{array}$

Thus, the geometric mean for the increase in stock over a 5 year period is 7.63%

To determine

The geometric mean of the data.

Answer to Problem 37EC

The geometric mean for the percentage increase in the cost of food for a specific geographical location over the past 3 years is 2.55%

Explanation of Solution

Given Info:

The percentages of increase in cost of food for a specific geographical location over the past 3 years are 1, 3 and 5%.

Calculation:

Substitute $3$ for $n$, $1$ for $X_1$, $3$ for $X_2$, $5.5$ for $X_3$.

Determine product of 1, 3 and 5.5, then find cube root of product.

$\begin{array}{c}\text{Geometric}\text{Mean=}\sqrt[3]{\left(1\times 3\times 5.5\right)}\\ =\sqrt[3]{16.5}\\ =2.55\end{array}$

Thus, the geometric mean for the percentage increase in the cost of food for a specific geographical location over the past 3 years is 2.55%.