The geometric mean (GM) is defined as the nth root of the product of n values. The formula is
GM = n √(X1) (X2) (X3). (Xn)
The geometric mean of 4 and 16 is
GM = √ (4) (16) = √64 = 8
The geometric mean of 1, 3, and 9 is
GM = 3√ (1)(3)(9) = 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 = √ (1.2) (1.1) = 1.1489
GM = √ (120) (110) = 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
$10,000 . 14.89% = $1489.00
$11,489 . 14.89% = 1710.71 / $3199.71 ≈ $3200
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%. 25.5%
b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. 5.7%
c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. 8.4%
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%. 3.2%