The mean defined in the text is sometimes called the arithmetic mean to distinguish it from other possible means. For example, a different mean, called the geometric mean (G.M.), is used in business and economics for finding average rates of change, average rates of growth, or average ratios. This mean is defined to be the nth root of the product of the numbers. For example, the geometric mean of 4, 5, and 6 is
G.M. = (4 · 5 · 6)1/3
=
3
120
≈ 4.9.
(a) Find the arithmetic and geometric means for the numbers 2, 4, 5, 7, 7, 9, and 10. (Round your answers to one decimal place.)
arithmetic mean
geometric mean
(b) The growth rates for three cities are 1.7%, 2.2%, and 0.9%. Compare the arithmetic and geometric means. (Round your answers to two decimal places.)
arithmetic mean
%
geometric mean
%
1. AM = (2+ 4+ 5+ 7+ 7+ 9+ 10)/7 = .....
GM = (2*4*5*7*9*10)^(1/7) =appr 4.3
2. follow the same steps
(a) To find the arithmetic mean, we need to add up all the numbers and then divide by the total count of numbers.
Sum of the numbers: 2 + 4 + 5 + 7 + 7 + 9 + 10 = 44
Count of numbers: 7
Arithmetic mean = Sum of numbers / Count of numbers
Arithmetic mean = 44 / 7 = 6.29 (rounded to one decimal place)
To find the geometric mean, we need to multiply all the numbers together and then take the nth root of the product, where n is the count of numbers.
Product of the numbers: 2 x 4 x 5 x 7 x 7 x 9 x 10 = 35,280
Count of numbers: 7
Geometric mean = nth root of the product of numbers
Geometric mean = 7√(35,280) ≈ 6.88 (rounded to one decimal place)
Therefore, the arithmetic mean of the numbers 2, 4, 5, 7, 7, 9, and 10 is 6.3 and the geometric mean is 6.9 (rounded to one decimal place).
(b) To compare the arithmetic and geometric means of the growth rates, we can calculate the average.
Growth rates: 1.7%, 2.2%, and 0.9%
Arithmetic mean = Sum of growth rates / Count of growth rates
Arithmetic mean = (1.7 + 2.2 + 0.9) / 3 ≈ 1.6% (rounded to two decimal places)
Geometric mean = nth root of the product of growth rates
Geometric mean = 3√(1.7 x 2.2 x 0.9) ≈ 1.4% (rounded to two decimal places)
Therefore, the arithmetic mean of the growth rates is 1.6% and the geometric mean is 1.4% (both rounded to two decimal places).