In this activity we explore the three different means of the ancient Greeks.
The arithmetic mean
The arithmetic mean is the good-old mean that we are all familiar with.
What is the mean that we are all familiar with? Explain how to compute the mean of \(a_1,a_2,\dots ,a_n\). Give some examples.
The geometric mean
The geometric mean is a bit different. The geometric mean of \(a_1,a_2,\dots , a_n\) is given by:
\[ \left (\prod _{i=1}^n a_i\right )^{1/n} \]
Can you explain the geometric mean in terms of geometry? First do it for 2 numbers. Next do it for three.
The harmonic mean
The harmonic mean might be the most mysterious of all. The harmonic mean of \(a_1,a_2,\dots , a_n\) is given by:
\[ \frac {n}{\sum _{i=1}^n \frac {1}{a_i}} \]
In the United States, the fuel efficiency of a car is usually given in the units:
\[ \frac {\text {miles}}{\text {gallon}} \]
However, in Europe, the fuel
efficiency of a car is usually given in the units: \[ \frac {\text {liters}}{100 \mathrm {km}} \]
Give some examples of fuel efficiency (both efficient and inefficient)
with each set of units. Now suppose that a car gets \(60\frac {\text {miles}}{\text {gallon}}\) and another car gets \(20\frac {\text {miles}}{\text {gallon}}\). What is the average fuel efficiency?
Now suppose that a car gets \(4\frac {\text {liters}}{100 \mathrm {km}} \) and another car gets \(20\frac {\text {liters}}{100 \mathrm {km}}\). What is the average fuel efficiency?
2025-01-06 15:53:03