Question

Harmonic Mean The harmonic mean is often used as a measure of center for data sets consisting of rates of change, such as speeds. It is found by dividing the number of values n by the sum of the reciprocals of all values, expressed as

$\frac{\mathbf{n}}{\sum \frac{\mathbf{1}}{\mathbf{x}}}$

(No value can be zero.) The author drove 1163 miles to a conference in Orlando, Florida. For the trip to the conference, the author stopped overnight, and the mean speed from start to finish was 38 mi/h. For the return trip, the author stopped only for food and fuel, and the mean speed from start to finish was 56 mi/h. Find the harmonic mean of 38 mi/h and 56 mi/h to find the true “average” speed for the round trip.

Short answer

The harmonic mean is 45.3 mi/h.

Step-by-step solution

Given information

The distance driven by the author is 1163 miles.

The speed measures are 38 mi/h and 56 mi/h.

Harmonic mean formula

If there are n counts of observations, the formula for harmonic mean is $\overline{x}=\frac{n}{\sum \frac{1}{x}}$forx observations.

Substitute the values to obtainthe harmonic mean for the speed of the round trip.

$$\begin{gathered} \overline{x}=\frac{2}{\frac{1}{38}+\frac{1}{56}} \\ =\frac{2}{0.0442} \\ =45.2766 \end{gathered}$$

Thus, the actual average value of the speed for the round trip is 45.3 mil/h.