วันพุธที่ 2 กรกฎาคม พ.ศ. 2568

harmonic mean vs arithmetic mean

 The harmonic mean is the appropriate average for ratios and rates because it gives equal weight to each "event" or "unit of work" rather than each individual number or time interval. Here's a deeper dive into why:

1. The Reciprocal (i.e. 1/x) Relationship

The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the data points. This "reciprocal of the average of the reciprocals" structure is key. When you're dealing with rates (like miles per hour, or words per minute), you're essentially looking at a ratio of two quantities (distance/time, words/time).

Consider the classic example of average speed:

 * Scenario 1: Equal Distances

   Imagine you drive from point A to point B at 60 km/h and return from point B to point A (the same distance) at 20 km/h. What's your average speed for the entire trip?

   * If you used the arithmetic mean: (60 + 20) / 2 = 40 km/h. This is incorrect.

   * Let's analyze it with the harmonic mean.

     * Distance (d) is constant.

     * Time for outbound trip = d/60

     * Time for return trip = d/20

     * Total distance = 2d

     * Total time = d/60 + d/20 = d(1/60 + 1/20) = d(4/60) = d/15

     * Average speed = Total Distance / Total Time = 2d / (d/15) = 2 \times 15 = 30 km/h.

   * Notice that the harmonic mean of 60 and 20 is: 2 / (1/60 + 1/20) = 2 / (4/60) = 2 \times 60 / 4 = 120 / 4 = 30 km/h.

   In this scenario, where the distance (the "work" done) is constant for each segment, the harmonic mean gives the correct average speed. The arithmetic mean would be too high because you spend more time traveling at the slower speed. The harmonic mean inherently accounts for the longer time spent at the lower rate.

 * Scenario 2: Equal Times

   If you drive for 1 hour at 60 km/h and then for another 1 hour at 20 km/h, your average speed would be the arithmetic mean: (60 \times 1 + 20 \times 1) / (1 + 1) = (60+20)/2 = 40 km/h. In this case, since the time (the denominator of the rate) is constant for each segment, the arithmetic mean is appropriate.

--gemini