1.16 Other Types of Means: Trimmed, Geometric, and Harmonic
A trimmed mean is simply an arithmetic mean that is calculated after some very high and very low values have been removed from the data. For instance, a mean could be calculated after removing the values below the 5th percentile and above the 95th percentile, or after removing those below the 10th percentile and above the 90th percentile. It is seen as a more accurate representation of data since outliers have been removed. A trimmed mean is used to calculate inflation rates from the Consumer Price Index or personal consumption expenditures.8 Please note that when presenting a trimmed mean, it should always be clearly stated that some parts of the dataset were not included in the calculation.
Next, to examine the concept of a geometric mean, let’s look at investment returns. Suppose that your friend speculated in the shares of a new, high-flying tech company that had recently hit the market with its Initial Public Offering (IPO).
The day after the IPO, he purchased $1000 worth of shares in the company. For the next 12 months, the shares rose by 20%. In the next year, the company’s performance improved even further, as did the share trajectory, as it posted a 50% gain.
What is his average annual return? It would be tempting to say 35%. After all, if we apply the arithmetic mean formula, we use (20 + 50) / 2 to arrive at this very value.
However, let’s take a closer look at his investment. At the end of the first year, his investment was worth $1200, after the 20% gain. During the second year, the 50% gain brought his total investment value up to $1800. That’s 80% more than the original investment.
To calculate a geometric mean, we will need to first find the percentage change across the entire period. Here, that’s a gain of 80%, as the net change across the holding period was $800, and the “base” value of the investment was $1000.
Next, we must bring 1 + that overall percentage change to the 1/n power, with n representing the number of periods. We subtract 1 from the resulting value to find the geometric mean. 1.8^(½) gives us 1.3416, so the average annual performance was 34.16%. That is the annual percentage gain that an investor would need for two consecutive years to arrive at the same ending value.
For an application of a harmonic mean, let’s consider the Lobster Junior roller coaster at Lobster Land. The Lobster Junior is a small, child-friendly roller coaster that provides thrills for younger park guests, in a very safe and controlled way. The entire length of track covered by the Lobster Junior is 4 miles. For two of those miles, the Lobster Junior moves at 4 miles per hour, and for the other two miles, it moves at 12 miles per hour.
What is a child’s average speed while riding on the Lobster Junior? It would be tempting to just average 4 and 12 together, and answer with 8. However, this would be incorrect. Since the slow phase and the fast phase were of equal lengths, the riders would have spent considerably more time on the ride in the slow phase.
We can solve this problem with a harmonic mean, found by taking the number of elements, and dividing by the sum of the reciprocals of the two speeds.
Our numerator here will be two, since we are averaging two separate speeds. In the denominator are the reciprocals of the two speeds: ¼ and 1/12, which will be added together. ¼ and 1/12 sum to 4/12, and 2 divided by 4/12 yields our harmonic mean of 6. The average speed of a rider on the Lobster Junior is 6 miles per hour.
A possibly more intuitive way to arrive at this solution of 6 miles per hour is to break the journey into two subcomponents, using the Distance = Rate * Time formula, or D = RT.
For the first leg of the journey, the distance is 2 miles, and the rate is 4 miles per hour. The time, therefore, must be 0.5 hours, or 30 minutes.
For the second leg of the journey, the distance is also 2 miles, but now, the rate is 12 miles per hour. The time, therefore, is ⅙ of an hour, or 10 minutes.
Summed up, we now know that the total elapsed time is 40 minutes, or ⅔ of an hour, and the total distance traveled is 4 miles. Again using D = RT, we can solve for the overall R. Since 4 = R(⅔), R must equal 6.
8 Kenton, W. (2021, July 17). ‘Trimmed mean definition’. Investopedia. https://www.investopedia.com/terms/t/trimmed_mean.asp