Additional Measures of Central Tendency

Prerequisites
Percentiles, Distributions, What is Central Tendency, Measures of Central Tendency, Mean and Median

Learning Objectives

  1. Compute the trimean
  2. Compute the geometric mean directly
  3. Compute the geometric mean using logs
  4. Use the geometric to compute annual portfolio returns
  5. Compute a trimmed mean

Although the mean, median, and mode are by far the most commonly used measures of central tendency, they are by no means the only measures. This section defines three additional measures of central tendency: the trimean, the geometric mean, and the trimmed mean. These measures will be discussed again in the section "Comparing Measures of Central Tendency."

Trimean

The trimean is a weighted average of the 25th percentile, the 50 percentile, and the 75th percentile. Letting P25 be the 25th percentile, P50 be the 50th and P75 be the 75th percentile, the formula for the trimean is:

Trimean = (P25 + 2P50 + P75)/4

As you can see from the formula, the median is weighted twice as much as the 25th and 75th percentiles.

Geometric Mean

The geometric mean is computed by multiplying all the numbers together and then taking the nth root of the product. For example, for the numbers 1, 10, and 100, the product of all the numbers is: 1 x 10 x 100 = 1,000. Since there are three numbers, we take the cubed root of the product (1,000) which is equal to 10. The formula for the geometric mean is therefore

where the symbol Π means to multiply. Therefore, the equation says to multiply all the values of X and then raise the result to the 1/Nth power. Raising a value to the 1/Nth power is, of course, the same as taking the Nth root of the value. In this case, 10001/3 is the cube root of 1,000.

The geometric mean is an appropriate measure to use for averaging rates. For example, consider a stock portfolio that began with a value of $1,000
and had annual returns of 13%, 22%, 12%, -5%, and -13%. Instead of using the percents, each return is represented as a multiplier indicating how much higher the value is after the year. This multiplier is 1.13 for a 13% return and 0.95 for a 5% loss. The geometric mean of these multipliers is 1.05 which is the annual rate of return. The portfolio described here and one that increased 5% a year for five years would both end up with a value of $1,276.

Trimmed Mean

To compute a trimmed mean, you remove some of the higher and lower scores and compute the mean of the remaining scores. A mean trimmed 10% is a mean computed with 10% of the scores trimmed off; 5% from the bottom and 5% from the top. A mean trimmed 50% is computed by trimming the upper 25% of the scores and the lower 25% of the scores and computing the mean of the remaining scores. The trimmed mean is similar to the median which, in essence, trims the upper 49+% and the lower 49+% of the scores.