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Additional Measures of Central Tendency

Module by: David Lane. E-mail the author

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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+P754 Trimean P25 2 P50 P75 4

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

Table 1 shows the number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season. The relevant percentiles are shown in Table 2

Table 1: Number of Touchdown Passes
37 33 33 32 29 28 28 23 22 22 22
21 21 21 20 20 19 19 18 18 18 18
16 15 14 14 14 12 12 9 6    
Table 2: Percentiles
Percentile Value
25 15
50 20
75 23

the trimean is therefore 15+2×20+234=784=19.5 15 2 20 23 4 78 4 19.5

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: 1×10×100=1000 1 10 100 1000 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 ∏=X1N X 1 N where the symbol ∏= 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, 100013 10001 3 is the cube root of 1,000.

The geometric mean has a close relationship with logarithms. Table 3 shows the logs (base 10) of these three numbers. The arithmetic mean of the three logs is 1. The anti-log of this arithmetic mean of 1 is the geometric mean. The anti-log of 1 is 101 = 10. Note that the geometric mean only makes sense if all the numbers are positive.

Table 3: Logarithms
X log10X 10 X
1 0
10 1
100 2

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. Therefore the trimmed mean is a hybrid of the mean and the median. To compute the mean trimmed 20% of the touchdown pass data shown in Table 1, you remove the lower 10% of the scores (6, 9, and 12) as well as the upper 10% of the scores (33, 33, and 37) and compute the mean of the remaining 25 scores. This mean is 20.16.

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