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
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 |
|
|
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.
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.