Statistical Glossary
Mean Values (Comparison):
The numerical example below illustrates basic properties of various descriptive statistics with “mean” in their name, like the arithmetic mean , the trimmed mean , the geometric mean , the harmonic mean , and several power mean . Because the last three statistics are defined only for non-negative values, all examples comprise only positive values.
| #
|
Sample
|
Type of the mean
|
|||||
| arithmetic
|
 -trimmed
|
geometric
|
harmonic
|
power
|
|||
| p=2
|
p=8
|
||||||
| 1
|
1, 1, 1, 1, 1
|
1.00
|
1.00
|
1.00
|
1.00
|
1.00
|
1.00
|
| 2
|
1, 1, 1, 1, 100
|
20.80
|
1.00
|
2.51
|
1.25
|
44.7
|
81.8
|
| 3
|
1, 1, 1, 1, 0.01
|
0.80
|
1.00
|
0.40
|
0.05
|
0.89
|
0.97
|
| 4
|
1, 10, 100, 1000, 10000
|
2222.2
|
370.0
|
100.0
|
4.50
|
4494.6
|
8177.7
|
| 5
|
1, 2, 3, 4, 5
|
3.00
|
3.00
|
2.61
|
2.19
|
3.32
|
4.18
|
| 6
|
8, 9, 10, 11, 12
|
10.00
|
10.00
|
9.90
|
9.80
|
10.10
|
10.61
|
Row 1 shows that for a set of equal values all mean values coincide with this value.
Row 2 shows the effect of a single extremely large observation (100) in a sample, e.g. because of a single anomalous measurement. The arithmetic mean is changed from 1.0 to 20, as compared to row 1; the trimmed mean is not affected at all (this illustrates the robustness of the trimmed mean). The harmonic mean has changed only from 1.0 to 1.25. The power mean values are affected most of all, and the power mean with 
has changed to a greater extent (from 1 to 81.8 – i.e. almost to the anomalous value 100 itself) than the power mean with 
(from 1 to 44.7).
Row 3 illustrates the effect of a single abnormally small value (0.01) in the sample. The harmonic mean is extremely sensitive to such a small element – it has changed from 1.0 to 0.05.
Row 4 includes 5 consecutive powers of 10 – 
. It illustrates the behavior of the geometric mean, that takes on the value with the average order of magnitude 
. In other words, the geometric mean involves averaging the powers, not the values themselves.
Row 5 comprises symmetrically distributed values, with 
as the center of symmetry. The arithmetic mean and the trimmed mean are equal to the central value, but other types of mean deviate from the central value.
Row 6 comprises values from row 5 shifted by 7.0 units upwards. This illustrates shift invariance of the arithmetic mean and the trimmed mean (which has changed their values by 7.0 units as compared to row 5). It also demonstrates the lack of shift invariance of other types of means (that changed by a value differing from 7.0).