Question about arithmetic mean?

Question

When we do an experiment, we get lots of numbers and normally we use arithmetic mean to deal with them. But why don't we use geometric mean or harmonic mean?

Answer 1

The arithmetical mean (sum / n) is most useful for linear regressions (i.e. adding the same amount to x adds the same amount to y). The geometric mean (n'th root of product) is most useful for exponential regressions (i.e. adding the same amount to x multiplies y by the same amount). The harmonic mean (reciprocal of the arithmetic mean of the reciprocals of the measurements -- think "resistors in parallel") is useful for inverse regressions (i.e. where y is 1/something and adding the same amount to x adds the same amount to 1/y).

Since we tend to prefer linear regressions, to the point where we engineer our measurements to produce linear regressions wherever possible, the arithmetic mean is usually the most representative one.

Sometimes, though, other averages are more representative. In the case of a blind range (i.e where there is a minimum but no maximum, or a maximum but no minimum; wages, for instance. Nobody earns less than nothing but some people earn tens of thousands of pounds a day), the median (obtained by lining up all the values in order from lowest to highest and picking the one closest to the middle) is generally most representative because it is less affected by outlying values. And sometimes, the mode (most frequently occurring value) is the most representative.

Answer 2

It depends on just what the data is and on how it is to be treated. Each has valid uses, but if you are looking for the "average" price of a new car, using the geometric or harmonic mean makes no sense to me.

Geometric mean is used in certain types of environmental monitoring and studies, notably in bacteria densities within samples. If you are doing a pH problem in chemistry and get an arithmetic mean of the pH values, you have actually gotten the negative geometric mean of the actual concentrations of ions.

I don't have a handy example for a harmonic mean although I remember there are some. But then I've been out of school for over 40 years.

There are other "averages" that are also used and fairly commonly. The median, or midvalue, of a collection of numbers is the one that has half the total number of values equal to or lower than it and also has half the values equal to or greater than it. That is certainly a valid concept of "average".

The mode is the most common value and, again, there can be a logical acceptance of that as "average".

Answer 3

The other two means are not for such situations. The geometric mean is used for geometry: the geometric mean of two numbers, a and b, is simply the length of the square whose area is equal to that of a rectangle with dimensions a and b. That is, what is n such that n² = a × b. The harmonic mean is also something totally diferent.

Answer 4

u should use arithmetric mean.