The Arithmetic Mean

Historical Background

Prior to the 17th century astronomers picked the value from their observations that seemed to suit their needs best [1963]. It was not until the solar-centric world view gained acceptance during the 16th century that astronomers adopted the principle of the arithmetic mean to get closer to the true position of a star [1890]. Earlier in the first half of the 16th century mariners used the arithmetic mean implicitly to determine the longitude at sea as the difference between the geographic and the magnetic north pole [1963]. Quêtelet [1846] transferred the concept from astronomy to other fields. Until then the arithmetic mean had been employed to eliminate errors in observations of a constant cause. Quêtelet distinguished between this typical mean and the untypical arithmetic mean. While the former would be that of multiple measurements of the height of the same house, the latter would be that of the heights of different houses [1913]. As the causes are constant for the same house, the differences to the true value follow the law of error [1809]. As they distribute symmetrically, the differences can be canceled out against each other, producing the arithmetic mean as the value most likely to be the closest to the true value. Following the law of large numbers [1713], the more often a measurement is repeated, the less the difference of the arithmetic mean to the true value will be. If one, however, considers the heights of different houses, the causes are no longer constant. Their differences do not distribute symmetrically any more. The arithmetic mean cannot represent a height of a house. It is rather an abstract value. Meyer [1891] wonders why the arithmetic mean is used in meteorology as a convention while the values regularly do not distribute symmetrically. He concludes the arithmetic mean cannot be considered in such cases as the most likely value. However, as everyone uses it without question, he assumes people to misunderstand the arithmetic mean as the most likely value in cases where it is not.

The Role of the School

School teaches us the arithmetic mean as the most common representative value for data [2007]. Its learning framework tacitly assumes a symmetric distribution as standard. It does not tell that many things in reality do not follow the symmetric distribution. This framework allows only for the conclusion, that the arithmetic mean is something like the most likely value.

It is the benevolent warning school adds not to trust the arithmetic mean in all situations that seems to encourage us that we can rely on it in most others. A vague feeling persists that the arithmetic mean is in most cases the most likely value – which it is not.

Representative for All Values

Each time we encounter many values, we desire to reduce them as our short-term memory is limited (TOC). We want to get an idea of the values by means of a representative value.

The arithmetic mean is the very value that all values had if they were equal. We only need to remember one value and the number of the values to imagine all. It is only the arithmetic mean that can be determined without having seen a single value itself. Usually we need the sum anyways for accounting purposes. We have only the arithmetic mean at our disposal if we are given the sum and the number of the values only (be it for space, privacy concerns or just by convention). We can neither determine the other measures of central tendency nor whether the values distribute symmetrically. Hence, one either miss-assumes the arithmetic mean as the most likely value or risks of not having any idea of the values at all. If man’s nature to diminish uncertainty encounters an option for more certainty in a rather ill-defined framework of reference not telling us sufficiently this certainty could be worse than the uncertainty, one is likely to fall prey to the misuse of the arithmetic mean.

The arithmetic mean will be for multiple measurements of constant causes the most likely value. If the causes are not constant, it can but usually will not be the most likely value.

630 exoplanets form a group of the same type. Most of their masses are similar. As the causes and the effects are similar, we tempt to attribute the force provoking the mass of an exoplanet to the cause, that is of being an exoplanet. Why don’t we see that 20 percent of exoplanets make up for 80 percent of the total mass? More than half of them do not belong to the group constituting for 80 percent of the total volume. One literally does not see for more than half of them any difference in volume compared to the 80 percent constituting only 20 percent of the total mass. As our view, one force provokes the mass for exoplanets, is based on the majority of values, we have absolutely no reason to expect a force for the outliers in their own regard, let alone be it superior.

We calculate the arithmetic mean, possibly under the assumption of having constant causes.

One centre of gravity of the values derives its force from the frequency of values while the other from their mass. As both forces relate inversely to each other, one force distorts the arithmetic mean to be a representative value for the other – and vice versa. The standard deviation of 2.7 times the arithmetic mean tells us not to trust it as the most likely value. We could calculate the most likely value directly or use the median as a measure of central tendency. However, both would fail to appreciate the second centre of gravity as it derives its force from mass rather than frequency. It seems paradoxical: the nature of any measure of central tendency, to divide the values into two groups, makes it impossible to recognise their two centres of gravity.

How, then can we find out the very centres of gravity, in other words that a minority of very similar causes provokes a majority of effects?

With effectus.