Applied Mathematics for Physical Chemistry

Scientific Folly of Averages

by James R. Barrante, Ph.D.

 

There is a misunderstanding among many scientists that using average values to do science is acceptable under any condition.  This is actually not the case.  To understand this, we must look at Error Theory.  Scientists are generally interested in two types of errors:  random error and systematic error in their measurements.  Random error is related to inherent errors in measuring devices, particularly when these measuring devices are being used near their limitations.  This type of error is associated with the precision of the measurement, how close individual measurements of the same parameter are to each other.  Systematic error is related to experimental design.  This type of error is associated with the accuracy of the measurement; that is, how close individual measurements of the same parameter are to the “true” value.   The precision of a measuring device can be increased by making the measurement more than once and taking an average only if the following is true:  the measurement must be made using the same measuring device on exactly the same sample over and again thousands of times.  Thus, the precision of the average or mean value will be greater than the precision of a single measured value.

No statistical analysis will affect the systematic error.  The average will be no more accurate than a single measurement.  What, then, is the advantage of measuring a sample several times and taking an average.  The reason is to insure that no systematic error was made in the measurement, a possibility if the measurement was made a single time.  For example, if you were determining the mass of an object, to insure that you did not misread the balance, you might take three consecutive readings, each one after zeroing the balance.  If they are all within the random error of the balance (generally supplied by the manufacturer), you can be sure that no error was made reading the scale of the balance.  Does this mean that the average of the three is accurate.  Not in the least.  If the balance had not been calibrated, the average could be precise (e.g., 24.55; 24.52; 24.54), but not accurate.  For example, the true weight could be 16.24.  Calibrating the balance is necessary to minimize systematic error.  It’s part of experimental design.

When do averages have no scientific meaning?  The study of climate change is a perfect example.  The idea of an average global temperature as it relates to an average global atmospheric CO2 level has about as much scientific meaning as the average diameter of a football has to its shape.  Knowing that the average diameter of a standard football is 9.00 inches tells you nothing about its shape.  In fact, the average presented as a single number would suggest that the football is spherical.  The average value of some measured parameter of a system (like pressure or temperature) is scientifically significant only if 1) the system is very small; 2) its boundaries are well-defined; and 3) measurements are made with the same measuring device on various points over the system where these measuring points have the same environment.  The globe satisfies none of these requirements.

You can certainly determine the temperature of the globe at various points around the globe and obtain an average that is valid as an average temperature.  But you cannot do anything scientifically meaningful with that number.  For example, suppose we wish to determine how atmospheric CO2 level affects that average.  For this to be scientifically meaningful, it would have to be true that only CO2 level could affect the temperature at every single measuring point over the entire system.  When the Soviet Union fell, a number of temperature measuring stations in Siberia were closed.  The average global temperature suddenly increased.  Climate change theory would have you believe that a sudden increase in CO2 level caused the sudden increase in average global temperature.  Another example, the average atmospheric CO2 level is found to suddenly increase.  It is assumed that this sudden increase in CO2 level was caused by an increase in burning fossil fuels.  In actuality, the opening of a coal-burning power plant occurred at one measuring point, while a volcanic eruption occurred at another measuring point.  Different causes, same effect.

Where cause and effect is meaningful is when the system is small with well-defined boundaries.  A small cylinder holding a gas is heated from 300 K to 400 K.  A pressure gauge on the cylinder measuring average pressure increases from 1.0 bar to 1.3 bar.  The average pressure change is consistent with the average temperature change, because the cylinder is a system that satisfies the three conditions mentioned above.  This is the nature of the natural sciences and it cannot be changed within the realm of the scientific method.  Unfortunately, many climate scientists do not seem to understand this.