Hitting a (mathematically) moving target

Many numbers are like a hummingbird's wings - they are constantly moving. The speed of the wind is a good example (the Blowin' in the Wind blog). Others are fuel economy of your car at any single moment, or your weight on a given day.

We plead, "Honest, Doctor, I normally weigh less."

We can measure things, but we don't know if a single sample reflects the normal, or the unusual.

Moving numbers prompt a math-oriented person to use averaging. This is taught in 4th grade Excel Math.

Average is one of many math terms used to describe choosing a single number out of a set of numbers:

• arithmetic mean (average) is the result you get when you add the sample numbers in a set, then divide that sum by the number of samples

• median is the center sample number, when a set is put in order from least to greatest; if there is no single center number, then the two numbers on either side of the center are added and the sum divided by 2 

• mode is the sample value that appears most frequently in a set of numbers; there is no mode if no value repeats

We choose different techniques for differing circumstances. For example, we might want to minimize the effect of outliers (numbers at the outer ranges of a set).

Set One: (15, 16, 18, 12, 11, 4, 10, 27, 15, 14, 23)

Here is an unsorted set of 11 numbers. Outliers are in red. 
The mean is 15 (165 ÷ 11) we add the numbers and divide the sum by 11
The median is 15 (4, 10, 11, 12, 14, 15, 15, 16, 18, 23, 27) we have sorted to find the median (middle)
The mode is 15 we notice that this value appears twice out of 11 samples

Set Two: (15, 16, 18, 12, 11, 2, 10, 44, 15, 14, 23)

Now I have changed 2 numbers - making the outliers further away from the center value.
The mean now is 16.36 (180 ÷ 11) we add and divide
The median is still 15 (2, 10, 11, 12, 14, 15, 15, 16, 18, 23, 44) we sort and find the middle
The mode is still 15 we notice it appears twice out of 11 samples

Set Three: (15, 16, 18, 12, 11, 10, 10, 10, 15, 14, 23)

I changed the previous outliers. Both became 10. Now the 10s and the 23 are the outliers.
The mean now is 14 (154 ÷ 11) we add and divide
The median is now 14 (10, 10, 10, 11, 12, 14, 15, 15, 16, 18, 23) we sort and find the middle
The mode is now 10 the 10 appears three times, so it displaces the two 15s

Which of the three averaging methods resulted in a value that changed the least? 

In these examples, it was the median.

The questions we consider when averaging are:

• how often do we take samples; how many do we need?
• how frequently do we average; how fast are we with our math?
• how do we account for outliers; do we include or ignore them?
• which method do we use; what kind of result is most useful?

There are no right answers to these questions.  But the answers DO make a difference.

Can mathematicians complicate these simple concepts in pursuit of more accuracy? YES they can!