Many

*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.*

**numbers**We plead,

*"Honest, Doctor, I normally weigh less."*

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

*, or the*

**normal***.*

**unusual**Moving numbers prompt a math-oriented person to use

*. This is taught in 4th grade Excel Math.*

**averaging**

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

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

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**median**

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

*(numbers at the outer ranges of a set).*

**outliers****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!