Math 101 '" Percent Change

The day before Thanksgiving (2010), my wife and I journeyed from Naples, FL to Minneapolis, MN. We were off to see my son and his family for the holiday. Yes, we questioned the wisdom of leaving temperatures in the mid 80's as we headed into an arrival temperature around 10:00 pm CST on Thanksgiving Eve of about 33 degrees. But, after all, we were going to see family!

Then on Thanksgiving morning, it was a nippy 11 degrees F in the Twin Cities (Minneapolis-St. Paul). Add in the wind and it felt somewhere between minus 10 and minus 5 degrees. Let's use minus 5 for purposes of what follows.

Doing the math for percentage change

[(Final value - original value)/original value]*100%

I obtained

[(- -5 - 83)/83]*100% = [-88/83]*100% = -106%

In other words the temperature change was minus 106%.

Here, the computation involved the ratio of change in apparent temperature divided by the starting apparent temperature, with the result then multiplied by 100%. But, the formula is used in many science and business applications. Just watch any of the business channels on TV or any stock market pages online and you'll see lots of percentage change information!

But, even though the formula used here was the correct one, the logic used was flawed (on purpose).

I'm sorry, but any way you look at it, it's pretty hard to state that minus 5 degrees is 106 percent less 83 degrees. How can a minus number be a percent less than a positive number? Or how can a positive number be a percent larger than a negative number? Think about this for a few minutes.

For the "what's it worth department," I have seen such incorrect comparisons made on The Weather Channel (paraphrasing, "It was 20 yesterday and its 40 degrees today, why the temperature has doubled...") and even in at least one 6^th grade math textbook (groan).

The reason the math doesn't work correctly is that temperature scale is "relative." Hence, when crossing zero, we get into trouble.

For example, suppose the temperature had been plus two degrees and the mercury dropped to minus 2 degrees. Performing the same mathematical process, we get minus 4-degree change, divided by 2 degrees and the temperature drop is 200%.

How can a change of 4 degrees from 2 degrees be so much more pronounced than an 88-degree change from 83 degrees?

Now imagine that the starting temperature was zero. Oops, the math becomes impossible. One can't divide by zero.

Finally, start at minus two degrees and let the temperature drop by 4 degrees. Minus 6 minus a minus 2 yields a temperature change of minus 4, but that has to be divided by -2, the starting value. This tells us that we have a positive temperature change (minus number divide by minus number yields a positive number).

It doesn't work for Fahrenheit temperatures and it doesn't work for Celsius either. To compute any type of temperature comparison, like this, one needs an ABSOLUTE temperature scale. That's where the Kelvin or Absolute scale comes into play (Fig.1). On that scale, everything is keyed to Absolute zero (-459.7 degrees F or -273.15 degrees C).

Then applying conversion scales (I like to do these manually rather than using online conversion programs), I have my key numbers in a framework by which I can make meaningful and correct temperature change comparisons.

So, to perform the initial comparison, one must first convert degrees Fahrenheit to degrees Celsius and then add 273.15 degrees to the answer.

An 83 degrees F yields 28.33 degrees C, which, in turn, yields 301.48 degrees Absolute.

A -5 degrees F yields -20.56 degrees C, which, in turn, yields 252.59 degrees Absolute.

Now the percent change becomes [(252.59 - 301.48)/301.48]*100%

= [-48.89/301.49]*100% = -16.22% change

Now, let's try the plus 2 degrees value dropping to minus 2 degrees.

Plus 2 degrees F = 256.48 degrees Absolute

Minus 2 degrees F = 254.26 degrees Absolute

Percentage change becomes [(254.26 - 256.48)/256.48]*100%

[-2.22/256.48]*100%

This yields a -0.87 % change in temperature

Finally, the -2 degree to -6 degree change

Minus 2 degrees F = 254.26 degrees Absolute

Minus 6 degrees F = 252.04 degrees Absolute

Percentage change becomes [(252.04-254.26)/254.26]*100%

[-2.22/254.26]*100%

This also yields a -0.87% change in temperature.

So, the next time you have to make percentage change comparisons and the scale you are using crosses zero, think long and hard about what you are trying to do. Logic, as well as math, needs to be considered.

© 2010, H. Michael Mogil