The Numbers Game

Consider the two headlines:

(1) Violent Crime Doubled in 10 Years

(2) Violent Crime Grew 7% Per Year over Decade

Which is more alarming?

My gut reaction is that (1) is more troubling. While (2) does not present great news, it is nothing to get that excited about.

The second speaker at the 2009 EntConnect conference, Dr. Albert Bartlett explained the meaning of growth statistics in his presentation: "Sustainability 101: Arithmetic, Population and Energy." Dr. Bartlett, an emeritus Professor of Physics at the University of Colorado at Boulder, has lectured on this topic over 1,600 times since September, 1969.

His message? Some very simple ideas about the problems we are facing are all tied together by arithmetic, and the arithmetic isn't very difficult.

For any entity growing at a steady rate, the doubling time T(double) = 70/(k%), where (k%) is the growth rate per unit time. (For you mathematical types, the derivation of this formula is done in the footnotes.)

Hence,

Growth rate ------Doubling Time
--per year ------of Initial Quantity
----k%--------T(double) = 70/(k%)--

---1%------------- 70 years----------
---5%-------------14 years-----------
---7%-------------10 years-----------
--10% -------------7 years-----------
--35%------------- 2 years-----------
--70%--------------1 year------------

In headline (2), the growth rate is 7% per year. Therefore its doubling time is 70/7 = 10 years. We see both headlines (1) and (2) reflect the same statistic. Yet, how these statistics are presented impact the emotions of the reader differently.

The following blogs will look at using arithmetic to write with more clarity and more accurately decipher what we are reading.

Quiz time:

If you now have $1000 in your savings account at a 1% interest per year, what year will it be when it doubles to $2000?

(a) 2019
(b) 2034
(c) 2079
(d) Time to get a new bank account with a better interest rate.


Related links:

EntConnect website: http://www.entconnect.org/

Dr. Bartlett links:
albartlett.org
en.wikipedia.org/wiki/Albert_Bartlett
globalpublicmedia.com/transcripts/645

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Footnotes:

For the mathematically curious, to show this formula, T(double) = 70 / k%, was not pulled out of a hat:

N = number or quantity of an entity, such as a population, statistic, resource, etc …
dN = differential change in the number
k = a constant
dt = differential change in time
T(double) = the doubling time
k% = constant percent, that is k x 100

The exponential function of the size of something growing at a steady rate of k:

(1/N) dN/dt = k or dN/N = k dt

Using calculus to find the doubling of quantity N, integrate N from N to 2N, over time period 0 to T(double):

ln(2) = k T(double)
100 ln(2) = k% T(double)
T(double) = 100 ln(2) / k%

Since,
100 ln(2) = 69.3 or approximately 70

Hence,
T(double) = 70 / k%

Q.E.D.