The Law of Continuous Improvement

Have you seen the following figure circulating around social media?

I will admit that I am not mathematically gifted.  Luckily, our children inherited their mother's mathematical abilities and not mine!  Regardless, even I can appreciate the poetic beauty here.  The figure is a great illustration (in mathematical terms) of the so-called Law of Continuous Improvement.  In this case, small, incremental improvement over time results in significant improvement of a process, as opposed to the "breakthrough" improvement that occurs all at once.  

W. Edwards Deming, an American industrial engineer and statistician who is one of the founding fathers of the modern quality improvement movement, preferred the term "continual improvement" as it was broader and more general in scope.  "Continual improvement" includes both "discontinuous improvement" and "continuous improvement."  The philosophy here is that while continuous improvement is preferable, any improvement is good.  I prefer to use the term "continuous improvement" as it aligns with other concepts, such as kaizen (the Japanese word meaning literally "change for the better" but generally referring to "continuous improvement"), operational excellence, and the High Reliability Organization principle of "Sensitivity to Operations".  

Continuous improvement is a scientific management philosophy embedded in quality improvement methodologies, such as Lean, Six Sigma, Zero Defects, Total Quality Leadership (TQL) / Total Quality Management (TQM), and the Model for Improvement.  Operational excellence has developed from concepts originally described by quality control engineers, management gurus, and scientists such as Walter Shewhart, Joseph Juran, Frederick Winslow Taylor, Taiichi Ohno, and the aforementioned W. Edwards Deming. 

As I stated above, I really like the mathematical illustration of "continuous improvement" above.  However, as I thought about it in greater detail, I realized that the mathematics behind this explanation is not accurate.  The mathematical equation above suggests that a little extra effort every day produces a much larger gain over the course of a year than the same level of effort applied every day.  I am reminded of the concept of simple interest here.  If we are improving in small, daily (i.e. "continuous") increments, the equation would look more like the equation for compound interest.  In other words, when we build upon our improvement every single day (even by just a tiny bit), we improve our performance by a much greater degree over the course of a year. 

We are essentially trying to calculate the sum of what is known in mathematics as an arithmetic sequence.  The sequence increases by 0.01 every day for 365 consecutive days:

1, 1.01, 1.02, 1.03, ...,

So, the sum of this sequence would be calculated by:

Sum = 1 + 1.01 + 1.02 + 1.03 + ...

In this case, the nth term in the sequence is 365 (if we improve every day for a year).  According to my mathematician wife, the formula for the sum of an arithmetic sequence (which is called an arithmetic series) is:

  

So, in our case, where a1=1, n=365, and d=0.01, the sum is then:

Sn = (365/2) x [(2 x 1.0) + (364)(0.01)] = 1,029.3

Wow! Performance significantly (maybe it's appropriate to say astronomically) improves over the course of a year, just by incrementally improving every day for the entire year!  Talk about the power of continuous improvement!  Incidentally, I butchered the calculation and came up with an even larger number during an initial draft.  Thanks to our math whiz kids who corrected my mistake!

The American author Mark Twain said it best, "Continuous improvement is better than delayed perfection."  Or, if you prefer the American statesman, Ben Franklin, "Without continual growth and progress, such words as improvement, achievement, and success have no meaning."  When you look at the concept of continuous improvement from a mathematical standpoint, both Twain and Franklin are correct!