Monday, October 27, 2014

Confession : The math I should have learned in school

Let’s face it; a lot of us in the IH profession are at some level intimidated by math.  I confess to the syndrome and I believe that  this fear is one of the reasons that modeling is still not generally done or even attempted by everyone.   Poor teachers, poor classes, and poor motivation as a teenager are all possible reasons.   I have experienced all of them, but significantly later in my professional life, well after the age of 30, I recognized the need.   I saw that math was the real basis of physical science and a tool I had to have if I was going to understand and work as a technologist.  At that point, I decided to back-fill my education and was fortunate to find some very good teachers in night school.

I have pushed modeling in the IH profession for many years and have seen a steady increase in its use as dedicated professional colleagues in the AIHA have picked up the banner and have written books and articles and given excellent courses.    I continue to believe, however, that there is only a relatively small percentage of IH professionals who might benefit from this tool are actually engaged.  My new hypothesis is that math intimidation is the reason.

The overarching purpose of this blog is as an “ An educational blog designed to introduce and facilitate industrial hygienists' involvement in quantitative risk assessment - especially exposure assessment and the specific area of exposure modeling.”   If foggy math concepts are keeping folks from engaging then I want to try and help.   I would like to help and back-fill your education relative to some basic mathematical concepts that will provide you with very useful tools.

At the end of this blog I am going to cut and paste some stuff from last week’s blog in which I tried to explain two-dimensional acute inhalation toxicity (both concentration and time).    To provide a good explanation of the subject I needed to explain PROBITS.   If your understanding of probits is a bit foggy, I think this explanation below could help.   If it doesn't explain it clearly please let me know where I lost you.   You can reply anonymously and, believe me, if you have the issue others will as well.  Indeed, please send me math topics that remain foggy to you.  Send me math topics that you just quit thinking about because you thought they were too hard.

Other areas I could try and provide some simple explanations as to how they work and why they are useful include:

  • Logarithms (done in a previous blog but it could be repeated and improved)
  • Calculus:  Area under the curve
  • Statistics:   correlation, curve fitting, standard deviation, linear regression
  • Mathematical definitions of inhalable, thoracic and respirable airborne concentrations and mass

PROBITS (and their use in acute toxicity – lethal dose-response example):

You may or may not remember what a “probit” is but it is a very useful mathematical construct.  It is directly related to the standard Gaussian bell-shaped curve with the area-under-the-curve (AUC) describing the portion of a population included on any part of that curve.  I know what some of you are thinking:  YIKES!  What is this guy talking about?  All I ask is that you Please stay with me on this for a few more paragraphs!  Like a professor of mine once said, “If it’s foggy you’re learning something!”   A few minutes of concentration can pay off in a lifetime of understanding. 

Here is an illustration of a Gaussian or bell-shaped curved:

The peak of the bell-shaped Gaussian curve is right at 50%.   That is, half the folks are in the area-under- the-curve (AUC) below (to the left of) the peak and half are in the AUC above (to the right) of the peak.   This is the average or mean value.   It is also Probit = 5.   Now I am going to ask you to remember a statistical construct you learned called the standard deviation .   The AUC from one standard deviation (or sigma on the above illustration) above the mean on the Gaussian curve is approximately 84%; that is, 84% of the population is in the AUC to the left of one standard deviation above the mean.   One standard deviation above the mean is also Probit = 6.   So 1 sigma above the mean  = probit = 6.   Because it symmetrical, Probit = 4 is one standard deviation below the mean and only 16% of the population are in the AUC to the left of this value.  

Still foggy?  Let’s put this in terms we all might understand:  SAT scores.    The average or mean SAT test score is a Probit multiplied by 100; that is, Probit 5 x 100 = 500.  Half the folks taking the test got a higher SAT score than 500 and half lower.   If you got an SAT of 600 you did better than 84% of the folks taking the test.    If you got a 733 on the SAT you were better than 99% of the folks who took that test.   The computer stops at 800 because you are getting so close to 100% that it does not matter any more.  How did I figure that a 733 SAT score beat 99% of the folks tested, it is a simple function in Excel.  Just put in (=NORMSINV(0.99)  + 5) and you will get 7.33, multiply by 100 to get the SAT score.

We all probably remember living and dying with “the curve” in college.  The raw tests score were converted to probits and, depending on the teacher, the grades assigned such that there was something like 10% “A”s, 25% “B”s, 55% “C”s and perhaps 10% “D”s or lower.   This is how you could get 40 out of 100 correct on a physics test and still get a “B”!    This is also why we all hated the person or persons who “killed the curve” by scoring very high and dragging the mean upward and everyone else's grade downward.

So let’s shift our thinking over to toxic or lethal dose-response which also follows this curve.   Probit = 7.33 (5 + 2.33) means that 99% of the population will respond with the toxic effect being measured, in this case death.   Since it is symmetrical, Probit = 2.67 (or 5 – 2.33) means that 1% of the population will be predicted to die and 99% predicted to live.    You can never get there but you can get as close to 0% deaths as you like with smaller and smaller Probit values.

Please let me know what you want to see and whether I am just talking to myself. 

Do you think I have it right about math intimidation in the IH ranks or do you think that I am all wet on this?


  1. I know a lot of IHs in my cohort (the late 1970s) came from biological rather than physical sciences and so had less of a math background. But I don't think math intimidation is an issue in the profession so much as how we were taught in the first place.

    My first physics courses were non-calculus and I got along fine. As a College Freshman, my physics course did use calculus, and I understood the physics both ways. But my training in IH, while it included a statistics course, did not address modeling.

  2. Remember a famous citation from Albertt Einstein :

    “Do not worry about your difficulties in Mathematics.
    I can assure you mine are still greater.”

    Daniel D.

  3. Daniel, I love the quote! He had some great ones. Mike

  4. Your probit explanation is quite clear. Thanks, Mike.

  5. The math intimidation factor is true for me. Always wished my physics and chemisry were stronger too as long as we're on the subject. Though I finisned Calculus, stopped short of differential equations, i lost focus when I got married and only pulled "Cs". Consequently, along with limited use, I have gaps. I'm always on the lookout for tools/helps to strengthen my foundations. And yes, there are aspects of statistics and modeling I never learned or applied. Many of my classmates never moved beyond College Algebra and elementary stats. While you can accomplish a great deal with that, it also leaves shortcomings. So, thanks for your concern and efforts.

  6. Dear Anonymous,

    Many thanks for your comments. I am proof it is never too late. If you provide some specifics perhaps I can help. If not, are you familiar with the Khan Academy? - a free online school that has tons of wonderfully clear lessons in math from simple algebra through the most difficult calculus and vector analysis.

  7. Sometimes I do forget how I struggled with statistics at the beginning of while I was pursuing my MS degree in Statistics. Now I am an IH student, I did find statistics are much more than "calculate TWA and make a decision in compliance or not", although even just that most of my classmates are still having hard time dealing with statistics. However, I found that once you pass the beginning difficult phase, statistics is very powerful and useful. Like your previous article about"why we still need modeling when you can measure it". Big fun of your work, BTW:)