2022-07-07

What Implications Can be Drawn from a Binomial Distribution?
Hello everyone I understand how to calculate a binomial distribution or how to identify when it has occurred in a data set. My question is what does it imply when this type of distribution occurs?
Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?

Miguidi4y

Expert

Lets say for example you are a student in a physics class and the professor states that the distribution of grades on the first exam throughout all sections was a binomial distribution. With typical class averages of around 40 to 50 percent. How would you interpret that statement?
Most likely the professor was talking loosely and his statement means that the histogram of percentage scores resembled the bell-shaped curve of a normal density function with average or mean value of 40% to 50%. Let us assume for convenience that the professor said the average was exactly 50%. The standard deviation of scores would have to be at most 16% or so to ensure that only a truly exceptional over-achiever would have scored more than 100%.
As an aside, in the US, raw scores on the GRE and SAT are processed through a (possibly nonlinear) transformation so that the histogram of reported scores is roughly bell-shaped with mean 500 and standard deviation 100. The highest reported score is 800, and the smallest 200. As the saying goes, you get 200 for filling in your name on the answer sheet. At the high end, on the Quantitative GRE, a score of 800 ranks only at the 97-th percentile.
What if the professor had said that there were no scores that were a fraction of a percentage point, and that the histogram of percentage scores matched a binomial distribution with mean 50 exactly? Well, the possible percentage scores are $0\mathrm{%}$, $1\mathrm{%},\dots ,100\mathrm{%}$ and so the binomial distribution in question has parameters $\left(100,\frac{1}{2}\right)$ with $P\left\{X=k\right\}=\left(\genfrac{}{}{0}{}{100}{k}\right)/{2}^{100}$.
So, if N denotes the number of students in the course, then for each k,$k,0\le k\le 100$, $N\cdot P\left\{X=k\right\}$ students had a percentage score of k%. Since $N\cdot P\left\{X=k\right\}$ must be an integer, and $P\left\{X=0\right\}=1/{2}^{100}$, we conclude that N is an integer multiple of ${2}^{100}$. I am aware that physics classes are often large these days, but having ${2}^{100}$ in one class, even if it is subdivided into sections, seems beyond the bounds of plausibility! So I would say that your professor had his tongue firmly embedded in his cheek when he made the statement.

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