# 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?

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Dear mysticxhobo, I edited your question a bit according to the standards here. Hope you don't mind. –  Rasmus Oct 8 '11 at 21:24
no i do not mind –  user17321 Oct 8 '11 at 21:30
Suppose the Physics test had $30$ questions, and each student guessed on every question, with the same probability of success $p$. Then individual total scores would come from a binomial distribution. But I very much doubt that is what the instructor meant. –  André Nicolas Oct 8 '11 at 23:01

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 $0\%$, $1\%, \ldots, 100\%$ and so the binomial distribution in question has parameters $(100, \frac{1}{2})$ with $P\{X = k\} = \binom{100}{k}/2^{100}$. So, if $N$ denotes the number of students in the course, then for each $k, 0 \leq k \leq 100$, $N\cdot P\{X = k\}$ students had a percentage score of $k\%$. Since $N\cdot P\{X = k\}$ must be an integer, and $P\{X = 0\} = 1/2^{100}$, 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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The binomial distribution $B(n,p)$ is the probability distribution of the number of successes in $n$ independent Bernoulli trials with probability $p$ of success on each trial. (A Bernoulli trial is a random experiment that has just two possible outcomes, often called "success" and "failure". Tossing a coin is a Bernoulli trial.)

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okay. i understood. but if you have a binomial distro of test grades. what does that mean? is there any way to relate the two? –  user17321 Oct 8 '11 at 21:37
Dear Michael, I don't think your answer as it stands answers the question satisfactorily. The information you provide is certainly a subset of the information we can find at wikipedia. –  Rasmus Oct 8 '11 at 21:41