# In-class demo: The binomial distribution and the bell curve

Many years ago, the only available in-class technology at my university was the Microsoft Office suite — probably Office 95 or 98. This placed severe restrictions on what I could demonstrate in my statistics class, especially when I wanted to have an interactive demonstration of how the binomial distribution gets closer and closer to the bell curve as the number of trials increases (as long as both $np$ and $n(1-p)$ are also decently large).

The spreadsheet in the link below is what I developed. It shows

• The probability histogram of the binomial distribution for $n \le 150$
• The bell curve with mean $\mu = np$ and standard deviation $\sigma = \sqrt{np(1-p)}$
• Also, the minimum and maximum values on the $x-$axis can be adjusted. For example, if $n = 100$ and $p = 0.01$, it doesn’t make much sense to show the full histogram; it suffices to have a maximum value around 5 or so.

In class, I take about 3-5 minutes to demonstrate the following ideas with the spreadsheet:

• If $n$ is large and both $np$ and $n(1-p)$ are greater than 10, then the normal curve provides a decent approximation to the binomial distribution.
• The probability distribution provides exact answers to probability questions, while the normal curve provides approximate answers.
• If $n$ is small, then the normal approximation isn’t very good.
• If $n$ is large but $p$ is small, then the normal approximation isn’t very good. I’ll say in words that there is a decent approximation under this limit, namely the Poisson distribution, but (for a class in statistics) I won’t say much more than that.

Doubtlessly, there are equally good pedagogical tools for this purpose. However, at the time I was limited to Microsoft products, and it took me untold hours to figure out how to get Excel to draw the probability histogram. So I continue to use this spreadsheet in my classes to demonstrate to students this application of the Central Limit Theorem.