The shape of a binomial distribution

What you will be able to do

• describe the shape of $B(n, p)$ for small, middle and large $p$, and as $n$ grows;
• give the usual range of a count, from about two standard deviations below the mean to two above;
• run a simulation and compare it with the formula.

Three shapes

Here are three binomial distributions with $n = 10$.

• With $p = 0.1$, most of the probability is at $0$, $1$ and $2$. A thin tail stretches to the right. We say the shape is skewed to the right.
• With $p = 0.5$, the bars are symmetric about the mean, $5$.
• With $p = 0.9$, the picture is the mirror image of $p = 0.1$. It is skewed to the left.

Why? A count cannot go below $0$ or above $n$. When the mean is near $0$, the bars pile up against that wall and spread out the other way.

As $n$ grows

Keep $p = 0.1$ and raise $n$ to $100$. The mean is $10$ and the standard deviation is $3$, so the counts sit well away from $0$. The wall stops mattering, and the bars form a smooth, nearly symmetric bell. The last course of the chapter is about that bell, the normal curve.

The usual range

For most binomial counts, almost all the probability lies within two standard deviations of the mean. The usual range runs from $np - 2\sigma$ to $np + 2\sigma$. A count outside it is unusual. It is not impossible, but it is worth a second look.

You should find that the band is widest at $p = 0.5$ and narrows towards $p = 0$ and $p = 1$. Raise $n$, and the band grows, but more slowly than the mean. Four times as many trials make the band only twice as wide.

Counts and proportions

Survey results are often given as a proportion, the count divided by $n$. A proportion has mean $p$ and standard deviation $\sqrt{p(1 - p)/n}$. This one shrinks as $n$ grows. A larger survey pins down $p$ more closely. The last chapter of this book builds on this idea.

A simulation that matches

In a made-up school, each student cycles to school with probability $0.3$. Take a class of $10$. The program simulates $10\,000$ such classes, counts the cyclists in each, and compares the share of each count with the formula.

Your turn

The usual mistakes

• Calling every binomial symmetric. Only $p = 0.5$ gives a symmetric shape for a small $n$.
• Mixing up the direction of skew. The skew is named after the side of the long tail.
• Expecting a simulation to match exactly. It matches closely, and more closely for more repeats.
• Using the count's standard deviation for a proportion. Divide by $n$: $\sqrt{p(1 - p)/n}$.

Remember these

• Small $p$: tail to the right. $p = 0.5$: symmetric. Large $p$: tail to the left.
• As $n$ grows, the shape becomes a bell.
• Usual range: $np \pm 2\sqrt{np(1 - p)}$.
• A proportion has mean $p$ and standard deviation $\sqrt{p(1 - p)/n}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Describe the shape of $B(20, 0.05)$.

2. For $n = 12$, which $p$ gives a symmetric distribution?

3. For a fixed $n$, which $p$ gives the largest standard deviation?

4. A survey of $100$ people has $p = 0.5$. Find the standard deviation of the proportion who say yes.

5. A simulation of $500$ classes gives shares that differ from the formula by up to $0.03$. Is something wrong?

References

• OpenStax, Introductory Statistics, Chapter 4, section on the binomial distribution, and Chapter 7 on the central limit theorem for sums.