When the normal model fits

What you will be able to do

• say when a binomial count is close to normal;
• check whether a data set is close to normal;
• explain why sums and averages of many small random parts tend to be normal.

A binomial count becomes a bell

In the course on the binomial model you saw the bars of $B(n, p)$ form a bell as $n$ grows. Here is $B(100, 0.5)$ with the normal curve of the same mean and standard deviation, $\mu = 50$ and $\sigma = 5$.

The fit is close. The exact chance of at most $55$ successes is $0.8644$. The normal curve gives $\Phi\!\left(\dfrac{55.5 - 50}{5}\right) = \Phi(1.1) = 0.8643$. The $55.5$ is there because each bar is $1$ wide, so the bar for $55$ reaches to $55.5$. This step is called the continuity correction. The Maths course on random variables and the binomial distribution works with it in exam style.

A common rule says the normal curve fits when both $np \ge 10$ and $n(1 - p) \ge 10$. Some books use $5$. For $B(100, 0.5)$ both are $50$. For $B(30, 0.1)$, $np = 3$, and the bars are far too skewed for a bell.

Not everything is normal

A made-up bus stop records how long people wait. Most waits are short and a few are long, so the histogram has a long tail to the right. The mean wait is about $4$ minutes and the standard deviation is about $4$ minutes too.

A normal curve with that mean and standard deviation goes wrong in two ways. It puts about $16\%$ of waits below $0$ minutes, which is impossible. And it places the middle of the data in the wrong spot.

Many real data sets are like this.

• Waiting times and lifetimes cannot go below $0$ and have long right tails.
• Incomes and house prices have a few very large values, so the mean is far above the median.
• Marks in an easy test pile up against the top mark.
• Small counts, such as the number of goals in a match, are whole numbers near $0$.

Three quick checks

1. Draw a histogram. Is it a single, roughly symmetric hump?
2. Compare the mean with the median. For a normal shape they are close.
3. Count the shares within one and two standard deviations. For a normal shape they are near $68\%$ and $95\%$.

You should find that with one customer the curve is far from a bell, and the normal curve puts about $16\%$ of its area below zero. By $12$ customers the two curves almost agree, and the impossible share has fallen to about $0.0003$.

Why so many things are normal

That simulation shows a big idea. A total of many small, independent random parts tends to follow a normal curve, even when each part is skewed. An average does the same, because it is a total divided by a number. This result is called the central limit theorem. It is why heights, which depend on many small causes, are close to normal, while a single waiting time is not. The last chapter of this book uses it.

Try it in Python

The program makes two made-up data sets with a fixed seed: $2000$ heights from a normal curve, and $2000$ waiting times with a long right tail. It runs two of the quick checks on each.

Your turn

The usual mistakes

• Using the normal model without looking at the data. Draw the histogram first.
• Trusting the normal curve for a small binomial count. Check that $np$ and $n(1 - p)$ are both at least $10$.
• Ignoring impossible values. A model that puts real area below zero for times or masses is a poor model.
• Thinking "normal" means usual or correct. It is only the name of one shape.

Remember these

• $B(n, p)$ is close to normal when $np \ge 10$ and $n(1 - p) \ge 10$.
• Skewed data, bounded data and small counts are often not normal.
• Check with a histogram, the mean against the median, and the 68-95 shares.
• Totals and averages of many independent parts tend to be normal.

Check yourself

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

1. Is $B(30, 0.1)$ close to normal?

2. A test is so easy that most students score between $90$ and $100$ marks out of $100$. Why is a normal model poor here?

3. Waiting times have a mean of $6$ minutes and a standard deviation of $6$ minutes. What share does a normal model put below $0$ minutes?

4. In a made-up data set, the mean is much larger than the median. What does that suggest about the shape?

5. Why are heights closer to normal than single waiting times?

References

• OpenStax, Introductory Statistics, Chapter 6, section on using the normal distribution, and Chapter 7 on the central limit theorem.