13.4 Links to the binomial, histograms and points of inflection — Learn

Prerequisites: the binomial model from Sections 13.1 and 13.2; the Normal model from Section 13.3; points of inflection from Section 7.4; histograms and the standard deviation from Sections 11.3 and 11.5.

By the end of this section you can: explain how the area of a histogram approaches the area under a curve; describe the shape of the Normal curve and its points of inflection; use the one-, two- and three-standard-deviation bands; and approximate a binomial probability with a Normal variable when the conditions are met.

1. From histogram to curve

In Section 11.3 you drew a histogram. The area of each bar is proportional to the frequency of the class. When the sample is large and the classes are narrow, the tops of the bars lie close to a smooth curve.

The total area of the histogram bars is the total frequency. Divide every frequency by the total, and the area of each bar becomes a relative frequency. The relative frequencies add to 1, just like probabilities. In the limit, the histogram becomes a probability density curve, and the area under the curve is a probability.

This is the link between the two halves of the chapter. The binomial bar chart is a histogram of a count. The Normal curve is the smooth limit of a histogram of a measurement.

2. Points of inflection

In Section 7.4 you met the second derivative. A curve is concave where $f''(x) < 0$ and convex where $f''(x) > 0$. A point of inflection is where the curve changes from concave to convex, or the other way.

The Normal curve is concave in the middle and convex in the two tails. The change happens at exactly

These are the points where the curve is steepest. The figure in Section 13.3 marks them. They are the reason $\sigma$ is the natural measure of spread for this curve: it fixes where the curve turns.

3. The one-, two- and three-standard-deviation bands

The symmetry of the curve and the standard Normal table give three useful probabilities.

So about 68% of a Normal variable lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three. These bands are a quick check on any Normal calculation. If an answer says that 40% of a Normal variable lies within one standard deviation, something is wrong.

4. The binomial link

The binomial bar chart has a centre and a spread. For $X \sim B(n, p)$, the centre sits at

and the spread is

When $n$ is large and $p$ is not close to 0 or 1, the bar chart is close to a Normal curve with these two parameters. We write

The approximation turns a sum of many binomial terms into one normal probability. Two conditions keep it honest.

• Both $np$ and $n(1-p)$ are greater than 5.
• The bars are not too far from the curve, so $p$ is not close to 0 or 1.

5. The continuity correction

A binomial variable takes whole-number values. A Normal variable is continuous. To match the two, move each whole number to the interval of width 1 around it.

For an at-most probability, the bar for the value $r$ covers the interval $[r - 0.5, r + 0.5)$. So

where $Y$ is the matching Normal variable. The $0.5$ is the continuity correction. It moves the boundary to the right.

For an at-least probability,

Example 1. The random variable $X \sim B(50, 0.4)$. Use a Normal approximation to estimate $P(X \le 15)$.

Check the conditions first:

Both are greater than 5, so the approximation is reasonable. The matching Normal variable has

Apply the continuity correction: $X \le 15$ becomes $Y \le 15.5$. Standardise:

Then

The exact binomial probability is $0.0955$, so the approximation is close.

Example 2. The random variable $X \sim B(100, 0.5)$. Estimate $P(X \le 40)$.

Here $np = 50$ and $n(1-p) = 50$, so the conditions hold. The matching Normal variable has

Apply the continuity correction and standardise:

Then

6. Try it: binomial bars and the Normal curve

Predict first. The figure shows the bars of $X \sim B(n, p)$ with a Normal curve drawn over them. The curve has mean $np$ and standard deviation $\sqrt{np(1-p)}$.

Before you move anything, predict how the bars change shape when $n$ increases while $p$ stays at 0.5. Write down whether the bars become closer to the curve or further from it.

Test. Set $p = 0.5$ and increase $n$ from 10 to 100. Watch how the bars move towards the curve. Then set $p = 0.1$ and watch the curve again. Note where the curve fits well and where it does not.

Explain. Write two sentences. The first explains why a larger $n$ makes the curve fit the bars better. The second explains why a small $p$ makes the fit worse, even when $n$ is large.

Watch out. Three mistakes are common.

Approximating when the conditions fail. If $n$ is small, or $p$ is close to 0 or 1, the bars are far from the curve. Check $np$ and $n(1-p)$ before you approximate.

Forgetting the continuity correction. Use $r + 0.5$ for $P(X \le r)$ and $r - 0.5$ for $P(X \ge r)$. Rounding a whole number straight into a continuous curve shifts the boundary by half a unit.

Saying the count is Normal. The count is binomial. It is approximately Normal when the conditions hold. The approximation is a modelling choice, not a fact about the variable.

Re-check: state the matching $\mu$ and $\sigma$, apply the continuity correction, and check the answer against the size you expected.

You should now be able to

• explain how a histogram approaches a probability density curve;
• locate the points of inflection of a Normal curve at $\mu \pm \sigma$;
• use the one-, two- and three-standard-deviation bands;
• find the matching Normal parameters $\mu = np$ and $\sigma = \sqrt{np(1-p)}$ for a binomial variable;
• apply the continuity correction and estimate a binomial probability;
• state when the Normal approximation to the binomial is reasonable.