Areas from z-scores

What you will be able to do

• turn a value into a z-score, and a z-score into an area;
• find the share below, above and between values of a normal distribution;
• find the value with a given share below it.

Bolts from a made-up machine

A made-up machine makes bolts. Their lengths follow a normal curve with mean $50.0\ \text{mm}$ and standard deviation $0.2\ \text{mm}$. What share of bolts is shorter than $49.7\ \text{mm}$? The 68-95-99.7 rule cannot answer, because $49.7\ \text{mm}$ is not a whole number of standard deviations from the mean.

The z-score

The z-score of a value says how many standard deviations it lies from the mean:

For $49.7\ \text{mm}$, $z = \dfrac{49.7 - 50.0}{0.2} = -1.5$. A negative z-score means below the mean. You met z-scores in the chapter on describing distributions. Here they do more: every normal distribution, turned into z-scores, becomes the standard normal distribution, $N(0, 1)$. So one curve answers every question.

The area to the left of $z$ under $N(0, 1)$ is written $\Phi(z)$. The Greek letter $\Phi$ is called phi.

A short table

• $\Phi(0) = 0.5$ and $\Phi(0.5) = 0.6915$
• $\Phi(1) = 0.8413$ and $\Phi(1.5) = 0.9332$
• $\Phi(2) = 0.9772$ and $\Phi(2.5) = 0.9938$
• $\Phi(3) = 0.9987$

For a negative $z$, use the symmetry of the curve: $\Phi(-z) = 1 - \Phi(z)$. Some books print a table of the area between $0$ and $z$ instead. That is $\Phi(z) - 0.5$, so check the heading of a table first.

Below, above and between

• Below. Shorter than $49.7\ \text{mm}$: $\Phi(-1.5) = 1 - 0.9332 = 0.0668$, about $6.7\%$ of bolts.
• Above. Longer than $50.5\ \text{mm}$: $z = 2.5$, so $1 - \Phi(2.5) = 1 - 0.9938 = 0.0062$.
• Between. From $49.8\ \text{mm}$ to $50.3\ \text{mm}$: $z = -1$ and $z = 1.5$, so $\Phi(1.5) - \Phi(-1) = 0.9332 - 0.1587 = 0.7745$.

You should find that the area between two ends is always the larger left area minus the smaller one. Put the ends at $-2$ and $2$, and the area is $0.9545$, the middle of the 68-95-99.7 rule.

Backwards: from a share to a value

Which length has $90\%$ of bolts below it? This value is the 90th percentile. Find the z-score with $\Phi(z) = 0.90$. It is about $1.2816$. Then turn it back into a length: $x = \mu + z\sigma = 50.0 + 1.2816 \times 0.2 \approx 50.256\ \text{mm}$.

Try it in Python

Python's statistics module has a ready normal distribution. cdf(x) gives the area to the left of x, and inv_cdf(share) runs backwards.

Your turn

The usual mistakes

• Using $\Phi(z)$ for the share above. The table gives the area to the left. For above, use $1 - \Phi(z)$.
• Dropping the sign of $z$. A value below the mean has a negative z-score.
• Adding areas to find "between". Take the smaller left area away from the larger one.
• Reading the wrong kind of table. Check whether it gives the area to the left of $z$ or from $0$ to $z$.

Remember these

• $z = \dfrac{x - \mu}{\sigma}$, and $\Phi(z)$ is the area to the left of $z$.
• Below: $\Phi(z)$. Above: $1 - \Phi(z)$. Between: $\Phi(z_2) - \Phi(z_1)$.
• Backwards: find $z$ from the share, then $x = \mu + z\sigma$.
• In Python: NormalDist(mu, sigma).cdf(x) and .inv_cdf(share).

Check yourself

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

1. What is $\Phi(0)$, and why?

2. Use the table to find the share of a normal distribution more than $1.5$ standard deviations above the mean.

3. Marks in a made-up test have $\mu = 64$ and $\sigma = 8$. What is the z-score of a mark of $52$?

4. Which z-score has $97.5\%$ of the area to its left?

5. Use the table to find $P(-1 < Z < 2)$.

References

• OpenStax, Introductory Statistics, Chapter 6, sections on the standard normal distribution and on using the normal distribution.
• The Python Standard Library documentation, the statistics module, class NormalDist.