z-scores and the normal model

Warm-up

Three quick questions from earlier lessons.

Discover

The blue curve is a normal distribution with mean $m$ and standard deviation $s$ . Move them.

A normal curve with mean m and standard deviation s: symmetric and bell-shaped, centered at m, wider and lower as s grows.

What happens to the curve when $s$ increases?

It gets wider and lower
It moves to the right

Key idea

Standardize to compare, and use the 68–95–99.7 rule.

z-score:

z = \dfrac{x - \mu}{\sigma}

, the number of SDs above the mean. For a normal distribution, about 68% of values lie within 1 SD of the mean, 95% within 2, 99.7% within 3.

Worked example

SAT maths: mean 520, SD 110. A score of 685.

z = 1.5

(685 − 520)/110.

Between 300 and 740: about 95% of scores

Within 2 SDs: about 95%.

Above 740: about 2.5%

Half of the remaining 5%.

Your turn, guided

Your turn: mean 70, SD 5. The z-score of 62.

z = -1.6

(62 − 70)/5.

Practice ★

Test scores have mean 75 and standard deviation 3. Find the z-score of a score of 52.
Test scores have mean 55 and standard deviation 2. Find the z-score of a score of 81.
Test scores have mean 66 and standard deviation 10. Find the z-score of a score of 94.

Watch out!

Practice ★★

Heights are approximately normal with mean 76 and standard deviation 4. By the 68–95–99.7 rule, what percent are between 72 and 84?
Heights are approximately normal with mean 69 and standard deviation 6. By the 68–95–99.7 rule, what percent are between 63 and 81?

Mean 71, standard deviation 8. Which value has z-score $-3$ ?
Mean 79, standard deviation 6. Which value has z-score $3$ ?

Exit ticket

Three questions to check you have got it.

Test scores have mean 75 and standard deviation 3. Find the z-score of a score of 52.
Test scores have mean 55 and standard deviation 2. Find the z-score of a score of 81.
Test scores have mean 66 and standard deviation 10. Find the z-score of a score of 94.

Summary card

z-scores and the normal model

$z = (x - \mu)/\sigma$ ; $x = \mu + z\sigma$ .
68–95–99.7 within 1, 2, 3 SDs.

Watch out: the percentages are for both sides together.

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