Statistics

Normal Distribution and z-scores

Lesson

The normal distributionis the classic bell-curve shape that turns up everywhere — heights, test scores, measurement errors. It’s described by just two numbers: the mean $\mu$ (the peak) and the standard deviation $\sigma$ (the spread).

The bell curve, with the x-axis in z-score units (standard deviations from the mean).

A z-score rescales any value to its number of standard deviations above or below the mean:

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

$z = 0$ means $x$ is exactly at the mean.
$z > 0$ means above the mean; $z < 0$ means below.
The size of $|z|$ tells you how unusual the value is. $z = 2$ is two standard deviations above the mean.

To go the other direction — find the raw value $x$ for a given $z$ :

x = \mu + z\,\sigma

Worked example 1 — z-score from raw value

Test scores have $\mu = 80$ , $\sigma = 8$ . A student scored 96. What’s her z-score?

z = \frac{96 - 80}{8} = \frac{16}{8} = 2

Two standard deviations above the mean.

Worked example 2 — raw value from z-score

IQ scores have $\mu = 100$ , $\sigma = 15$ . What raw score corresponds to z = -1.4?

\begin{aligned} x &= 100 + (-1.4)(15) \\ &= 100 - 21 \\ &= 79 \end{aligned}

How to type your answer

A single number — z-score or raw value, as the question asks. Use a decimal point if needed. Examples: 2, -1.5, 87, 0.6.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\mu = 50,\ \sigma = 10.\ z \text{ for } x = 60

Problem 2

\mu = 50,\ \sigma = 10.\ z \text{ for } x = 40

Problem 3

\mu = 100,\ \sigma = 15.\ z \text{ for } x = 130

Problem 4

\mu = 100,\ \sigma = 15.\ x \text{ for } z = 1

Practice

Standard problems matching the lesson.

Problem 5

\mu = 80,\ \sigma = 8.\ z \text{ for } x = 96

Problem 6

\mu = 80,\ \sigma = 8.\ z \text{ for } x = 64

Problem 7

\mu = 500,\ \sigma = 100.\ z \text{ for } x = 650

Problem 8

\mu = 500,\ \sigma = 100.\ x \text{ for } z = -2

Problem 9

\mu = 25,\ \sigma = 5.\ z \text{ for } x = 35

Problem 10

\mu = 25,\ \sigma = 5.\ x \text{ for } z = 0

Problem 11

\mu = 70,\ \sigma = 12.\ z \text{ for } x = 88

Problem 12

\mu = 70,\ \sigma = 12.\ x \text{ for } z = -1.5

Problem 13

\mu = 200,\ \sigma = 25.\ z \text{ for } x = 237.5

Problem 14

\mu = 200,\ \sigma = 25.\ x \text{ for } z = 2.4

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\mu = 68,\ \sigma = 4.\ z \text{ for } x = 70.4

Problem 16

\mu = 68,\ \sigma = 4.\ x \text{ for } z = -2.5

Problem 17

\mu = 72,\ \sigma = 6.\ z \text{ for } x = 63

Problem 18

\mu = 72,\ \sigma = 6.\ x \text{ for } z = 2.5

Problem 19

\mu = 15,\ \sigma = 3.\ z \text{ for } x = 21

Problem 20

\mu = 100,\ \sigma = 15.\ z \text{ for } x = 122.5

Problem 21

\mu = 100,\ \sigma = 15.\ x \text{ for } z = -1.4

Problem 22

\text{Heights } \mu = 66\text{ in},\ \sigma = 3.5.\ z \text{ for } x = 73

Practice

Standard problems matching the lesson.

Problem 23

SAT section scores are roughly normal with μ = 500, σ = 100. A student scored 650. What is the z-score?

Problem 24

Adult male heights are roughly normal with μ = 68 in, σ = 3 in. What height corresponds to z = 2?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Test scores have μ = 75, σ = 8. A student's z-score is -1.25. What is the raw score?

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