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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 μ\mu (the peak) and the standard deviation σ\sigma (the spread).

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

z=x−μσz = \frac{x - \mu}{\sigma}
  • z=0z = 0 means xx is exactly at the mean.
  • z>0z > 0 means above the mean; z<0z < 0 means below.
  • The size of ∣z∣|z| tells you how unusual the value is. z=2z = 2 is two standard deviations above the mean.

To go the other direction — find the raw value xx for a given zz:

x=μ+z σx = \mu + z\,\sigma

Worked example 1 — z-score from raw value

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

z=96−808=168=2z = \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 μ=100\mu = 100, σ=15\sigma = 15. What raw score corresponds to z = -1.4?

x=100+(−1.4)(15)=100−21=79\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

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

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

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

Problem 14

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

Challenge

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

Problem 15

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

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

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

Problem 20

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

Problem 21

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

Problem 22

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