Quick questions on The normal distribution: z-scores, the empirical rule, probabilities and percentiles

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the normal distribution?

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A continuous random variable

X

is normally distributed with mean

\mu

and standard deviation

\sigma

, written

X \sim N(\mu, \sigma^2)

, if its probability density function is

What are z-scores?

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The z-score of a value

x

measures how many standard deviations it is from the mean:

What are computing probabilities as areas?

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For

X \sim N(\mu, \sigma^2)

and

a < b

, the probability of landing between

a

and

b

is the area under the curve over that interval, and standardising turns it into a standard-normal area:

What is a single cut-off?

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The cumulative probability

P(Z \le z)

is the whole area to the left of

z

. For example the area left of

z = 1

is about

0.84

(half the curve,

0.5

, plus the

0.34

between

0

and

1

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