3.4StatisticsStretch

The Standard Normal Distribution

The standard normal distribution Z has mean 0 and standard deviation 1. Standardising — subtracting the mean and dividing by σ — converts any normal variable to Z, which is essential for finding unknown parameters.

24 min Video by Zeeshan Zamurred The Normal Distribution

Edexcel A Level Maths: 3.4 The Standard Normal DistributionWatch the full walkthrough before the notes below.

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What you'll be able to do

Define Z ~ N(0, 1)
Standardise with z = (x − μ)/σ
Convert between X and Z
Use Z for unknown-parameter problems

Standardising

The standard normal variable is $Z \sim N (0, 1)$ . Any value $x$ of $X \sim N (μ, σ^{2})$ converts to a $z$ -value by $z = \frac{x - μ}{σ}$ — the number of standard deviations from the mean.

z = \frac{x - μ}{σ}

Standardising formula.

z = \frac{58 - 50}{4}

= 2

Answer

z = 2

(two sd above the mean).

Tip — A z-value is just “how many standard deviations from the mean”.

Formula recap

Z \sim N (0, 1)

Standard normal.

z = \frac{x - μ}{σ}

Standardise.

Common mistakes to avoid

Dividing by σ² instead of σ when standardising.

Divide by the standard deviation σ.

Forgetting to subtract the mean first.

z = (x − μ)/σ — subtract μ, then divide.

Key takeaways

Z ~ N(0, 1) is the standard normal.
z = (x − μ)/σ: standard deviations from the mean.
Standardising underpins finding μ or σ.

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