Back to course
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 The Normal Distribution
Edexcel A Level Maths: 3.4 The Standard Normal DistributionCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

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
1

Standardising

The standard normal variable is Z∼N(0,1)Z \sim N(0, 1). Any value xx of X∼N(μ,σ2)X \sim N(\mu, \sigma^2) converts to a zz-value by z=x−μσz = \dfrac{x - \mu}{\sigma} — the number of standard deviations from the mean.

z=x−μσz = \frac{x - \mu}{\sigma}
Standardising formula.
Example 1For X∼N(50,42)X \sim N(50, 4^2), standardise x=58x = 58.
1z=58−504z = \dfrac{58 - 50}{4}.
2=2= 2.
Answer
z=2z = 2 (two sd above the mean).

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

Formula recap

Z∼N(0,1)Z \sim N(0, 1)
Standard normal.
z=x−μσz = \frac{x - \mu}{\sigma}
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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The standard normal distribution is:

    • N(0,1)N(0,1)
    • N(1,0)N(1,0)
    • N(μ,σ2)N(\mu,\sigma^2)
    • N(0,0)N(0,0)
    Show answer and explanation

    Answer: N(0,1)N(0,1)

    Mean 0, sd 1.

  2. The standardising formula is:

    • z=x−μσz = \frac{x-\mu}{\sigma}
    • z=x−μσ2z = \frac{x-\mu}{\sigma^2}
    • z=x−μz = x - \mu
    • z=μ−xσz = \frac{\mu - x}{\sigma}
    Show answer and explanation

    Answer: z=x−μσz = \frac{x-\mu}{\sigma}

    Subtract mean, divide by sd.

  3. For N(50,42)N(50, 4^2), x=58x = 58 gives z =

    • 22
    • 88
    • 0.50.5
    • 11
    Show answer and explanation

    Answer: 22

    (58−50)/4.

  4. A z-value measures:

    • standard deviations from the mean
    • the probability
    • the variance
    • the area
    Show answer and explanation

    Answer: standard deviations from the mean

    Definition.

  5. When standardising you divide by:

    • σ\sigma
    • σ2\sigma^2
    • μ\mu
    • zz
    Show answer and explanation

    Answer: σ\sigma

    Standard deviation.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.