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3.5StatisticsStretch

Finding the Mean and Standard Deviation

When μ or σ is unknown, you work backwards. Convert a probability to a z-value using the inverse standard normal, then set up an equation with the standardising formula and solve.

26 min The Normal Distribution
Edexcel A Level Maths: 3.5 Finding the Mean and Standard Deviation for a Normal DistributionCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Find an unknown μ or σ
  • Convert a probability to a z-value
  • Set up and solve the standardising equation
  • Handle two unknowns with simultaneous equations
1

The method

Find the zz-value matching the given probability (inverse standard normal). Then substitute into z=x−μσz = \dfrac{x - \mu}{\sigma} and solve for the unknown.

z=x−μσ ⇒ solve for μ or σz = \frac{x - \mu}{\sigma} \ \Rightarrow\ \text{solve for } \mu \text{ or } \sigma
Use a known z-value.
Example 1X∼N(μ,52)X \sim N(\mu, 5^2) and P(X<20)=0.9P(X < 20) = 0.9. Find μ.
1z=InvNorm(0.9)≈1.2816z = \text{InvNorm}(0.9) \approx 1.2816.
21.2816=20−μ5⇒μ=20−5(1.2816)1.2816 = \dfrac{20 - \mu}{5} \Rightarrow \mu = 20 - 5(1.2816).
Answer
μ≈13.6\mu \approx 13.6

Tip — Two unknowns (both μ and σ) need two probability statements → two equations.

Formula recap

z=x−μσz = \frac{x - \mu}{\sigma}
Rearrange for the unknown.
z=InvNorm(p)z = \text{InvNorm}(p)
z-value from probability.

Common mistakes to avoid

Using the X-value where a z-value is needed.
Find the z-value first via the inverse standard normal.
Trying to find two unknowns from one equation.
Two unknowns need two probability statements.

Key takeaways

  • Convert the probability to a z-value (InvNorm).
  • Substitute into z = (x − μ)/σ and solve.
  • Two unknowns ⇒ two equations.

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. To find an unknown μ, first find the:

    • z-value
    • variance
    • area
    • mode
    Show answer and explanation

    Answer: z-value

    From the probability.

  2. The equation you set up is:

    • z=x−μσz = \frac{x-\mu}{\sigma}
    • z=xμz = x\mu
    • z=σxz = \frac{\sigma}{x}
    • z=μ−σz = \mu - \sigma
    Show answer and explanation

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

    Standardising.

  3. Finding both μ and σ needs:

    • two equations
    • one equation
    • no equations
    • a graph
    Show answer and explanation

    Answer: two equations

    Two unknowns.

  4. For N(μ,52)N(\mu, 5^2) with P(X<20)=0.9P(X<20)=0.9 (z≈1.2816), μ ≈

    • 13.613.6
    • 2020
    • 26.426.4
    • 1515
    Show answer and explanation

    Answer: 13.613.6

    20 − 5(1.2816).

  5. The z-value comes from the:

    • inverse standard normal
    • binomial
    • mean
    • variance
    Show answer and explanation

    Answer: inverse standard normal

    InvNorm of p.

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.