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 Video by Zeeshan Zamurred The Normal Distribution

Edexcel A Level Maths: 3.5 Finding the Mean and Standard Deviation for a Normal DistributionWatch the full walkthrough 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

The method

Find the $z$ -value matching the given probability (inverse standard normal). Then substitute into $z = \frac{x - μ}{σ}$ and solve for the unknown.

z = \frac{x - μ}{σ} \Rightarrow solve for μ or σ

Use a known z-value.

z = InvNorm (0.9) \approx 1.2816

1.2816 = \frac{20 - μ}{5} \Rightarrow μ = 20 - 5 (1.2816)

Answer

μ \approx 13.6

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

Formula recap

z = \frac{x - μ}{σ}

Rearrange for the unknown.

z = 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.

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