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Statistics · Statistical distributions

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Finding μ and σ

Working backwards from given probabilities to find an unknown mean or standard deviation (or both) by standardising to a z-value and solving — including a pair of simultaneous equations when both are unknown.

Statistics · Statistical distributions

Finding μ and σ

Working backwards from given probabilities to find an unknown mean or standard deviation (or both) by standardising to a z-value and solving — including a pair of simultaneous equations when both are unknown.

Why it works

Sometimes the distribution's parameters are what you're after: you know some probabilities and must find μ\mu, or σ\sigma, or both. The key is that every probability statement pins down a zz-value, and zz is tied to xx, μ\mu and σ\sigma by the standardising equation z=x−μσ.z = \frac{x - \mu}{\sigma}. So you turn each probability into a zz (using the inverse normal), write down the equation, and solve.

One unknown. If only μ\mu (or only σ\sigma) is unknown, one probability is enough. Suppose X∼N(μ,16)X \sim N(\mu, 16) and P(X<30)=0.90P(X < 30) = 0.90. The inverse normal gives z=1.2816z = 1.2816, and since σ=4\sigma = 4, 1.2816=30−μ4  ⇒  μ=30−4(1.2816)=24.9.1.2816 = \frac{30 - \mu}{4} \;\Rightarrow\; \mu = 30 - 4(1.2816) = 24.9.

Both unknown — simultaneous equations. If μ\mu and σ\sigma are unknown you need two probabilities, giving two equations. Each probability →\to a zz-value →\to an equation x=μ+zσx = \mu + z\sigma. Subtracting the two equations eliminates μ\mu and gives σ\sigma; back-substitute for μ\mu.

Two cautions:
  • Get the sign of zz right. A probability below 0.50.5 sits in the lower half, so
its zz is negative; turn "P(X>x)=0.30P(X > x) = 0.30" into "P(X<x)=0.70P(X < x) = 0.70" first so you always look up a left-area.
  • Keep the zz-values to several decimals while solving — rounding early throws off
μ\mu and σ\sigma.