Solve two quantile equations for an unknown mean and standard deviation.
Notes and key methods
If \(\displaystyle P(X\le a)=\Phi (z_{1})\) and \(\displaystyle P(X\le b)=\Phi (z_{2})\) under a normal model, then \(\displaystyle a=\mu +z_{1}\sigma\) and \(\displaystyle b=\mu +z_{2}\sigma\). Subtract to obtain \(\displaystyle \sigma = \frac{b- a}{z_{2}- z_{1}}\), then find μ.
Use the fitted distribution only after checking σ>0 and whether the normal model is plausible for the context. Quantiles of the data and z quantiles are on different scales.
Check the conditions
The two cumulative probabilities correspond to different z values.
The fitted standard deviation must be positive.
Three worked examples
Example 1 · Fitting a normal model
A normal X satisfies \(\displaystyle P(X\le \frac{59}{2})= \Phi (- \frac{1}{2}), \ P(X\le \frac{173}{5})= \Phi (\frac{6}{5})\). Here Φ is the standard normal CDF.
A normal X satisfies \(\displaystyle P(X\le 30)= \Phi (- \frac{1}{2}), \ P(X\le \frac{184}{5})= \Phi (\frac{6}{5})\). Here Φ is the standard normal CDF.
A normal X satisfies \(\displaystyle P(X\le \frac{61}{2})= \Phi (- \frac{1}{2}), \ P(X\le 39)= \Phi (\frac{6}{5})\). Here Φ is the standard normal CDF.