How Do You Use Z-Values to Find an Unknown Mean or Standard Deviation?

Use the standardization formula $z=\frac{x-\mu}{\sigma}$, substitute the known value of $x$ and the z-value corresponding to the given probability, then rearrange for the unknown mean $\mu$ or standard deviation $\sigma$.

For a normally distributed variable $X\sim N(\mu,\sigma^2)$, a z-value measures how many standard deviations $x$ lies above or below the mean:

$$z=\frac{x-\mu}{\sigma}.$$

First use the GDC's inverse-normal function to convert the cumulative probability $P(X<x)=p$ into $z=\Phi^{-1}(p)$. Then form an equation using the standardization formula.

Check the probability direction before using inverse normal. For a left-tail statement $P(X<x)=p$, use $z=\Phi^{-1}(p)$. For a right-tail statement , first convert it to . A value below the mean must produce , while a value above the mean must produce ; this sign check catches many calculator errors.

Example 1: unknown mean. Suppose $P(X<72)=0.90$ and $\sigma=8$. Using inverse normal, $z=1.28155$, so

$$\mu=72-(1.28155)(8)=61.7$$

to three significant figures.

Example 2: unknown standard deviation. Suppose $\mu=50$ and $P(X<62)=0.95$. Since $z=1.64485$,

$$\sigma=\frac{62-50}{1.64485}=7.30.$$

When both parameters are unknown, suppose $P(X<40)=0.20$ and $P(X<70)=0.80$. The corresponding z-values are approximately $-0.8416$ and , giving and . Subtracting gives , so , and substitution gives . This illustrates why two probability statements produce two simultaneous equations.

Always retain unrounded z-values during working, because premature rounding can shift the final mean or standard deviation beyond the accepted accuracy.

A common misconception is that the z-value is the original data value. It is not: $x$ is measured in the variable's units, while $z$ is a dimensionless standardized value. Also remember that a GDC normally requires $\sigma$, not the variance $\sigma^2$.

Exam technique: This is SL 4.12 and is normally tested on Paper 2 because inverse-normal calculations require technology. Show the z-value, write the standardization equation, rearrange clearly, and give an exact answer or three significant figures unless instructed otherwise.