The z-score of a data element is z = (x - m)/d, where x is the value, m is the mean and d is the standard deviation. Solve the formula for m.

Step-by-step solution

1. 1

   Clear the denominator first. The unknown $m$ is trapped inside a fraction, so multiply both sides by $d$ (a standard deviation is positive, so this is safe):

   $zd=x-m.$

2. 2

   Get m by itself on one side. Add $m$ to both sides and subtract $zd$ — or, equivalently, swap $m$ and $zd$ across the equals sign:

   $m=x-zd.$

   Working this way avoids the sign slip you get by trying to move $-m$ directly.

3. 3

   Check the algebra by substituting back.

   $\frac{x-m}{d}=\frac{x-(x-zd)}{d}=\frac{zd}{d}=z\ \checkmark$

4. 4

   Read the formula for meaning. $m=x-zd$ says the mean sits $z$ standard deviations below the value. If $z>0$ the value is above the mean, so subtracting $zd$ moves you down to it; if $z<0$ the subtraction of a negative moves you up. The sign works out automatically.

5. 5

   Try a numerical example. If $x=85$, $z=1.5$ and $d=6$, then $m=85-1.5(6)=85-9=76$. Checking forwards: $\dfrac{85-76}{6}=\dfrac{9}{6}=1.5$ ✓.