Finding μ and σ from probabilities
A reverse normal question gives an area and asks for a boundary, mean or standard deviation. Convert the area into a signed -value before doing the algebra.
Convert the area into a signed z-value
Left area decides the sign of z
left area 0.90 > 0.5 → z = +1.282
left area 0.20 < 0.5 → z = −0.842
- Rewrite the probability as a left area.
- Decide whether the boundary is left or right of the mean.
- Use inverse Normal or the MF19 critical-value table to find the magnitude and sign of .
- Write . Do not put the probability itself into this equation.
Key idea
The MF19 critical-value table contains only selected areas. For an arbitrary area, use inverse Normal with mean 0 and standard deviation 1.
One unknown parameter needs one equation
Worked example
Find a boundary
Walking times have mean 14.8 minutes and standard deviation 1.5 minutes. Find if 90% of the times are greater than .
A right area of 0.90 means the left area at is 0.10, so .
(9709/53/O/N/25 Q7(b))
Worked example
Find a standard deviation
If and , the left area at 112 is 0.90, so .
Common mistake
Two unknown parameters need two equations
Convert each probability separately. Subtracting the two standardisation equations removes and gives .
Worked example
Use two non-symmetric probabilities
Chocolate-bar masses have unknown mean and standard deviation . Suppose 8% exceed 114.0 g and 20% are below 99.5 g.
(9709/51/O/N/25 Q6(b))
Let . Given and , find and .
Show worked answer
The second statement gives a left area of 0.78. The signed inverse-normal values are -1.0364 and +0.7722.