Inverse normal calculator
Give the probability and the side it is on. The tool finds the value of x, or works backwards to an unknown mean or standard deviation, and shows each step.
- z
- 1.28
- x
- 180
Show the working
z = Φ−1(0.900) = 1.28
x = μ + zσ = 170 + (1.28) × 8 = 180
Keep the unrounded z in your calculator for the last step; rounding it first can change the third significant figure.
Answers are rounded to 3 significant figures. A Level questions usually say how accurate to be: follow the question.
How to do it by hand
- Turn the question into an area to the left. If the probability is above x, the area to the left is 1 − p; for the middle p, each tail has (1 − p) ÷ 2.
- Find z with Φ(z) equal to that area: the inverse normal function, Φ⁻¹.
- Undo the standardising: x = μ + zσ. To find an unknown μ or σ, write (x − μ) ÷ σ = z and solve for the unknown (two conditions give two simultaneous equations).
Worked example
The lengths of a species of fish are normally distributed with mean 24 cm. 15% of the fish are longer than 30 cm. Find the standard deviation.
Solution
P(X > x) = 0.15 means P(X < x) = 1 − 0.15 = 0.850: inverse normal always works with the area to the left.
z = Φ−1(0.850) = 1.04
(30 − 24) ÷ σ = 1.04, so σ = (30 − 24) ÷ 1.04 = 5.79
Keep the unrounded z in your calculator for the last step; rounding it first can change the third significant figure.
Answer: σ = 5.79
Every number in this example is worked out by the same code as the tool above, and the code is tested against independent results from Python's SciPy library.
Where you need it
A Level Statistics (and IAL S1): probabilities, percentage points, finding μ or σ, and the normal as an approximation to the binomial (with a continuity correction).
Practise it: Statistical distributions · IAL S1.
On your calculator
In the exam you use your own calculator, so practise it too. Our guides give the exact keystrokes for the Casio fx-CG50, TI-Nspire CX II, TI-84 Plus CE, Casio fx-991EX ClassWiz and Casio fx-991CW ClassWiz, with a worked example to check against:
Common mistakes
- Putting the probability above x straight into inverse normal. Most calculators default to the area to the left: use 1 − p, or set the tail to the right.
- Losing the sign of z. If x is below the mean, z is negative, and so is x − μ.
- Using a rounded z (such as 1.04) and then giving the answer to 3 s.f.: the rounding can change the last figure. Store z.
- For an unknown σ, dividing by a z that has the wrong sign gives a negative σ: check that x is on the side of the mean you expect.
Notes and practice
Questions students ask
What is inverse normal?
It runs the normal distribution backwards: instead of turning a value into a probability, it turns a probability (an area under the curve) into the value with that area to its left.
How do I find μ and σ when both are unknown?
Each piece of information gives an equation of the form (x − μ) ÷ σ = z. Find each z with inverse normal, then solve the two equations simultaneously. Use the tool's “Find μ” or “Find σ” to check each one once you know the other.
What is a percentile?
The value with that percentage of the distribution below it. The 90th percentile is the x with P(X < x) = 0.9, which is exactly what this tool finds.