Normal distribution calculator
Type the mean, the standard deviation and the values. You get the probability, the z-scores that lead to it and the shaded area under the curve.
- z for a
- −1.00
- z for b
- 1.33
Show the working
X ~ N(100, 15²): the mean is 100 and the standard deviation is 15 (so the variance is 225).
Standardise: z = (x − μ) ÷ σ turns each value into a number of standard deviations from the mean.
za = (85 − 100) ÷ 15 = −1.00
zb = (120 − 100) ÷ 15 = 1.33
P(85 < X < 120) = Φ(1.33) − Φ(−1.00) = 0.750
Φ is the standard normal cumulative probability, what your calculator's normal CD (normalcdf) works out. With a continuous distribution, < and ≤ give the same answer.
Answers are rounded to 3 significant figures, which is what IB papers ask for unless a question says otherwise.
How to do it by hand
- Draw a quick sketch of the bell curve, mark the mean in the middle and shade the area the question wants. It tells you straight away whether the answer is more or less than 0.5.
- Standardise each value: z = (x − μ) ÷ σ is how many standard deviations the value is from the mean.
- Read Φ(z), the area to the left of z, from your calculator's normal CD function (or a table), and combine the areas: P(X > a) = 1 − Φ(z), and P(a < X < b) = Φ(z_b) − Φ(z_a).
Worked example
The time a bus journey takes is normally distributed with mean 42 minutes and standard deviation 6 minutes. Find the probability that a journey takes between 35 and 50 minutes.
Solution
X ~ N(42, 6²): the mean is 42 and the standard deviation is 6 (so the variance is 36.0).
Standardise: z = (x − μ) ÷ σ turns each value into a number of standard deviations from the mean.
za = (35 − 42) ÷ 6 = −1.17
zb = (50 − 42) ÷ 6 = 1.33
P(35 < X < 50) = Φ(1.33) − Φ(−1.17) = 0.787
Φ is the standard normal cumulative probability, what your calculator's normal CD (normalcdf) works out. With a continuous distribution, < and ≤ give the same answer.
Answer: P(35 < X < 50) = 0.787
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
| AA SL | Normal probabilities and inverse normal. |
|---|---|
| AA HL | Also standardising (z-scores) to find an unknown μ or σ. |
| AI SL | Normal probabilities and inverse normal. |
| AI HL | As SL, plus linear combinations of normal variables. |
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 and TI-84 Plus CE, with a worked example to check against:
Common mistakes
- Using the variance instead of the standard deviation. N(50, 16) has σ = 4, not 16: take the square root before you standardise.
- Forgetting which side is shaded. Φ(z) is always the area to the left; for “more than”, subtract from 1.
- Rounding z to two decimal places before finding the probability. Keep the full value in your calculator and round only the final answer.
- Writing calculator syntax as working. Write P(35 < X < 50) = 0.814 with X ~ N(42, 6²), not normalcdf(35, 50, 42, 6).
Notes and practice
Questions students ask
Is P(X < a) the same as P(X ≤ a) for a normal distribution?
Yes. A continuous distribution gives zero probability to any single value, so including or leaving out the end point makes no difference. That is not true for the binomial or Poisson distributions, which are discrete.
What does the z-score tell me?
How many standard deviations a value is above (positive) or below (negative) the mean. About 68% of values have z between −1 and 1, about 95% between −2 and 2 and nearly all between −3 and 3.
Why does my answer differ slightly from a printed table?
Tables give Φ(z) for z rounded to two decimal places, so they can be out in the third significant figure. This tool and your calculator use the exact z.