Normal distribution calculator
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Result
Find probabilities for any normal distribution N(μ, σ²): the area to the left or right of a value, between two values, or in both tails. The inverse modes work backwards from an area to the value x, which is how you find percentiles and cut-off scores. Every answer shows the z-scores and the Φ values used, so you can check them against a standard normal table.
Step-by-step solution
Standardise
Areas under the standard normal curve
Key formulas
- z-score
- z = (x − μ)/σ
- Below x
- P(X < x) = Φ(z)
- Above x
- P(X > x) = 1 − Φ(z)
- Between
- Φ(z2) − Φ(z1)
- Percentile
- x = μ + z·σ
How to find a normal distribution probability
- Write down the mean μ and the standard deviation σ. If you are given the variance σ², take its square root first.
- Standardise each value with z = (x − μ)/σ. The z-score says how many standard deviations the value lies above or below the mean.
- Look up Φ(z), the area to the left of z under the standard normal curve, in a table or with a calculator.
- Combine the areas: P(X < a) = Φ(z), P(X > a) = 1 − Φ(z) and P(a < X < b) = Φ(z_b) − Φ(z_a).
- To work backwards from an area, find the z with Φ(z) equal to the left-tail area, then convert it back with x = μ + z·σ.
Questions and answers
What is the difference between P(X < a) and P(X ≤ a)?
For a continuous distribution such as the normal distribution there is none. The probability of one exact value is zero, so P(X < a) = P(X ≤ a). This is different for discrete distributions such as the binomial and Poisson distributions.
How do I find a percentile of a normal distribution?
Use the inverse mode with a left-tail area. The 90th percentile is the value x with P(X < x) = 0.90: z = Φ⁻¹(0.90) = 1.2816, so x = μ + 1.2816σ. For a top-10% cut-off, use the right-tail area 0.10, which gives the same value.
What does the 68–95–99.7 rule say?
About 68% of a normal distribution lies within one standard deviation of the mean, about 95% within two (more precisely 1.96σ) and about 99.7% within three. It is a quick way to check whether an answer is in the right range.
Why does my z-table give a slightly different answer?
A printed table rounds z to two decimals and Φ(z) to four. The calculator evaluates Φ numerically, far more precisely than a printed z-table, also far out in the tails, and shows the result to 4 significant figures, so table answers can differ in the last digit.
When can I use the normal distribution for a binomial problem?
When np and n(1 − p) are both at least about 10, a binomial distribution is close to a normal one with μ = np and σ = √(np(1 − p)). Add or subtract 0.5 at the boundary (the continuity correction). For an exact answer, use the binomial distribution calculator.
Practise it yourself
Practise Normal distribution
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.