Normal Distribution Calculator: Probability & Inverse

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Formula at a glance

Normal Distribution Calculator: Probability — P(X < x) = Φ((x − μ)/σ). Example: IQ between 85 and 115 → P(85 < X < 115): 0.682689 = 68.2689% — How the Normal Distribution Calculator works: Probability — P(X < x) = Φ((x − μ)/σ)

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📘 About the Normal Distribution Calculator

The normal distribution is the bell-shaped curve that describes heights, measurement errors, exam scores, IQ and many natural quantities. It is fixed by its mean (the centre) and standard deviation (the spread). Probabilities are areas under the curve.

Choose the mean and SD, then find the probability below a value, above it, between two values or outside them — or work backwards to the value with a given probability below it (for example the 95th percentile). Results are computed precisely, not read from a z-table.

🧮 Formula

Probability

P(X < x) = Φ((x − μ)/σ)

Between

P(a < X < b) = Φ(z_b) − Φ(z_a)

Inverse

x = μ + σ·Φ⁻¹(p)

✏️ Worked examples

IQ between 85 and 115

You enter
Mean μ	100
Standard deviation σ	15
Find	P(x₁ < X < x₂) — between
x₁	85
x₂	115
Probability for the inverse (%)	95

You get
P(85 < X < 115)	0.682689 = 68.2689%
z-scores	z₁ = −1, z₂ = 1
Out of 1,000 people	about 683 would be in this range
Distribution	Normal, mean μ = 100, SD σ = 15

0.682689 = 68.2689%.

Height above 180 cm (mean 170, SD 7)

You enter
Mean μ	170
Standard deviation σ	7
Find	P(X > x₁) — above
x₁	180
x₂	190
Probability for the inverse (%)	95

You get
P(X > 180)	0.0765637 = 7.65637%
z-scores	z₁ = 1.42857
Out of 1,000 people	about 77 would be in this range
Distribution	Normal, mean μ = 170, SD σ = 7

0.0765637 = 7.65637%.

🧭 How to use this calculator

What you enter

Mean μ
Standard deviation σ
Find
x₁
x₂
Probability for the inverse (%)

What you get

P(85 < X < 115)
z-scores
Out of 1,000 people
Distribution

❓ Frequently asked questions

What percentage of values fall within one standard deviation?

About 68.27%. Within two SDs about 95.45%, within three about 99.73%.

How do I find the value for the top 10%?

Use the inverse mode with 90%: the value with 90% below it. For IQ that is about 119.

Is my data normal?

Check a histogram — roughly symmetric and bell-shaped. Skewed data (incomes, waiting times) are not normal.

🔔 Normal Distribution Calculator (Probability & Inverse)