How to use the Normal Distribution Calculator
- 1. Set mean and SDEnter the average and standard deviation of the distribution.
- 2. Choose what to findPick below, above, between, outside, or a percentile value.
- 3. See the areaRead the probability and see it shaded on the curve.
From value to probability
Every normal question starts with a z-score: z = (x − μ) ÷ σ. The probability below x is the area under the standard normal curve to the left of z, written Φ(z). Above x is 1 − Φ(z), and between two values is Φ(z₂) − Φ(z₁).
Worked example: exam marks are normal with mean 70 and SD 10. For a mark of 85, z = 1.5 and P(X < 85) = Φ(1.5) = 0.9332, so about 93.32% of students score below 85 and 6.68% above. The inverse mode works backwards: the 95th percentile is 70 + 1.6449 × 10 ≈ 86.45.
Handy facts and limits
- 68–95–99.7 rule: about 68% of values lie within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
- Exact areas: the tool computes Φ from the error function, not a rounded z-table, so results are accurate to four or more decimals.
- Continuous only: P(X = exactly 85) is zero for a normal curve. For counts, use the binomial calculator.
Real data is rarely perfectly normal. Heights and measurement errors come close; incomes, website times and house prices are usually skewed, and normal probabilities for them can be badly off. Check the shape with a histogram before relying on these numbers, and use the z-score calculator to standardise a whole list.
Frequently asked questions
What is the difference between this and a z-table?
A z-table only covers the standard normal (mean 0, SD 1) and is rounded. This tool works with any mean and SD directly.
Can I find the middle 95%?
Yes. Use the inverse mode for the 2.5th and 97.5th percentiles, or between mode with those two values.
Is the Normal Distribution Calculator free?
Yes, the Normal Distribution Calculator is free. The first time you see your first result, you are asked for your name and WhatsApp number once. After that, this and every other Uncodemy tool opens straight away without the form.