About the Normal Distribution Calculator
The normal distribution is the familiar bell-shaped curve. It describes heights, blood pressure, measurement errors, exam scores on well-designed tests and countless other quantities, and it is the foundation of most statistical methods. A normal distribution is completely described by two numbers — its mean, which sets the centre, and its standard deviation, which sets the width — and from them you can work out what share of values falls in any range.
This normal distribution calculator finds the probability below a value, above a value or between two values, for any mean and standard deviation. It also works in reverse, finding the value at any percentile. Each answer comes with the z-score and a plain-language reading, together with the ranges that hold the central 68 and 95 percent of values.
How to Use the Normal Distribution Calculator
Enter the mean and standard deviation of the distribution.
Choose what you want to find:
- P(X < x) — the share of values below x
- P(X > x) — the share above x
- Between two values — the share between x₁ and x₂
- Value at a percentile — the value with a given share below it
Enter the value, values or percentile, and the answer appears below.
The Formulas
z = (x − μ) ÷ σ
P(X < x) = Φ(z) Φ = standard normal cumulative distribution
P(X > x) = 1 − Φ(z)
P(x₁ < X < x₂) = Φ(z₂) − Φ(z₁)
value at percentile p: x = μ + σ × Φ⁻¹(p)Every normal distribution becomes the standard normal (mean 0, SD 1) after converting to z-scores, so a single function Φ answers every question.
Step-by-Step Examples
IQ-style scores with mean 100 and standard deviation 15.
Below 115: z = (115 − 100) ÷ 15 = 1 Φ(1) = 0.8413 → 84.13%
Between 85, 115: z = −1 to 1 0.8413 − 0.1587 = 0.6827 → 68.27%
Above 130: z = 2 1 − 0.9772 = 0.0228 → 2.28%
90th percentile: z = 1.2816 100 + 1.2816 × 15 = 119.22About 84% of people score below 115, 68% between 85 and 115, and just over 2% above 130. A score of about 119 marks the top 10 percent.
The 68–95–99.7 Rule
For any normal distribution:
within 1 SD of the mean: 68.27%
within 2 SD: 95.45%
within 3 SD: 99.73%This empirical rule gives quick mental estimates. If adult male heights in a country have a mean of 175 cm and SD of 7 cm, about two thirds of men are between 168 and 182 cm, and only about 1 in 370 is more than 21 cm from the average in either direction.
Why the Normal Distribution Is Everywhere
The central limit theorem explains much of its importance: when many small, independent effects add together, the total tends to be normally distributed, whatever the individual effects look like. Height depends on many genes and environmental factors; measurement error comes from many tiny disturbances. The theorem also means that averages of large samples are approximately normal, which is why confidence intervals and many hypothesis tests rely on the normal curve even when the raw data is not normal.
A Manufacturing Example
A machine fills bottles with a mean of 502 ml and a standard deviation of 1.5 ml, and the label promises 500 ml. The share of bottles below 500 ml is Φ((500 − 502) ÷ 1.5) = Φ(−1.33), about 9.12 percent — nearly one bottle in eleven. To bring that below 1 percent, the z-score of 500 ml must be below −2.326, so the mean must rise to at least 500 + 2.326 × 1.5 ≈ 503.5 ml, or the machine's variation must fall to about 0.86 ml. Calculations like this help manufacturers balance the cost of overfilling against the risk of underfilled products.
Continuous Values and Rounding
The normal distribution is continuous, so the probability of any exact value is zero and it makes no difference whether you write "below" or "at or below". When the data is rounded or counted — test scores in whole marks, for instance — a continuity correction improves accuracy: to estimate the chance of a score of 115 or less, calculate P(X < 115.5) instead.
Checking for Normality
Not everything is normal. Incomes, house prices and waiting times are skewed to the right, with long tails of high values; using a normal model for them underestimates the chance of extreme values. Before relying on normal probabilities, look at a histogram of your data, compare the mean and median (they should be close), or use a normal probability plot. If the data is clearly skewed, work with percentiles of the data itself.
Understanding Your Result
The headline is the probability, or the value when finding a percentile.
The z-score line shows how the value was standardised.
The what it means line gives the shares below and above, or inside and outside the range.
The worth knowing line gives the central 68 and 95 percent ranges for your distribution.
When Should You Use This Calculator?
Use it to find the share of a population above or below a threshold.
Use it to set cut-offs, such as the score for the top 10 percent.
Use it for quality control, such as the share of parts outside tolerance.
Use it for statistics homework on the normal curve.
Common Mistakes
Using the variance instead of the standard deviation. Enter σ, not σ².
Forgetting to subtract from 1 for "above". Φ gives the area below.
Applying it to skewed data. Check the shape first.
Mixing up percentiles and probabilities. A percentile is a value; a probability is a share.
Ignoring the continuity correction for whole-number data. It matters most for small ranges.
Using a sample standard deviation from very few values. With a small sample the estimate of σ is rough, and tail probabilities can be badly off.