Normal Distribution Calculator
Enter a mean, standard deviation and a value (or two bounds) to get the probability density, cumulative probabilities P(X x), P(a < X < b) and the z-score for a normal distribution.
Find Normal Distribution Probabilities
Normal Distribution Formula
Where:
- \(\mu\) — mean, the center of the bell curve
- \(\sigma\) — standard deviation, the spread of the distribution (must be positive)
- \(x\) — the value at which the density and probabilities are evaluated
- \(f(x)\) — probability density function (PDF), the height of the curve at \(x\)
- \(F(x)\) — cumulative distribution function (CDF), the area under the curve to the left of \(x\)
- \(z\) — z-score, the distance from the mean measured in standard deviations
- \(\Phi\) — CDF of the standard normal distribution (\(\mu=0,\ \sigma=1\))
The first equation defines the bell curve itself. The second converts any value into a z-score and reads the probability from the standard normal curve, so one universal table (or error function) works for every mean and standard deviation.
Normal Distribution – Calculation Example
Example: IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. What is the probability that a randomly chosen person has an IQ below 110?
\[ z=\frac{x-\mu}{\sigma}=\frac{110-100}{15}=0.6667 \]
\[ P(X<110)=\Phi(0.6667)=\frac{1}{2}\left[1+\operatorname{erf}\!\left(\frac{0.6667}{\sqrt{2}}\right)\right]\approx 0.7475 \]
The probability density at that point is \( f(110)=\frac{1}{15\sqrt{2\pi}}e^{-0.6667^2/2}\approx 0.0213 \).
Result: P(X 110) ≈ 25.25%, and the z-score is 0.6667.
How to use the calculator
- Enter the Mean (μ) — the center of your distribution (e.g. 100).
- Enter the Standard deviation (σ) — it must be greater than 0.
- Choose the Probability you want: below a value, above a value, between two values, or outside a range (the two tails).
- For the "below" and "above" modes, enter the Value (x) where the probabilities should be evaluated.
- For the "between" and "outside" modes, enter the Lower bound (a) and Upper bound (b); the upper bound must be greater than the lower bound. You can still fill in x to also get the density and cumulative probabilities at that point.
- Read the results: probability density, cumulative probability, upper-tail probability, area between the bounds, z-score and a plain-language interpretation.
What is the normal distribution?
The normal distribution (also called the Gaussian distribution or bell curve) is the most widely used probability distribution in statistics. It is fully described by just two numbers: the mean \(\mu\), which locates the center of the curve, and the standard deviation \(\sigma\), which controls how wide or narrow it is. Many natural measurements — heights, measurement errors, test scores, blood pressure — are approximately normally distributed, largely because of the central limit theorem: when many small independent effects add up, the total tends toward a bell curve.
The curve is symmetric around the mean and never touches zero, stretching infinitely in both directions. The probability of any exact value is zero; instead, probabilities correspond to areas under the curve. The cumulative probability \(F(x)=P(Xx)=1-F(x)\) is the area to the right. The probability density \(f(x)\) is not itself a probability — it is the height of the curve, useful for comparing how likely different regions are relative to each other.
The z-score \(z=(x-\mu)/\sigma\) expresses a value as a number of standard deviations from the mean. A z-score of 2 means "two standard deviations above average." Converting to z-scores is the key trick: every normal distribution, whatever its mean and standard deviation, becomes the same standard normal curve, so probabilities can be read from one universal function.
Interpreting the result
The 68-95-99.7 rule (empirical rule) gives a quick sanity check for any answer:
| Range around the mean | Share of data inside | Share outside (both tails) |
|---|---|---|
| \(\mu \pm 1\sigma\) | 68.27% | 31.73% |
| \(\mu \pm 2\sigma\) | 95.45% | 4.55% |
| \(\mu \pm 3\sigma\) | 99.73% | 0.27% |
If your computed probability disagrees strongly with these benchmarks, double-check your inputs. For example, with the default settings the probability of a value below 110 (z = 0.67) should be somewhat more than 68%, since 110 is more than one standard deviation above the mean — and 74.75% fits that expectation.
Assumptions, limitations and practical tips
- Normality assumption: the calculator assumes your data truly follow a normal distribution. Real data are often skewed, bounded or heavy-tailed; probabilities for extreme values are especially sensitive to this assumption.
- Strict vs. non-strict inequality: for a continuous distribution, \(P(X<x)\) and \(P(X\le x)\) are identical, so it does not matter which convention you use.
- Small probabilities: values far in the tails (|z| beyond about 4) give probabilities close to 0 or 1; the calculator shows six decimal places so you can still read them.
- Sample vs. population: enter the population mean and standard deviation if you know them. If you only have sample estimates, the results are approximations — for small samples a t-distribution is more appropriate.
- Negative or zero σ: the standard deviation must be strictly positive; the calculator rejects anything else, as a zero spread would collapse the curve into a spike.
- Precision: probabilities are computed with an error-function approximation accurate to about 7 decimal places, which is more than enough for practical work.
Frequently Asked Questions
How do I calculate normal distribution probabilities by hand without a calculator?
First convert your value to a z-score: \(z = (x - \mu) / \sigma\). Then look up the z-score in a standard normal (Z) table, which gives you \(P(Z x) = 1 - \Phi(z)\). For a range, subtract the two table values: \(P(a < X < b) = \Phi(z_b) - \Phi(z_a)\). For negative z, use the symmetry of the curve: \(\Phi(-z) = 1 - \Phi(z)\).
What is a good or normal z-score?
A z-score tells you how many standard deviations a value is from the mean. Values between −2 and +2 cover about 95% of cases and are considered typical. A z-score beyond ±2 is unusual (roughly the top or bottom 2.5%), and beyond ±3 is very rare (about 0.27% of data lies outside ±3). There is no "good" z-score in general — it depends on context: in quality control, z-scores near 0 are desired, while in some tests a high z-score is better.
Why is my probability bigger than 1 or the density bigger than 1?
A probability can never exceed 1, so if you see that, check your inputs — most often the standard deviation was entered in the wrong field or is extremely small. A density \(f(x)\) greater than 1, however, is perfectly valid: density is the height of the curve, not a probability. When \(\sigma\) is small the curve is narrow and tall, and its peak height is \(1/(\sigma\sqrt{2\pi})\), which can easily exceed 1.
Can I use this calculator for sample data or a t-distribution?
Use it with sample estimates of the mean and standard deviation only as an approximation, ideally with at least 30 observations. For small samples (roughly n < 30) where the population standard deviation is unknown, the t-distribution is more appropriate — it has heavier tails, so extreme probabilities differ noticeably. If your data are counts, proportions, or strongly skewed, a different distribution (Poisson, binomial, log-normal) may fit better.
How do I find the value x that corresponds to a given percentile?
This calculator works in the forward direction, but you can go backwards manually. Take the percentile as a decimal (the 90th percentile is 0.90), find the z-score whose cumulative probability equals it (for 0.90, z ≈ 1.2816), then un-standardize: \(x = \mu + z \cdot \sigma\). For example, with μ = 100 and σ = 15, the 90th percentile is 100 + 1.2816 × 15 ≈ 119.2.
What's the difference between the normal distribution and the standard normal distribution?
The standard normal distribution is just the special case with \(\mu = 0\) and \(\sigma = 1\). Every normal distribution can be transformed into it by standardizing: subtract the mean and divide by the standard deviation. This is why one universal Z-table serves all normal distributions. If your mean is 0 and standard deviation is 1, entering them here gives you the standard normal directly.
Does it matter whether I ask for P(X < x) or P(X ≤ x)?
No. For a continuous distribution like the normal, the probability of any single exact value is zero — the area under the curve at one point is zero. Therefore \(P(X < x)\) and \(P(X \le x)\) are identical, and the same holds for the upper tail. This only becomes an issue when approximating discrete data (like exam scores rounded to whole numbers) with a normal curve, where a continuity correction of ±0.5 can improve accuracy.