Normal Distribution Calculator
Evaluate the probability density function f(x) = 1/(σ√(2π)) · e^−(x−μ)²/(2σ²), compute tail and interval probabilities via the cumulative distribution, and invert probabilities back to x-values — with a live bell-curve diagram, shaded probability regions, and a full step-by-step solution.
f(x) is a density, not a probability
The PDF f(x) tells you how concentrated the probability is near x, not the probability that X equals x (which is zero for any continuous distribution). Only areas under the curve correspond to probabilities. That is why f(x) can exceed 1 when σ is small — for σ = 0.1, the peak density is about 4.0, which is perfectly valid.
68–95–99.7 — the empirical rule
For any normal distribution, about 68.27% of the probability lies within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ. These numbers are universal — they do not depend on μ or σ. They follow directly from the CDF of the standard normal: Φ(1) − Φ(−1) = 0.6827, and so on. This is the source of the ubiquitous "6σ" quality target in manufacturing.
μ shifts, σ stretches
Changing μ slides the entire curve left or right without altering its shape. Changing σ stretches or compresses the curve horizontally and scales it vertically to preserve the total area of 1. The z-transform z = (x − μ)/σ converts any normal distribution to the standard normal (μ=0, σ=1), which is why a single z-table serves every normal distribution.
The central limit theorem makes it ubiquitous
Whenever many independent random effects add together, their sum tends toward a normal distribution regardless of the individual distributions — this is the central limit theorem. That is why measurement errors, heights, test scores, manufacturing tolerances, and financial returns (in calm markets) all look roughly normal. It is the single most important distribution in applied statistics.
How to Use This Calculator
The normal distribution is defined by two parameters (μ, σ) and described by two functions: the probability density f(x) and the cumulative distribution Φ(z). This tool evaluates both, plus tail and interval probabilities, and inverts them back to x-values.
1. Load a scenario preset
Click any preset — standard normal, IQ, SAT, adult height, body temperature, manufacturing tolerance, stock return — to load its μ, σ and x values in one click. Or type your own parameters directly.
2. Choose the calculation type
Pick f(x) for the density value, P(X ≤ x) for the lower tail, P(X ≥ x) for the upper tail, P(x₁ ≤ X ≤ x₂) for an interval, or x from p to invert a probability back to a value.
3. Enter μ and σ
μ is the mean (any real number), σ is the standard deviation (strictly positive). Everything is expressed in the same units as your data — cm, points, °C, dollars.
4. Enter x, x₁/x₂, or p
Depending on the mode, you enter a single x value, an interval (x₁ and x₂), or a probability p. The remaining inputs are hidden until they are relevant.
5. Read the result and the diagram
The result card shows the primary answer plus z, the complementary tail probability, and a colour-coded verdict. The diagram below the calculator draws the bell curve and shades the relevant probability region so you can see exactly what is being measured.
6. Switch to Advanced for depth
Advanced mode adds a density comparison chart (three σ values), a sensitivity check on μ, σ and x, and a metrics panel with the interquartile range, median, common confidence intervals, and a z-table reference.
What This Tool Offers
Built around the classic probability density f(x) = 1/(σ√(2π)) · e^−(x−μ)²/(2σ²) and the standard normal CDF Φ(z), the fundamental pair of functions in applied statistics.
Five calculation modes
PDF, lower tail, upper tail, interval, and inverse CDF. Every mode returns the same full set of derived values (z, complementary tail, percentile) so you never have to switch modes just to see an adjacent result.
Live bell-curve diagram
A scaled Gaussian curve with μ marked, ±1σ, ±2σ, ±3σ bands shaded, the current x marked, and the mode-specific probability region highlighted so you can see exactly what the number means.
Full step-by-step solution
Every mode shows the intermediate arithmetic — the z-score computation, the exponent, the PDF factor 1/(σ√(2π)), the CDF evaluation, and for inverse mode the probit computation — so you can verify the result by hand.
Multi-format probabilities
Each probability is shown as a decimal (0.9750), a percentage (97.50%), and a complement (1 − 0.9750 = 0.0250). Percentiles and common critical values (1.28, 1.645, 1.96, 2.576) are shown in the metrics panel.
Dual comparison charts
The result column shows the standard normal CDF Φ(z) with the current point marked; advanced mode adds a density-curve comparison showing how the bell changes when σ is scaled by 0.7 and 1.4.
Reference tables
A z-table lookup with the classic critical values (0.90, 0.95, 0.975, 0.99, 0.995, 0.999), typical real-world μ and σ values, and a full variable reference table.
Reference Tables & Conversions
A standard normal z-table, common critical values, typical real-world normal distributions, and the variable reference.
Standard Normal Z-Table — Φ(z) = P(Z ≤ z)
Cumulative probability of the standard normal. Φ(−z) = 1 − Φ(z). For any normal, P(X ≤ x) = Φ((x − μ)/σ).
| z | Φ(z) | P(Z ≥ z) | Two-tailed α |
|---|
Common Critical Values
The z-values that every statistics textbook cites. Multiply by σ and add μ to convert to any normal distribution.
| Confidence | z (one-tailed) | z (two-tailed) | x = μ + z·σ |
|---|
Typical Real-World Normal Distributions
Representative μ and σ values for commonly modelled quantities.
| Quantity | μ | σ | Unit | Typical range (μ ± 2σ) |
|---|
Variable Reference
| Symbol | Meaning | Formula | Notes |
|---|---|---|---|
| x | Value of the random variable | — | Any real number |
| μ | Mean (expectation) | E[X] | Location of the peak |
| σ | Standard deviation | √Var(X) | Strictly positive; width of the curve |
| σ² | Variance | Var(X) | σ squared |
| z | Standard score | (x − μ)/σ | Units of σ from the mean |
| f(x) | Probability density | 1/(σ√(2π))·e^−z²/2 | Not a probability; integrate to get one |
| Φ(z) | Standard normal CDF | ∫₋∞^z f(t)dt | Tabulated; used to get tail probabilities |
| P(X ≤ x) | Lower-tail probability | Φ(z) | Percentile of x in the distribution |
| P(X ≥ x) | Upper-tail probability | 1 − Φ(z) | Complement of the lower tail |
What Does f(x) = 1/(σ√(2π)) · e^−(x−μ)²/(2σ²) Measure?
The normal (Gaussian) distribution is the most important continuous probability distribution in statistics. Its probability density function f(x) describes how probability is concentrated around the mean μ, with the spread controlled by the standard deviation σ. The function is symmetric about μ, peaks at x = μ with value 1/(σ√(2π)), and decays to zero in both tails. Although f(x) itself is not a probability, its integral over any interval [a, b] gives the probability that the random variable falls in that interval.
The Normal Probability Density Function
The density of a normal random variable with mean μ and standard deviation σ is:
In terms of the standard score z = (x − μ)/σ, this simplifies to:
The cumulative distribution function (CDF), which gives probabilities directly, is:
The three probability statements that matter in practice are:
P(X ≥ x) = 1 − Φ((x − μ)/σ)
P(x₁ ≤ X ≤ x₂) = Φ((x₂ − μ)/σ) − Φ((x₁ − μ)/σ)
To invert a cumulative probability back to a value, use the probit function z = Φ⁻¹(p):
Normal Distribution Variables
| Symbol | Meaning | Notes |
|---|---|---|
| μ | Mean of the distribution | Location parameter — peak of the bell |
| σ | Standard deviation | Scale parameter — width of the bell |
| σ² | Variance | The square of σ |
| x | Point of interest | Any real number |
| z | Standard score | (x − μ)/σ; unitless |
| f(x) | Probability density | Peak value 1/(σ√(2π)) ≈ 0.3989/σ |
| Φ(z) | Standard normal CDF | Monotonically increasing from 0 to 1 |
| φ(z) | Standard normal PDF | Peak value 1/√(2π) ≈ 0.3989 at z = 0 |
How μ and σ Shape the Curve
The mean μ shifts the entire curve along the x-axis. Adding a constant c to μ moves the whole distribution right by c and leaves everything else unchanged — in particular, the shape, the width, and the total area of 1 are all preserved. This is why the z-score transformation z = (x − μ)/σ converts every normal distribution to the same standard normal, no matter what μ is.
The standard deviation σ stretches or compresses the curve horizontally. A larger σ makes the bell flatter and wider; a smaller σ makes it taller and narrower. The peak density scales as 1/(σ√(2π)), so doubling σ halves the peak height. Since the total area under the curve is fixed at 1, stretching horizontally necessarily flattens vertically — this is the conservation-of-probability rule.
The most important practical consequence is the empirical 68–95–99.7 rule. Regardless of μ and σ, about 68.27% of the probability lies within one σ of the mean, 95.45% within two σ, and 99.73% within three σ. If you remember nothing else about the normal distribution, remember those three numbers — they are what make confidence intervals, quality control charts, and hypothesis tests work.
Normal Distribution Calculation Examples
z = 1.96 · φ(z) = 1/√(2π) · e^−1.96²/2 = 0.3989 × 0.1468 = 0.0584
f(x) = φ(z)/σ = 0.0584 · P(X ≤ 1.96) = Φ(1.96) = 0.9750
z = (130 − 100)/15 = 2.00
P(X ≤ 130) = Φ(2.00) = 0.9772
P(X ≥ 130) = 1 − 0.9772 = 0.0228 → about 2.3% of the population
z₁ = (165 − 175.3)/7.1 = −1.451 · z₂ = (185 − 175.3)/7.1 = 1.366
P = Φ(1.366) − Φ(−1.451) = 0.9141 − 0.0735 = 0.8406 (about 84%)
z = Φ⁻¹(0.95) = 1.6449
x = 500 + 100 × 1.6449 = 664.49
So 95% of SAT scores fall below 664.49
Frequently Asked Questions
The probability density function (PDF) f(x) describes how concentrated the probability is at each point — it is the height of the bell curve. It is not a probability itself, and it can take values greater than 1 (for σ small enough). The cumulative distribution function (CDF) Φ(z) is the integral of the PDF from −∞ to z, and it gives the actual probability P(X ≤ x). The PDF tells you where the probability is concentrated; the CDF tells you how much of it lies to the left of a given point.
Because f(x) is a density, not a probability. Only the integral of f over an interval is a probability, and the integral of a valid PDF over the whole real line equals 1. For a normal with σ = 0.1, the peak density is 1/(0.1·√(2π)) ≈ 3.989. This is perfectly valid — the probability of being within a small interval around the peak is still a small number, because the interval must be multiplied by the density to get the probability. A density can be any positive number; only its integral is constrained.
The z-score z = (x − μ)/σ expresses how many standard deviations the value x lies from the mean. It is unitless, so it lets you compare values from different normal distributions on a common scale. A z-score of 2 means "two standard deviations above the mean" whether you are talking about IQ (μ=100, σ=15), adult height (μ=175cm, σ=7cm), or daily stock returns (μ=0.05%, σ=1.2%). The z-transform maps every normal distribution onto the same standard normal curve, which is why a single z-table can serve every normal distribution.
It states that for any normal distribution, about 68.27% of the probability lies within one σ of the mean, 95.45% within two σ, and 99.73% within three σ. These numbers come directly from the standard normal CDF: Φ(1) − Φ(−1) = 0.6827, Φ(2) − Φ(−2) = 0.9545, Φ(3) − Φ(−3) = 0.9973. They do not depend on μ or σ. The rule is the foundation for confidence intervals, control charts, and the "6σ" quality target in manufacturing — six σ corresponds to a defect rate of about 2 per billion.
A 95% confidence interval for the mean of a normal population is x̄ ± 1.96 · σ/√n, where x̄ is the sample mean, σ is the population standard deviation, and n is the sample size. If σ is unknown and n is small, replace 1.96 with the corresponding t-value from the Student's t-distribution and use the sample standard deviation s. In general, a (1 − α) confidence interval uses z = Φ⁻¹(1 − α/2) as the multiplier — 1.645 for 90%, 1.96 for 95%, 2.576 for 99%. This calculator's inverse mode can compute those multipliers directly.
Real data often deviates from normality. Common cases where the normal distribution is inappropriate: heavy-tailed data (financial returns, insurance claims), skewed data (income, house prices, reaction times), bounded data (percentages, proportions, counts), and any quantity that cannot be negative but has small mean relative to spread. Before applying a normal model, check the data with a histogram, Q-Q plot, or a normality test (Shapiro-Wilk, Anderson-Darling). If the data is clearly non-normal, consider a log-normal, gamma, Weibull, or other distribution instead.
The central limit theorem (CLT) states that the sum (or average) of many independent and identically distributed random variables tends toward a normal distribution, regardless of the shape of the underlying distribution, as the number of variables grows. This is why measurement errors (sums of many small independent errors), heights (sums of many genetic and environmental factors), and aggregated financial returns all look approximately normal. The CLT is the reason the normal distribution appears everywhere in applied statistics, and it is the justification for using z-tests and t-tests even when the raw data is not itself normal.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f(x) | Probability density at x | |
| x | Value of interest | |
| μ | Distribution mean | |
| σ | Distribution standard deviation |
What it means
The normal (Gaussian) distribution is the most important probability distribution in statistics. Its probability density function is symmetrical, bell‑shaped, with parameters μ (mean) and σ (standard deviation). It arises from the central limit theorem and is used to model many natural phenomena (heights, errors, test scores). It is the basis for many statistical tests (z‑tests, t‑tests). The density function gives the probability of a value in an infinitesimal interval. Understanding the normal distribution is essential for inference, confidence intervals, and for many parametric methods. The standard normal (z‑distribution) is a special case with μ=0, σ=1.
Worked example
Normal Distribution PDF – Two Examples
Real‑World| Parameter | Value |
|---|---|
| x | 100 |
| μ | 100 |
| σ | 15 |
| Parameter | Value |
|---|---|
| x | 115 |
| μ | 100 |
| σ | 15 |
Common mistakes
- Normal PDF: Probability density function – gives the height of the curve at x, not a probability.
- Mean μ and standard deviation σ: Parameters of the distribution.
- Symmetry: The normal distribution is symmetric about μ.
- Total area: The integral of f(x) over all x equals 1.
- Units: The density has units of 1/x (e.g., if x is in cm, f(x) has units of 1/cm).
Applications
The normal distribution probability density function describes the classic bell‑shaped curve, characterised by the mean μ and standard deviation σ. It is the most important distribution in statistics, used to model countless natural and engineered phenomena. Engineers use it for process capability analysis, tolerance design, and measurement uncertainty. In quality control, it underlies control charts. In finance, it models asset returns. In research, it is assumed for many parametric tests. By understanding the normal distribution, professionals can compute probabilities, set specification limits, and perform hypothesis tests. Its properties are the basis for many statistical methods, making it essential for anyone working with quantitative data.
- Process capability analysis and specification setting
- Measurement uncertainty and error propagation
- Control charts for statistical process control
- Financial modelling and risk assessment
- Parametric hypothesis testing (t‑tests, ANOVA)
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