Formula & Calculator
Normal Distribution Probability Density
Calculates the relative likelihood of a continuous random variable taking a specific value under a normal (bell curve) distribution.
Interpretation
f(x) = (1/(σ√(2π)))·e^[−(x−μ)²/(2σ²)]. Bell‑shaped density for continuous data. Central to statistics. Used in many natural and social science applications.
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)
Frequently Asked Questions
f(x) = (1/(σ√(2π))) · e^(−(x−μ)²/(2σ²)). It describes the relative likelihood of a continuous random variable taking a specific value, forming the well‑known bell curve.
For continuous variables, the PDF is not a probability – it is a density. The actual probability is the area under the curve over an interval (integral). The PDF can exceed 1 for narrow distributions.
μ is the mean (centre), and σ is the standard deviation (spread). They completely define the normal distribution.
It is the foundation for many statistical tests (z‑tests, t‑tests, ANOVA, regression), and is used to model natural phenomena, measurement errors, and many more.
The standard normal has μ=0 and σ=1. It is denoted as Z and is obtained by standardising any normal variable: Z = (X−μ)/σ.
You integrate the PDF over a range. Because the integral has no closed form, we use Z‑tables or software to find cumulative probabilities (CDF).
About 68% of data lies within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. This is a quick guide to spread.
Many processes are assumed to be normally distributed. Control charts use this to monitor means and ranges, and to set specification limits.
The t‑distribution is used when the population standard deviation is unknown and estimated from the sample. It has heavier tails, especially for small sample sizes, to account for extra uncertainty.
The CLT states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population distribution (under certain conditions). This justifies many inferential procedures.