Enter a mean (μ), standard deviation (σ), and random variable (x) to compute the Probability Density Function (PDF) for the normal distribution. Your results include the PDF value and the corresponding standard normal (z-score) — giving you a precise measure of the likelihood density at any point along the curve.
PDF Value f(x)
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Standard Normal (z-score)
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Variance (σ²)
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Peak PDF at Mean
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The PDF describes the relative likelihood of a continuous random variable taking on a given value. Unlike discrete probabilities, the PDF does not give a direct probability at a single point — instead, the area under the curve between two values gives the probability that the variable falls in that range. For the normal distribution, the PDF is bell-shaped and symmetric around the mean.
The normal distribution PDF is computed as f(x) = (1 / (σ√(2π))) × e^(−(x−μ)² / (2σ²)), where μ is the mean, σ is the standard deviation, and x is the point of evaluation. The z-score is calculated as z = (x − μ) / σ, which tells you how many standard deviations x is from the mean.
The PDF value f(x) represents the density of probability at the specific point x — it is not a probability itself, but a density. Higher f(x) values indicate regions of the distribution where outcomes are more concentrated. The area under the entire PDF curve always equals 1, representing 100% probability.
A standard deviation of zero would imply all values are concentrated at a single point, which is a degenerate (non-continuous) distribution and makes the PDF formula undefined due to division by zero. Valid standard deviations are strictly positive numbers greater than zero.
The z-score (standard normal value) measures how many standard deviations the point x is from the mean μ. A z-score of 0 means x equals the mean, +1 means one standard deviation above, and −1 means one standard deviation below. It allows comparison across different normal distributions by standardizing the values.
The PDF of the normal distribution reaches its peak (maximum value) exactly at x = μ (the mean). This peak value equals 1 / (σ√(2π)). The curve then falls off symmetrically in both directions from the mean.
The normal distribution PDF appears in statistics, quality control, finance, natural sciences, and machine learning. It is used to model height, test scores, measurement errors, stock returns, and many other real-world phenomena. Computing the PDF at a specific point helps evaluate the relative likelihood of observing that value under the assumed distribution.
The PDF (Probability Density Function) gives the density at a single point x, while the CDF (Cumulative Distribution Function) gives the total probability that a variable falls below x. The CDF is the integral of the PDF from −∞ to x. If you want the probability of an interval, you integrate the PDF (or take the difference in CDF values) between the two endpoints.