Normal Distribution PDF Equation, Derivation & Rearrangements

Core idea

Overview

The Normal Distribution Probability Density Function (PDF) describes the distribution of a continuous random variable characterized by a symmetric, bell-shaped curve. It defines the relative likelihood of a variable taking on a specific value based on the mean (center) and standard deviation (spread) of the dataset.

When to use: Apply this equation when modeling natural phenomena like human height, blood pressure, or measurement errors that tend to cluster around a central average. It is essential when data follows the Central Limit Theorem, indicating that the sum of independent factors leads to a normal distribution.

Why it matters: It is the foundational distribution in statistics, enabling the calculation of z-scores, confidence intervals, and p-values. Understanding this curve allows researchers to predict the probability of specific outcomes in fields ranging from quality control to financial risk management.

Symbols

Variables

x = x Value, $μ$ = Mean, $σ$ = Std Dev, f(x) = Density

x

x Value

Variable

μ

Mean

Variable

σ

Std Dev

Variable

f(x)

Density

Variable

Walkthrough

Derivation

Formula: Normal Distribution PDF

The normal (Gaussian) distribution is a continuous distribution determined by mean $μ$ and standard deviation $σ$ , with a bell-shaped density.

$μ$ $\in$ $R$ .
$σ$ >0.
The density is normalized so total probability is 1.

1

State the Probability Density Function:

The parameter $μ$ shifts the center and $σ$ controls the spread. The constant 1/( $σ$ $2 π$ ) ensures the total area under the curve is 1.

f (x) = \frac{1}{σ 2 π} exp (- \frac{1}{2} (\frac{x - μ}{σ})^{2})

Note: This is usually introduced as a definition; deriving the normalization constant involves evaluating a Gaussian integral.

Result

f (x) = \frac{1}{σ 2 π} exp (- \frac{1}{2} (\frac{x - μ}{σ})^{2})

Source: Standard curriculum — Mathematical Statistics

Free formulas

Rearrangements

Solve for $y$

Normal Distribution PDF (Standardized Form)

y = \frac{1}{σ 2 π} e^{- 0.5 Z^{2}}

Transform the Probability Density Function of the Normal Distribution into its standardized form by substituting the Z-score and renaming the density function.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The normal distribution plot features a symmetric, bell-shaped curve that reaches its peak at the mean and tapers off toward the x-axis in both directions. It exhibits a central turning point (the mode/mean) and horizontal asymptotes as the curve approaches zero on either side of the horizontal axis. This shape illustrates the principle of probability density, where outcomes closer to the mean are significantly more likely to occur than those in the distant tails.

Graph type: exponential

Why it behaves this way

Intuition

A symmetric, bell-shaped curve centered at the mean $μ$ , where the height of the curve at any point x represents the relative likelihood of observing that value, and the curve's spread is determined by the standard

f(x)

The probability density at a specific value x for a continuous random variable.

A higher value indicates that x is more likely to occur relative to other values.

x

The specific value of the continuous random variable for which the probability density is being evaluated.

The particular outcome or measurement whose likelihood we are interested in.

μ

The mean (average) of the distribution, which determines its central location.

The center point of the bell curve, where the probability density is highest.

σ

The standard deviation, a measure of the typical spread or dispersion of data points around the mean.

Controls the width of the bell curve; a larger

σ

means a wider, flatter curve.

(x - μ) / σ

The z-score, representing how many standard deviations x is away from the mean \mu.

Standardizes the value x, allowing comparison of its relative position across different normal distributions.

Signs and relationships

e^{-...}: The negative sign in the exponent ensures that the probability density f(x) decreases as x moves further away from the mean $μ$ , both positively and negatively, creating the characteristic bell shape with a peak at $μ$ .
(x-μ)^2: Squaring the difference (x- $μ$ ) ensures that deviations from the mean, whether positive or negative, contribute equally to reducing the probability density, making the distribution symmetric around $μ$ .

Free study cues

Insight

Canonical usage

The random variable (x), mean (Î¼), and standard deviation (Ïƒ) must all share the same units, resulting in the probability density function f(x) having units inverse to those of x.

Common confusion

A common mistake is to confuse the probability density `f(x)` (which has units of `1/x`) with a dimensionless probability. `f(x)` can be greater than 1, while a probability cannot.

Dimension note

The exponent `- $\frac{1}{2}$ ( $\frac{x - μ}{σ}$ )^2` is dimensionless. This is because `(x- $μ$ )` and ` $σ$ ` must have the same units, making their ratio `( $\frac{x - μ}{σ}$ )` dimensionless, and thus the entire

Unit systems

$x$ Any consistent unit (e.g., m, kg, s, years) · The continuous random variable whose distribution is being described.

Î¼Same unit as x · The mean (expected value) of the distribution, representing its central tendency.

ÏƒSame unit as x · The standard deviation of the distribution, representing its spread or variability.

f(x)Inverse of the unit of x (e.g., m^-1, kg^-1, s^-1, years^-1) · The probability density at point x. It is not a probability itself; integrating f(x) over a range yields a dimensionless probability.

One free problem

Practice Problem

Calculate the probability density (y) at the mean (u) of 10, assuming a standard deviation (s) of 2.

x Value10

Mean10

Std Dev2

Solve for: $y$

Hint: When x equals u, the exponent term e to the power of zero becomes 1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Height distribution in population.

Study smarter

Tips

The value of y is highest when x equals the mean (u).
The total area under the density curve is always exactly 1.0.
The term (x-u)/s represents the z-score, or the number of standard deviations x is from the mean.
A smaller standard deviation (s) results in a taller, narrower peak.

Avoid these traps

Common Mistakes

Thinking this gives P(X<x) (CDF).
Forgetting the 1/sqrt(2pi) term.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The normal (Gaussian) distribution is a continuous distribution determined by mean \mu and standard deviation \sigma, with a bell-shaped density.

Apply this equation when modeling natural phenomena like human height, blood pressure, or measurement errors that tend to cluster around a central average. It is essential when data follows the Central Limit Theorem, indicating that the sum of independent factors leads to a normal distribution.

It is the foundational distribution in statistics, enabling the calculation of z-scores, confidence intervals, and p-values. Understanding this curve allows researchers to predict the probability of specific outcomes in fields ranging from quality control to financial risk management.

Thinking this gives P(X<x) (CDF). Forgetting the 1/sqrt(2pi) term.

Height distribution in population.

The value of y is highest when x equals the mean (u). The total area under the density curve is always exactly 1.0. The term (x-u)/s represents the z-score, or the number of standard deviations x is from the mean. A smaller standard deviation (s) results in a taller, narrower peak.

References

Sources

Wikipedia: Normal distribution
A First Course in Probability by Sheldon Ross
Wikipedia: Probability density function
Casella and Berger, Statistical Inference
Wikipedia: Central Limit Theorem
Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and Statistics for Engineers and Scientists (9th ed.). Pearson.
Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
Standard curriculum — Mathematical Statistics

Normal Distribution PDF

Overview

Variables

Derivation

State the Probability Density Function:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Mean Value

Standard Deviation

Z-Score

Frequently Asked Questions

Sources