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Normal PDF

Probability Density Function of the Normal Distribution, defined by mean (μ) and standard deviation (σ).

Understand the formulaSee the free derivationOpen the full walkthrough

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

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Core idea

Overview

The Normal PDF defines the probability density of a continuous random variable that follows a symmetric, bell-shaped distribution. It is uniquely determined by the mean, which sets the location of the peak, and the standard deviation, which controls the spread or width of the curve.

When to use: Apply this equation when modeling natural phenomena like heights, weights, or test scores that cluster around a central average. It is also the appropriate choice when the Central Limit Theorem suggests that the sum of independent random variables tends toward a normal distribution.

Why it matters: This distribution is the foundation of frequentist statistics, enabling the calculation of P-values, Z-scores, and confidence intervals. Its mathematical properties allow scientists to quantify uncertainty and predict the likelihood of outcomes in everything from engineering to psychology.

Symbols

Variables

f(x) = Probability Density, x = Value, \mu = Mean, \sigma = Std Deviation

f (x)

Probability Density

V a r iab l e

x

Value

V a r iab l e

μ

Mean

V a r iab l e

σ

Std Deviation

V a r iab l e

Walkthrough

Derivation

Formula: Normal Probability Density Function (PDF)

The normal PDF defines the bell-shaped curve for a normal distribution; its mean $μ$ and standard deviation $σ$ control location and spread.

x is continuous.
The distribution is symmetric about $μ$ .

Identify Parameters:

These two parameters fully determine the position and width of the normal curve.

μ (mean), σ (standard deviation)

State the PDF:

The total area under the curve is 1, representing total probability.

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

Result

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

Source: Edexcel A-Level Mathematics — Statistics (Normal Distribution)

Free formulas

Rearrangements

Solve for $z$

Make z the subject

z = \frac{x - μ}{σ}

Start from the Normal Probability Density Function. The task is to identify and explicitly state the definition of the standardized variable `z` (z-score) as it appears within the exponent of the PDF.

Difficulty: 4/5

Solve for $f (x)$

Normal PDF with Standard Score

f (x) = \frac{1}{σ} ϕ (z)

This derivation demonstrates the relationship between the general Normal Probability Density Function and the standard normal distribution using the z-score substitution.

Difficulty: 2/5

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Visual intuition

Graph

The graph of the normal probability density function (PDF) is a symmetric, bell-shaped curve that is centered at the mean. It features a single global maximum at the mean and approaches the horizontal axis asymptotically as it extends toward positive and negative infinity. This shape indicates that data points are most concentrated near the mean, with probabilities decreasing symmetrically as values move further away from the center.

Graph type: exponential

Why it behaves this way

Intuition

A symmetric, bell-shaped curve centered at the mean, whose width is controlled by the standard deviation, representing the relative likelihood of observing different values of a continuous random variable.

μ

Mean of the distribution

Determines the central location or peak of the bell curve along the x-axis.

σ

Standard deviation of the distribution

Controls the spread or width of the bell curve; a larger

σ

results in a wider, flatter curve.

x - μ

Deviation of the random variable x from the mean \mu

Quantifies how far a specific observation x is from the average value.

(\frac{x - μ}{σ})^{2}

Squared standardized deviation (Z-score squared)

Measures how many standard deviations x is from

μ

, squared, emphasizing larger deviations and ensuring symmetry.

Signs and relationships

e^{-\frac{1}{2}\left(\frac{x-μ}{σ}\right)^2}: The negative exponent ensures that the probability density is highest at the mean (x= $μ$ ) and decreases as x moves away from $μ$ . The square \left( $\frac{x - μ}{σ}$ \right)^2 makes the decrease symmetric for values
\frac{1}{σ}: This factor ensures that the total area under the probability density curve integrates to 1. A larger standard deviation $σ$ means the distribution is more spread out, so the peak height must be proportionally lower

Free study cues

Insight

Canonical usage

The equation requires the random variable (x), mean (μ), and standard deviation (σ) to have consistent 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 function f(x) with a probability. f(x) has units of 1/unit(x) and does not directly give a probability unless integrated over an interval.

Dimension note

The term (x-μ)/σ is dimensionless, representing how many standard deviations a value is from the mean. The mathematical constant 2π is also dimensionless.

Unit systems

$x$ e.g., meters, kg, seconds, dollars · Represents the value of the continuous random variable.

$μ$ same as x · Represents the mean of the distribution.

$σ$ same as x · Represents the standard deviation of the distribution.

$f (x)$ 1/unit(x) · Represents the probability density at x.

One free problem

Practice Problem

In a distribution with a mean (μ) of 50 and a standard deviation (σ) of 10, calculate the probability density (y) at the point where x = 50.

Mean50

Std Deviation10

Value50

Solve for: $y$

Hint: When x equals the mean, the exponent term becomes zero, making the entire exponential part of the equation equal to 1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Distribution of heights in a population.

Study smarter

Tips

The total area under the entire density curve must always equal 1.
The function reaches its absolute maximum value when x is equal to μ.
The curve is perfectly symmetrical, meaning the mean, median, and mode are all located at μ.
Nearly 68% of the distribution lies within one standard deviation (σ) of the mean.

Avoid these traps

Common Mistakes

Evaluating PDF instead of CDF for probability.
Sigma in denominator.

Common questions

Frequently Asked Questions

The normal PDF defines the bell-shaped curve for a normal distribution; its mean $\mu$ and standard deviation $\sigma$ control location and spread.

Apply this equation when modeling natural phenomena like heights, weights, or test scores that cluster around a central average. It is also the appropriate choice when the Central Limit Theorem suggests that the sum of independent random variables tends toward a normal distribution.

This distribution is the foundation of frequentist statistics, enabling the calculation of P-values, Z-scores, and confidence intervals. Its mathematical properties allow scientists to quantify uncertainty and predict the likelihood of outcomes in everything from engineering to psychology.

Evaluating PDF instead of CDF for probability. Sigma in denominator.