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Normal Distribution



Key Terms
The Probabilistic Experiment Behind normal distribution
Normal Distribution Notation
Parameters
Probability Density Function (PDF) and Support (Range)
Cumulative Distribution Function (CDF)
Expected Value (Mean)
Variance and Standard Deviation
Mode and Median
Quantiles/Percentiles
Real-World Examples and Common Applications
Interactive Calculator
Special Cases
Properties
Normal Distribution at a Glance
Normal Distribution FAQ



Normal Distribution: Natural Variability Around a Typical Value


Many real-world measurements arise from the accumulation of many small, independent influences acting together. The probabilistic experiment behind the normal distribution describes observing a quantity that fluctuates continuously around a typical or central value, with deviations in either direction becoming less likely as they grow larger. The random variable represents a measurement, not a count, and extreme values are possible but increasingly rare.

Key Terms

Normal Distribution— symmetric bell-shaped distribution with parameters μ\mu, σ2\sigma^2
Expected Value— E[X]=μE[X] = \mu
Variance— Var⁡(X)=σ2\operatorname{Var}(X) = \sigma^2
Standard Deviation— σ\sigma, the spread parameter
Probability Density Function— the bell curve f(x)f(x) defining the distribution

See All Probability Definitions →


The Probabilistic Experiment Behind normal distribution


The probabilistic experiment underlying the normal distribution arises when an outcome is shaped by many small, independent influences, none of which dominates the result. Each influence nudges the outcome slightly upward or downward, and the final value reflects the combined effect of all these contributions.

This experiment is not defined by repetition of identical trials, but by aggregation. The core assumption is that deviations occur naturally in both directions, are roughly symmetric, and tend to cancel out when added together. Extreme outcomes are possible, but increasingly unlikely because they require many influences to align in the same direction.

The normal distribution emerges as a consequence of stability: when numerous independent factors interact, the resulting variability concentrates around a central value. The spread reflects how strong those individual influences are, while the center reflects their average balance point.

This experiment explains why many natural and measurement-based quantities cluster around a typical value with gradual falloff on both sides.

Example
Human height results from genetics, nutrition, environment, and random biological variation. No single factor determines the outcome, but together they produce values concentrated around an average, with fewer extremely short or tall individuals.

Normal Distribution Notation

Notation

Normal Distribution Notation

The normal is the only distribution with reserved letters: a capital for its standard variable, a Greek pair for its density and accumulation, and a declaration whose second slot is the most-checked convention in probability. The general warning points here — this is the minefield's home ground. All marks are catalogued among the probability symbols.
The declaration tilde and the capital/lowercase split come from random variable notation and distribution notation; μ\mu and σ2\sigma^2 from expected value and variance notation; FF and F−1F^{-1} from CDF notation.
X∼N(μ,σ2)X \sim N(\mu, \sigma^2) vs N(μ,σ)N(\mu, \sigma)
normal with mean mu and — check the source — variance or standard deviation
The second slot is a convention, not a fact: most textbooks put the variance there, while much software and many applied texts put the standard deviation — R's and Python's normal functions take the SD. The declaration form itself follows the family pattern; N\mathcal{N} and the spelled-out Normal⁡\operatorname{Normal} are typographic variants, and the naked N(μ,σ2)N(\mu,\sigma^2) names the distribution while X∼X \sim attaches it.
CasesMagnitude sometimes betrays the convention — a height model reading N(175,49)N(175, 49) is carrying σ=7\sigma = 7 squared, one reading N(175,7)N(175, 7) is carrying the SD raw — but only the source's definition settles it; the standard slots N(0,1)N(0, 1) agree in both conventions, which is why the ambiguity hides so well.
Do not confuseNothing louder than itself: reading the wrong convention silently rescales every interval and every z-score by a square root — the exact trap the standard-error slot inherits one level up.
Z∼N(0,1)Z \sim N(0,1), z=x−μσz = \dfrac{x - \mu}{\sigma}
capital Z — the standard normal variable; little z — a score in sigma units
ZZ is reserved: by convention the letter means a standard normal variable unless a text says otherwise — no other distribution owns a letter. Lowercase zz is the score: a data value recentred and rescaled into σ\sigma-units, the capital/lowercase convention carrying the variable/value split one more time.
CasesThe subscripted zpz_p marks standard-normal quantiles — Quantiles/Percentiles below builds every percentile from them via xp=μ+σzpx_p = \mu + \sigma z_p; the sample-mean cousin Zn=(Xˉn−μ)/(σ/n)Z_n = (\bar{X}_n - \mu)/(\sigma/\sqrt{n}) belongs to central limit theorem notation — same recipe, standard error in the denominator.
Do not confuseA distance. z=−2z = -2 is two standard deviations below the mean, not "a distance of −2" — the sign carries direction, and dropping it collapses the two tails the score exists to tell apart.
φ(z)\varphi(z), Φ(z)\Phi(z)
little phi — the standard normal density; capital Phi — its CDF
The Greek pair is the normal's second reserved privilege: lowercase φ\varphi (or ϕ\phi) names the standard density, capital Φ\Phi its accumulation — the ff/FF case convention transposed into Greek. Cumulative Distribution Function above defines Φ\Phi and the reduction F(x)=Φ(x−μσ)F(x) = \Phi\big(\tfrac{x-\mu}{\sigma}\big) that makes one table serve every normal.
CasesΦ\Phi has no closed form — it is defined by its integral and evaluated by table or software; its inverse Φ−1\Phi^{-1} is the probit, the function behind every zpz_p lookup; when a text writes φ\varphi and Φ\Phi without introduction, the standard normal is being assumed silently.
Do not confuseThe letter's other tenants. Handwritten φ\varphi collides with the empty set's ∅\emptyset — a loop through a stroke either way — and ϕ\phi moonlights as an angle and as the golden ratio; inside probability, though, the pair is unambiguous: density below, accumulation above.

Parameters


μ (mu): mean or center of the distribution, where μ∈R\mu \in \mathbb{R}

σ (sigma): standard deviation, measuring spread around the mean, where σ>0\sigma > 0

The normal distribution is fully characterized by these two parameters.

μ determines the location (where the peak sits on the number line), while σ controls the spread (how wide or narrow the bell curve is).

Variance is σ2\sigma^2, but we typically use σ\sigma as the primary parameter since it's in the same units as the data.
Normal curves N(0, 1), N(3, 1) and N(0, 2^2): mu moves the bell, sigma widens and lowers itxy−6−5−4−3−2−11234567blue: μ = 0, σ = 1amber: μ = 3, σ = 1navy: μ = 0, σ = 2μ moves itσ widens itμ slides the bell; σ widens it and lowers the peak
What μ and σ each do

Three normal curves. Changing μ from 0 to 3 (blue to amber) slides the whole bell to the right without changing its shape. Changing σ from 1 to 2 (blue to navy) keeps the centre but spreads the bell out, and since the area must stay 1 the peak drops to half its height. Location and spread are controlled separately, which is why two numbers describe every normal distribution.

Doubling σ halves the peak height: the total area under every normal curve is 1.

Probability Density Function (PDF) and Support (Range)


Probability Density Function (PDF)


The probability density function (PDF) of a normal distribution is given by:

Normal PDF
f(x)=1σ2πe−(x−μ)22σ2,−∞<x<∞f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < \infty
Learn more about this formula: Normal PDF →


Intuition Behind the Formula


Bell-Shaped Curve: The normal distribution creates a symmetric, bell-shaped curve centered at μ\mu.

Parameters:
• μ\mu: The mean determines where the center of the bell sits
• σ\sigma: The standard deviation controls the width of the bell
• σ2\sigma^2: The variance (σ\sigma squared) appears in the exponent

Support (Range of the Random Variable):
• The random variable XX can take any real value: (−∞,+∞)(-\infty, +\infty)
• While theoretically unbounded, approximately 99.7% of values fall within μ±3σ\mu \pm 3\sigma
• The support is the entire real line

Logic Behind the Formula:
• 1σ2π\frac{1}{\sigma\sqrt{2\pi}}: Normalization constant ensuring total area equals 1
• (x−μ)2(x-\mu)^2: Squared distance from the mean (makes curve symmetric)
• e−(x−μ)22σ2e^{-\frac{(x-\mu)^2}{2\sigma^2}}: Exponential decay as you move away from μ\mu
• The total area under the curve equals 1:

PDF Normalization
∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x)\,dx = 1
Learn more about this formula: PDF Normalization →


Practical Example: Human heights follow approximately a normal distribution. If adult male heights have μ=175\mu = 175 cm and σ=7\sigma = 7 cm, then X∼N(175,49)X \sim N(175, 49). The PDF tells us the relative likelihood of observing different heights, with the peak at 175 cm and decreasing probability as we move toward very short or very tall individuals.

Normal (Gaussian) Distribution

Bell-shaped curve, symmetric around mean

Explanation

The normal distribution, also known as the Gaussian distribution, is the most important probability distribution in statistics. Its probability density function is f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, where μ\mu is the mean and σ\sigma is the standard deviation. The expected value is E[X]=μE[X] = \mu and the variance is Var(X)=σ2\text{Var}(X) = \sigma^2. The normal distribution appears naturally in many phenomena due to the Central Limit Theorem. Applications include measurement errors, heights and weights in populations, IQ scores, financial returns, and any process that results from many small independent random effects.

0.000.080.160.240.320.40-4-3.0-2.0-1.0-0.01.02.03.04.0xf(x)
Normal density, mean 0, standard deviation 1

The bell is symmetric about the mean, peaks there at about 0.4 for unit standard deviation, and has inflection points one standard deviation to each side. The total area is 1, with about 68 percent of it inside one standard deviation of the mean. Change the parameters and see the bell shift and stretch on the PDF visualizer.

Its cumulative form has no closed formula, as the next section explains.

Cumulative Distribution Function (CDF)

Cumulative Distribution Function (CDF)


The cumulative distribution function (CDF) gives the probability that XX is less than or equal to a specific value:

Cumulative Distribution Function (CDF)
F(x)=P(X≤x)=∫−∞x1σ2πe−(t−μ)22σ2 dtF(x) = P(X \leq x) = \int_{-\infty}^{x} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(t-\mu)^2}{2\sigma^2}}\,dt

Standard Normal CDF: For the standard normal distribution N(0,1)N(0, 1), the CDF is traditionally denoted by Φ(z)\Phi(z):

Cumulative Distribution Function (CDF)
Φ(z)=P(Z≤z)=∫−∞z12πe−t22 dt\Phi(z) = P(Z \leq z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}}\,dt

Key Properties:
• F(−∞)=0F(-\infty) = 0 and F(+∞)=1F(+\infty) = 1
• F(μ)=0.5F(\mu) = 0.5 (the mean is the 50th percentile)
• The CDF is strictly increasing and S-shaped
• Any normal CDF can be expressed using the standard normal: F(x)=Φ(x−μσ)F(x) = \Phi\left(\frac{x-\mu}{\sigma}\right)

Practical Use: The normal CDF has no closed-form expression, so we use tables, calculators, or software. To find P(X≤180)P(X \leq 180) when X∼N(175,49)X \sim N(175, 49), convert to standard normal: Z=(180−175)/7≈0.714Z = (180-175)/7 \approx 0.714, then look up Φ(0.714)≈0.762\Phi(0.714) \approx 0.762, meaning about 76.2% of values fall below 180.

Normal Distribution CDF

Visualizing probability accumulation for normal (Gaussian) distribution

Normal (Gaussian) - CDF

S-shaped curve, steepest at mean

CDF Explanation

The cumulative distribution function (CDF) of the normal distribution is F(x)=12[1+erf(x−μσ2)]F(x) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right], where erf is the error function. The CDF gives the probability P(X≤x)P(X \leq x) that a normally distributed random variable is less than or equal to xx. The S-shaped curve is symmetric around the mean μ\mu, where F(μ)=0.5F(\mu) = 0.5. The curve is steepest at the mean and flattens out in the tails. About 68% of values fall within one standard deviation of the mean (F(μ+σ)−F(μ−σ)≈0.68F(\mu + \sigma) - F(\mu - \sigma) \approx 0.68), 95% within two standard deviations, and 99.7% within three standard deviations.
0.00.20.40.60.81.0-4-3.0-2.0-1.0-0.01.02.03.04.0xF(x) = P(X ≤ x)
Normal CDF, mean 0, standard deviation 1

The S-curve passes through one half at the mean and approaches 0 and 1 in the tails; its slope at any point is the density there, which is why it is steepest at the centre. Values are read from tables or software because the integral has no elementary form. Move along the curve and read cumulative probabilities on the continuous CDF visualizer.

The mean and variance can nevertheless be computed exactly.

Expected Value (Mean)


The mean of a continuous random variable requires integrating across all possible values, with each value weighted by its density. For the normal distribution, this computation follows the standard continuous expected value formula:

Expected Value (Continuous)
E[X]=∫−∞∞x⋅f(x) dxE[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx
Learn more about this formula: Expected Value (Continuous) →


Formula


Normal Mean
E[X]=μE[X] = \mu
Learn more about this formula: Normal Mean →


Where:
μ\mu = the location parameter of the distribution

Derivation and Intuition


The normal distribution has PDF:

Expected Value (Mean)
f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Computing the expected value:

Expected Value (Mean)
E[X]=∫−∞∞x⋅1σ2πe−(x−μ)22σ2 dxE[X] = \int_{-\infty}^{\infty} x \cdot \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \, dx

Using the substitution z=x−μσz = \frac{x-\mu}{\sigma}, we get x=μ+σzx = \mu + \sigma z and dx=σ dzdx = \sigma \, dz:

Expected Value (Mean)
E[X]=∫−∞∞(μ+σz)⋅12πe−z22 dzE[X] = \int_{-\infty}^{\infty} (\mu + \sigma z) \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \, dz

Expected Value (Mean)
E[X]=μ∫−∞∞12πe−z22 dz+σ∫−∞∞z⋅12πe−z22 dzE[X] = \mu \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \, dz + \sigma \int_{-\infty}^{\infty} z \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \, dz

The first integral equals 1 (total probability of standard normal). The second integral equals 0 by symmetry (integrating an odd function over a symmetric interval).

Therefore: E[X]=μ⋅1+σ⋅0=μE[X] = \mu \cdot 1 + \sigma \cdot 0 = \mu

The result E[X]=μE[X] = \mu reveals a fundamental property of the normal distribution: the parameter μ\mu directly represents the mean. The distribution is perfectly symmetric around μ\mu, making it simultaneously the center of mass, the balance point, and the most likely region of values.

Example


Consider human heights modeled as normally distributed with μ=170\mu = 170 cm and σ=10\sigma = 10 cm:

Expected Value (Mean)
E[X]=170 cmE[X] = 170 \text{ cm}

The expected height is exactly 170 cm, which is the center of the distribution. Half of all heights fall above this value, and half fall below it.

To compute the expected value of a normal distribution for your own parameters, step by step, use the normal expected value calculator.

Variance and Standard Deviation


The variance of a continuous random variable quantifies the spread of values around the mean. For continuous distributions, it is calculated through integration:

Variance and Standard Deviation
Var(X)=E[(X−μ)2]=∫−∞∞(x−μ)2f(x) dx\mathrm{Var}(X) = \mathbb{E}[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) \, dx

Alternatively, using the computational formula:

Variance and Standard Deviation
Var(X)=E[X2]−μ2\mathrm{Var}(X) = \mathbb{E}[X^2] - \mu^2

For the normal distribution, this calculation yields a direct relationship with the distribution parameter.

Formula


Normal Variance
Var(X)=σ2\mathrm{Var}(X) = \sigma^2
Learn more about this formula: Normal Variance →


Where:
σ\sigma = the scale parameter of the distribution

Derivation and Intuition


The normal distribution has PDF:

Variance and Standard Deviation
f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

We know from the expected value section that μ=E[X]\mu = E[X]. Computing the variance:

Variance and Standard Deviation
Var(X)=∫−∞∞(x−μ)2⋅1σ2πe−(x−μ)22σ2 dx\mathrm{Var}(X) = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \, dx

Using the substitution z=x−μσz = \frac{x-\mu}{\sigma}, we get (x−μ)=σz(x-\mu) = \sigma z and dx=σ dzdx = \sigma \, dz:

Variance and Standard Deviation
Var(X)=∫−∞∞(σz)2⋅12πe−z22 dz=σ2∫−∞∞z2⋅12πe−z22 dz\mathrm{Var}(X) = \int_{-\infty}^{\infty} (\sigma z)^2 \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \, dz = \sigma^2 \int_{-\infty}^{\infty} z^2 \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} \, dz

The integral equals 1 (this is the variance of the standard normal distribution).

Therefore: Var(X)=σ2⋅1=σ2\mathrm{Var}(X) = \sigma^2 \cdot 1 = \sigma^2

The result Var(X)=σ2\mathrm{Var}(X) = \sigma^2 reveals that the parameter σ\sigma directly controls the spread of the distribution. Just as μ\mu determines the center, σ2\sigma^2 determines how dispersed values are around that center. A larger σ\sigma produces a wider, flatter bell curve; a smaller σ\sigma produces a narrow, peaked distribution.

Standard Deviation


Variance and Standard Deviation
σ\sigma

The standard deviation is simply the parameter σ\sigma itself, which is why it's called the standard deviation parameter. This makes interpretation straightforward: about 68% of values fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations.

Example


Consider human heights modeled as normally distributed with μ=170\mu = 170 cm and σ=10\sigma = 10 cm:

Variance and Standard Deviation
Var(X)=(10)2=100 cm2\mathrm{Var}(X) = (10)^2 = 100 \text{ cm}^2

Variance and Standard Deviation
σ=10 cm\sigma = 10 \text{ cm}

The variance of 100 cm² and standard deviation of 10 cm indicate that most heights cluster within 10 cm of the mean. We expect about 68% of heights to fall between 160 cm and 180 cm, and about 95% between 150 cm and 190 cm.
Heights N(170, 10^2): about 68% within 160-180 cm, 95% within 150-190 cmxy140150160170180190200210blue: within 1σ, 160–180 cm, about 68%navy: 1σ to 2σ, together about 95%amber: beyond 2σ, about 5%Heights N(170, 10²): the 68–95 rule as shaded areas
The 68–95 rule for heights

The section’s heights, normal with mean 170 cm and standard deviation 10 cm. The blue area between 160 and 180 cm, within one standard deviation, is about 68% of the total. Adding the navy bands out to 150 and 190 cm, two standard deviations, brings it to about 95%. Only the amber tails, about 5%, lie further out.

Out to three standard deviations, 140 to 200 cm, the share rises to about 99.7%.

Mode and Median

Mode


The mode is the value where the probability density function reaches its maximum—the peak of the distribution curve.

For the normal distribution, the mode is:

Mode and Median
Mode=μ\text{Mode} = \mu

Intuition: The normal PDF is:

Mode and Median
f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

The exponential term e−(x−μ)22σ2e^{-\frac{(x-\mu)^2}{2\sigma^2}} is maximized when the exponent equals zero, which occurs when (x−μ)2=0(x-\mu)^2 = 0, giving x=μx = \mu.

The normal distribution is perfectly symmetric around μ\mu, making this point simultaneously the center of mass, the highest point, and the balance point. No other value has higher probability density.

Example
For human heights with μ=170\mu = 170 cm and σ=10\sigma = 10 cm:
Mode = 170 cm
This is the most common height—the peak of the bell curve. Heights become progressively less common as you move away from 170 cm in either direction.

Median


The median is the value mm such that P(X≤m)=0.5P(X \leq m) = 0.5—the point that divides the distribution's probability in half.

For the normal distribution, the median is:

Mode and Median
Median=μ\text{Median} = \mu

Intuition: Because the normal distribution is perfectly symmetric around μ\mu, exactly half the probability mass lies below μ\mu and half lies above. The CDF evaluated at μ\mu gives:

Mode and Median
F(μ)=0.5F(\mu) = 0.5

This symmetry means that mode = median = mean = μ\mu, a unique property that holds only for symmetric distributions.

Example
For human heights with μ=170\mu = 170 cm and σ=10\sigma = 10 cm:
Median = 170 cm
Half of all heights fall below 170 cm, and half fall above. This coincides with both the mean and mode.

Properties:
• For the normal distribution: mode = median = mean (all equal μ\mu)
• The parameter σ\sigma controls spread but doesn't affect the location of mode or median
• This triple equality holds for all symmetric distributions but is particularly important for the normal distribution
• Unlike skewed distributions where mean, median, and mode diverge, the normal distribution's symmetry keeps them aligned

Quantiles/Percentiles


A quantile is a value that divides the distribution at a specific probability threshold. The pp-th quantile xpx_p satisfies:

Quantiles/Percentiles
P(X≤xp)=pP(X \leq x_p) = p

where 0<p<10 < p < 1.

Percentiles are quantiles expressed as percentages: the kk-th percentile corresponds to the quantile at p=k/100p = k/100. For example, the 25th percentile is the 0.25 quantile, the 50th percentile is the median, and the 75th percentile is the 0.75 quantile.

Quantiles are found by inverting the CDF: if F(xp)=pF(x_p) = p, then xp=F−1(p)x_p = F^{-1}(p).

Finding Quantiles for the Normal Distribution


For a normal distribution with mean μ\mu and standard deviation σ\sigma, the pp-th quantile is:

Quantiles/Percentiles
xp=μ+σ⋅zpx_p = \mu + \sigma \cdot z_p

where zpz_p is the pp-th quantile of the standard normal distribution (mean 0, standard deviation 1).

The standard normal quantiles zpz_p cannot be expressed in closed form and must be obtained from:
• Statistical tables (z-tables)
• Software functions (e.g., qnorm() in R, norm.ppf() in Python)
• Numerical approximations

Common Percentiles


25th Percentile (First Quartile, Q1):

Quantiles/Percentiles
x0.25=μ+σ⋅z0.25=μ−0.674σx_{0.25} = \mu + \sigma \cdot z_{0.25} = \mu - 0.674\sigma

About 25% of values fall below this point.

50th Percentile (Median, Q2):

Quantiles/Percentiles
x0.50=μ+σ⋅z0.50=μx_{0.50} = \mu + \sigma \cdot z_{0.50} = \mu

This is the median, dividing the distribution in half.

75th Percentile (Third Quartile, Q3):

Quantiles/Percentiles
x0.75=μ+σ⋅z0.75=μ+0.674σx_{0.75} = \mu + \sigma \cdot z_{0.75} = \mu + 0.674\sigma

About 75% of values fall below this point.

Interquartile Range (IQR):

Quantiles/Percentiles
IQR=Q3−Q1=1.349σ\text{IQR} = Q3 - Q1 = 1.349\sigma

The IQR contains the middle 50% of the distribution.

Example


For human heights with μ=170\mu = 170 cm and σ=10\sigma = 10 cm:

25th percentile: 170+10(−0.674)=170−6.74=163.26170 + 10(-0.674) = 170 - 6.74 = 163.26 cm

25% of people are shorter than 163.26 cm.

50th percentile: 170+10(0)=170170 + 10(0) = 170 cm

Half of people are shorter than 170 cm (the median).

75th percentile: 170+10(0.674)=170+6.74=176.74170 + 10(0.674) = 170 + 6.74 = 176.74 cm

75% of people are shorter than 176.74 cm.

IQR: 176.74−163.26=13.48176.74 - 163.26 = 13.48 cm

The middle 50% of heights span about 13.5 cm.

Other Notable Percentiles


90th percentile: x0.90=μ+1.282σx_{0.90} = \mu + 1.282\sigma (only 10% exceed this value)

95th percentile: x0.95=μ+1.645σx_{0.95} = \mu + 1.645\sigma (only 5% exceed this value)

99th percentile: x0.99=μ+2.326σx_{0.99} = \mu + 2.326\sigma (only 1% exceed this value)

These percentiles are commonly used in hypothesis testing and confidence interval construction.

Real-World Examples and Common Applications


The normal distribution appears throughout nature and human activity whenever many small, independent factors combine to produce a measurement.

Common Applications


Measurement and Physical Sciences:
• Human heights, weights, and other biological measurements
• Measurement errors in scientific instruments
• IQ scores and standardized test results
• Blood pressure readings in healthy populations

Finance and Economics:
• Asset returns over short time periods (approximately normal)
• Pricing models for options and derivatives
• Portfolio risk analysis
• Economic indicators like inflation rates

Quality Control and Manufacturing:
• Product dimensions and tolerances
• Production process variations
• Six Sigma methodologies
• Control chart limits

Natural Phenomena:
• Temperature variations around seasonal averages
• Rainfall amounts in many regions
• Particle velocities in gases (Maxwell-Boltzmann distribution)

Why It Appears


The Central Limit Theorem explains why the normal distribution is ubiquitous: when many independent random effects add together, their sum tends toward normal regardless of the individual distributions. This makes it the natural model for aggregate phenomena.

Example Application


A factory produces bolts with target diameter 10 mm. Due to manufacturing variation, actual diameters follow N(10,0.12)N(10, 0.1^2). Quality standards require diameters between 9.8 mm and 10.2 mm.

Using the normal distribution, we can calculate that approximately 95.4% of bolts meet specifications, helping determine acceptable defect rates and production costs.

Interactive Calculator

Normal Distribution Calculator

Calculate probabilities and distribution properties

Center of the distribution

Spread of the distribution


Special Cases


The normal distribution exhibits several important special cases and limiting behaviors that connect it to other distributions and reveal its mathematical structure.

Standard Normal Distribution


When μ=0\mu = 0 and σ=1\sigma = 1, we obtain the standard normal distribution Z∼N(0,1)Z \sim N(0, 1):

Standard Normal PDF
f(z)=12πe−z22f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}
Learn more about this formula: Standard Normal PDF →


Any normal random variable X∼N(μ,σ2)X \sim N(\mu, \sigma^2) can be standardized:

Normal Standardization
Z=X−μσ∼N(0,1)Z = \frac{X - \mu}{\sigma} \sim N(0, 1)
Learn more about this formula: Normal Standardization →


This transformation is the foundation for z-scores, hypothesis testing, and normal probability tables.

Degenerate Case


As σ→0\sigma \to 0, the normal distribution converges to a point mass at μ\mu:

Special Cases
lim⁡σ→0N(μ,σ2)=δμ\lim_{\sigma \to 0} N(\mu, \sigma^2) = \delta_{\mu}

All probability concentrates at a single value, and the distribution becomes deterministic. This represents the limiting case of no variability.

As σ Increases


As σ→∞\sigma \to \infty, the distribution spreads out indefinitely. The density becomes flatter and approaches zero everywhere, though it never becomes truly uniform over the real line (total probability remains 1).

Limiting Behavior


Central Limit Theorem Connection:
The normal distribution emerges as the limit of many other distributions. For example:

• Binomial B(n,p)B(n, p) approaches N(np,np(1−p))N(np, np(1-p)) as n→∞n \to \infty with fixed pp
• Sum of nn independent identically distributed variables approaches normal (under mild conditions)
• Sample means from any distribution approach normal as sample size grows

Sum of Normals:
If X1∼N(μ1,σ12)X_1 \sim N(\mu_1, \sigma_1^2) and X2∼N(μ2,σ22)X_2 \sim N(\mu_2, \sigma_2^2) are independent, then:

Special Cases
X1+X2∼N(μ1+μ2,σ12+σ22)X_1 + X_2 \sim N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)

The normal distribution is closed under addition—sums of normal variables remain normal.

Practical Implications


Measurement Precision:
In metrology, as measurement precision improves (σ\sigma decreases), measurements cluster more tightly around the true value. The standard normal case (σ=1\sigma = 1) provides a natural reference scale.

Model Robustness:
Many statistical procedures assume normality. When nn is large (typically n>30n > 30), the Central Limit Theorem justifies this assumption even when underlying data aren't normal—this is why normal-based inference is so widely applicable.

Properties


The normal distribution possesses several distinctive mathematical properties that make it central to probability theory and statistics.

Symmetry


The normal distribution is perfectly symmetric around its mean μ\mu:

Properties
f(μ+x)=f(μ−x) for all xf(\mu + x) = f(\mu - x) \text{ for all } x

This symmetry implies:
• Mean = Median = Mode = μ\mu
• Odd central moments equal zero: E[(X−μ)2k+1]=0E[(X - \mu)^{2k+1}] = 0
• The distribution is mirror-symmetric about the vertical line at x=μx = \mu

Skewness


Properties
Skewness=0\text{Skewness} = 0

The skewness coefficient measures asymmetry. Zero skewness confirms perfect symmetry—there is no tendency toward either tail.

Kurtosis


Properties
Kurtosis=3\text{Kurtosis} = 3

Properties
Excess Kurtosis=0\text{Excess Kurtosis} = 0

Kurtosis measures tail weight and peakedness. The normal distribution defines the baseline (kurtosis = 3), so excess kurtosis is zero by definition. Distributions with kurtosis > 3 have heavier tails than normal; kurtosis < 3 indicates lighter tails.

Tail Behavior


The normal distribution has exponentially decaying tails:

Properties
f(x)∼e−(x−μ)22σ2 as ∣x−μ∣→∞f(x) \sim e^{-\frac{(x-\mu)^2}{2\sigma^2}} \text{ as } |x - \mu| \to \infty

Probabilities in the tails decrease rapidly:
• P(∣X−μ∣>2σ)≈0.05P(|X - \mu| > 2\sigma) \approx 0.05 (5%)
• P(∣X−μ∣>3σ)≈0.003P(|X - \mu| > 3\sigma) \approx 0.003 (0.3%)
• P(∣X−μ∣>4σ)≈0.00006P(|X - \mu| > 4\sigma) \approx 0.00006 (0.006%)

The tails are "thin"—extreme values are increasingly rare. This is lighter than polynomial tails but heavier than exponential cutoffs.

Unique Mathematical Properties


Standardization:
Every normal variable converts to the standard normal by subtracting its mean and dividing by its standard deviation:

Normal Standardization
Z=X−μσ∼N(0,1)Z = \frac{X - \mu}{\sigma} \sim N(0, 1)
Learn more about this formula: Normal Standardization →


Closure Under Linear Transformation:
If X∼N(μ,σ2)X \sim N(\mu, \sigma^2), then for constants aa and bb:

Normal Linear Transform
aX+b∼N(aμ+b,a2σ2)aX + b \sim N(a\mu + b, a^2\sigma^2)
Learn more about this formula: Normal Linear Transform →


Closure Under Addition:
If X1∼N(μ1,σ12)X_1 \sim N(\mu_1, \sigma_1^2) and X2∼N(μ2,σ22)X_2 \sim N(\mu_2, \sigma_2^2) are independent:

Sum of Independent Normals
X1+X2∼N(μ1+μ2,σ12+σ22)X_1 + X_2 \sim N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)
Learn more about this formula: Sum of Independent Normals →


Sums of independent normal variables remain normal—a rare property called stability or reproductive property.

Maximum Entropy:
Among all continuous distributions on (−∞,∞)(-\infty, \infty) with specified mean μ\mu and variance σ2\sigma^2, the normal distribution has the highest entropy. It represents maximum uncertainty given only these two constraints.

Characteristic Function:
Properties
ϕ(t)=E[eitX]=eiμt−σ2t22\phi(t) = E[e^{itX}] = e^{i\mu t - \frac{\sigma^2 t^2}{2}}

This simple exponential form makes analytical work tractable.

Useful Identities


68-95-99.7 Rule (Empirical Rule):
• Approximately 68% of values lie within μ±σ\mu \pm \sigma
• Approximately 95% of values lie within μ±2σ\mu \pm 2\sigma
• Approximately 99.7% of values lie within μ±3σ\mu \pm 3\sigma

Moment Generating Function:
Properties
M(t)=E[etX]=eμt+σ2t22M(t) = E[e^{tX}] = e^{\mu t + \frac{\sigma^2 t^2}{2}}

Linear Combinations:
For independent normal variables Xi∼N(μi,σi2)X_i \sim N(\mu_i, \sigma_i^2) and constants aia_i:

Properties
∑i=1naiXi∼N(∑i=1naiμi,∑i=1nai2σi2)\sum_{i=1}^n a_i X_i \sim N\left(\sum_{i=1}^n a_i \mu_i, \sum_{i=1}^n a_i^2 \sigma_i^2\right)

Central Limit Theorem:
For i.i.d. random variables X1,X2,…,XnX_1, X_2, \ldots, X_n with mean μ\mu and variance σ2\sigma^2:

Properties
Xˉn−μσ/n→dN(0,1) as n→∞\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0, 1) \text{ as } n \to \infty

This convergence explains why the normal distribution appears so frequently in nature and statistics.

Normal Distribution at a Glance

The table below collects the full anatomy of the normal distribution into a single reference card — its parameters and support, the PDF and CDF, the mean and variance, mode and median, the standardization rule that produces z-scores, the 68-95-99.7 empirical rule, closure properties and the Central Limit Theorem connection, and a canonical human-heights example.
Aspect Formula / statement Note / example
Parameters μ ∈ ℝ (mean / location); σ > 0 (standard deviation / scale) bell curve fully determined by these two
Support x ∈ (−∞, ∞) unbounded; tails decay exponentially fast
PDF f(x) = (1 / (σ · √(2π))) · exp(−(x − μ)2 / (2σ2)) symmetric bell curve centered at μ
CDF F(x) = Φ((x − μ) / σ); no closed form evaluated via z-tables, software, or numerical methods
Expected value E[X] = μ the location parameter is the mean
Variance Var(X) = σ2; standard deviation = σ scale parameter directly gives the standard deviation
Mode and median mode = median = μ perfect symmetry collapses all three measures of center
Standardization Z = (X − μ) / σ ~ N(0, 1) basis of z-scores, probability tables, and hypothesis testing
68-95-99.7 rule ≈68% within μ ± σ; ≈95% within μ ± 2σ; ≈99.7% within μ ± 3σ quick empirical estimate without computing the CDF
Closure & CLT independent normals sum to normal: ∑aiXi ~ N(∑aiμi, ∑ai2σi2); CLT: sample means of any distribution → normal why the normal appears in measurement, finance, and the sciences
Canonical example human heights: μ = 170 cm, σ = 10 cm ≈68% in [160, 180] cm, ≈95% in [150, 190] cm

Normal Distribution FAQ

Does N(μ, σ²) use variance or standard deviation?

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It depends on the source, and nothing in the notation says which. Most textbooks put the variance in the second slot; much software puts the standard deviation — R's and Python's normal functions both take the SD. Magnitude sometimes betrays it: N(175, 49) carries σ = 7 squared, N(175, 7) the SD raw. N(0, 1) agrees under both, which is why the ambiguity hides so well.Read more →

What is the 68-95-99.7 rule?

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For a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a fast way to read a normal curve without tables. Note it holds only for normal data — a distribution with a variance but no bell shape guarantees far less.Read more →

Why is there no formula for normal probabilities?

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Because Φ, the standard normal CDF, has no closed form. It is defined by its integral, and that integral has no elementary antiderivative — so values come from tables or software rather than algebra. Its inverse Φ⁻¹, the probit, is what every z-quantile lookup computes. This is why normal problems route through z-scores and a single table.Read more →

What does a negative z-score mean?

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Below the mean, by that many standard deviations. z = −2 means two standard deviations below, not a distance of two — the sign carries direction, and dropping it collapses the two tails the score exists to tell apart. Capital Z is the standard normal variable; lowercase z is one computed score.Read more →