Normal Distribution Explorer

Set a mean and a spread, shade any band, and read the probability that a value lands in it.

Starting values — Normal Distribution Explorer

Mean (μ)	0
Std dev (σ)	1

Change any control above and the results recompute from these.

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the 68-95-99.7 rule 🖖

These percentages aren't rounded guesses — they fall directly out of the Gaussian integral, each one the area under e^(−z²/2) between ±1, ±2, and ±3 standard deviations, exactly what the error-function steps below compute.

Physicists borrow this scale for 'discovery' thresholds: the 2012 Higgs boson announcement required 5σ evidence, a one-sided false-positive rate of roughly 1 in 3.5 million, precisely because a real signal has to clear a bar that high before it's trusted over statistical noise.

The rule also explains why 'unlikely' events aren't actually rare in large datasets — with a million independent measurements you should expect about 63 to land beyond ±4σ purely by chance, while fewer than one would clear ±5σ. That gap is most of the reason the bar sits at five rather than four, the rest being a field that had already watched three-sigma results evaporate.

Why area, not height, is the answer 🖖

The bell curve plots probability density, not probability itself — so the height at any single point is not a probability. It compares values — twice the height means twice as likely per unit of x — but the chance of landing exactly there is zero. What matters is area: the shaded strip between x₁ and x₂ is the chance a value lands in that range, and the whole curve always encloses an area of exactly 1. Sliding μ moves the peak left or right; raising σ flattens and widens the curve, yet that total area never changes.

The bell curve that appeared on money 🖖

The curve was famous enough to be printed on cash: Germany's 10-Deutsche-Mark note, in circulation from 1991 to 2001, showed Carl Friedrich Gauss beside this very curve and its formula f(x). He refined it while modelling measurement errors in astronomy, which is where the curve earns its other name of the error function. Yet Abraham de Moivre had derived the shape back in 1733, decades before Gauss was born — a textbook case of Stigler's law, which jokes that nothing is named after its actual discoverer.

Problem solved in full

Two randomly chosen adults with μ = 170 cm 5 steps
How far apart are two randomly chosen adults? This is Heights: μ = 170 cm with σ = 10 cm, and the shaded band runs from x₁ = 160 to x₂ = 180 cm.
1. Standardising turns centimetres into standard deviations, which is the only currency the normal table accepts. The band is exactly one σ either side of the mean.
  
  $z_{1} = \frac{160 - 170}{10} = - 1, z_{2} = \frac{180 - 170}{10} = + 1$
2. The curve is symmetric about the mean, so the area below −1 equals the area above +1 and the shaded middle is whatever is left. One table value does the whole calculation.
  
  $P = Φ (1) - Φ (- 1) = 2Φ (1) - 1 = 2 (0.8413) - 1 = 0.6827 = 68.3%$
3. Now change the question: not one person against the average, but two people against each other. Variances add for independent quantities even when the quantities are subtracted, so the difference is spread wider than either height is.
  
  $D = X_{1} - X_{2} : E [D] = 0, Var (D) = σ^{2} + σ^{2} ⟹ σ_{D} = 102 = 14.142 cm$
4. A 10 cm difference between two people is therefore only 0.71 standard deviations of D, where the same 10 cm was a full standard deviation for one person.
  
  $P (∣ D ∣ \leq 10) = 2Φ (\frac{10}{14.142}) - 1 = 2Φ (0.7071) - 1 = 2 (0.7603) - 1 = 52.1%$
5. The far tail of that same difference distribution gives the chance a pair look strikingly unalike.
  
  $P (∣ D ∣ \geq 20) = 2 [1 - Φ (1.4142)] = 2 (0.0787) = 15.7% \approx \frac{1}{6}$
Answer

68.3% of adults fall between 160 and 180 cm, and the 15.9% above 180 is what is left once the tool's Φ(z₂) = 84.1% below it is taken away. The difference between two adults is spread by 10√2 = 14.142 cm, so barely half of all pairs (52.1%) differ by less than 10 cm, while about one pair in six differs by more than 20 cm. Those two one-in-six figures are not the same fact, and it is worth separating them: one person exceeds the mean by 10 cm about one time in six, and one pair differs by 20 cm about one time in six. The σ written on a distribution measures distance from the mean, but the differences you actually notice are between two draws, and those run 1.414 times wider.

$σ_{X_{1} - X_{2}} = σ 2 = 1.414 σ$

Learning path

From counts to curves

Leads to Central limit theorem

References (3)

Insight block 1 — the 5σ bar, and the announcement that had to clear it: ATLAS Collaboration, "Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC." Physics Letters B 716(1), 1–29, 2012.
CMS Collaboration, "Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC." Physics Letters B 716(1), 30–61, 2012.
Insight block 3 — the law that says the curve is misnamed: S. M. Stigler, "Stigler's Law of Eponymy." Transactions of the New York Academy of Sciences 39(1), 147–157, 1980.

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Standard normal - The textbook case - μ = 0, σ = 1. s = 1 means values typically sit about 1 unit away from the mean. The graph always shows ±4.5σ - here -4.5 to +4.5 - to capture 99.999% of the distribution. Every normal distribution can be rescaled to this. Drag the sliders to see how μ shifts the curve and σ stretches or squeezes it.
IQ scores - IQ tests are normed so the population averages 100, and the spread of 15 is a choice rather than a measurement: the scale was defined to have it, which is what lets a score mean the same thing on any properly normed test. Scores beyond three standard deviations — below 55 or above 145 — are real but rare, at about 1 in 370 people across both ends together. Try ±2σ to see how uncommon extreme scores really are.
Heights - Adult heights in many populations follow a normal distribution. Height is the textbook example, though the textbook reason for it is thinner than it sounds: many genes and many environmental effects each nudge it a little, and no single one dominates. That is why the curve fits well in the middle and badly at the ends, where growth disorders live. The graph spans 125-215 cm (μ ± 4.5σ, or 170 ± 45) - heights above 200 cm exist but are uncommon, occurring in roughly 1 in 750 people. The shaded band (160-180 cm) is what most people would call 'average height'.
Exam scores - A class where the average score is 65. Exam marks are the case where the model fits worst: they are bounded at both ends and often bunched near a pass mark, so the tails are a convenience rather than a description. The graph spans about 11-119 (μ ± 4.5σ, or 65 ± 54) and extends past 100 - a normal curve has no hard ceiling; those tails above 100 are very flat and represent rare outcomes. Adjust x1 and x2 to any grade band to see the percentage.