Concept units

The bell curve (normal distribution)

Basic

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The idea in one paragraph

Many measurements cluster around a typical value and thin out evenly on both sides: heights of adults, errors in a measurement, test scores. Their histogram has a bell shape. The normal distribution is the exact bell. Two numbers fix it: the mean puts its center, and the standard deviation sets its width. About 68% of draws land within one standard deviation of the mean, and about 95% within two. Programs that need random noise usually draw it from this bell.

Values pile up in the middle and thin out evenly on both sides

Measure the height of a thousand adults and sort them into bins, one bin per centimeter. Most heights fall near the average, fewer a little away from it, and very few far away. The bars form a hill with a rounded top and two tails that fade out. The same shape appears whenever a value is the sum of many small, separate effects that push it up or down.

The normal distribution, also called the Gaussian or the bell curve, is the smooth curve that such histograms approach. It is a distribution: it says how likely each value is. The height of the curve at a value says how densely the draws pile up there, and the area under the curve between two values is the chance that a draw lands between them. The total area is 1.

Two numbers describe a normal distribution completely. The mean μ (mu) is the center, where the curve peaks. The standard deviation σ (sigma) is the width: the typical distance of a draw from the mean. The mean-and-spread unit shows how to compute both from a list of numbers.

The mean moves the bell; the standard deviation widens or narrows it
Of 2000 draws, 68% landed within one σ of the mean and 95% within two.
Try this. The grey bars are 2000 draws from the bell; the curve is the bell itself; the shaded band runs from μ − σ to μ + σ. Before you move σ, predict whether the peak will rise or fall as the bell widens. Then check whether the 68% share changes.

68, 95 and 99.7 percent fall within one, two and three standard deviations

Every normal distribution has the same proportions once you measure in standard deviations. About 68% of draws land within 1 σ of the mean, about 95% within 2 σ, and about 99.7% within 3 σ. So a value 3 standard deviations from the mean is rare: about 3 draws in 1000 are that far out.

Worked example

Test scores follow a bell with mean 70 and standard deviation 10.

  1. One σ each side: 70 − 10 = 60 to 70 + 10 = 80. About 68% of students score in this range.
  2. Two σ each side: 50 to 90. About 95% score in this range.
  3. So about 5% score outside 50 to 90. The bell is symmetric, so half of those, about 2.5%, score above 90.

Your turn. Adult heights in some group have mean 170 cm and standard deviation 7 cm. What range holds the middle 95%? About what share is taller than 177 cm?

Answer

170 ± 14, so 156 cm to 184 cm. 177 cm is one σ above the mean. 68% lie within one σ, so 32% lie outside, and half of that, about 16%, lie above 177 cm.

For readers who want the formula, the height of the curve at x is:

p(x) = 1 / (σ√(2π)) × e−(x − μ)² / (2σ²)

The number e ≈ 2.718 is explained in the exponentials unit. You do not need the formula to use the bell: code draws from it directly. The normal distribution with mean 0 and standard deviation 1 is called the standard normal. A draw from any bell is a standard normal draw times σ, plus μ.

This is a simplification

Real data are only roughly normal. The rule "a sum of many small, separate effects is close to a bell" is a theorem, the central limit theorem, but it needs conditions this page skips. Some data are not bell-shaped at all: incomes and waiting times have one long tail. Using the 68–95–99.7 rule on such data gives wrong answers.

bell.py: 100,000 draws, and the share within one and two standard deviations

import numpy as np

rng = np.random.default_rng(seed=0)
scores = rng.normal(loc=70, scale=10, size=100_000)

print(round(scores.mean(), 2), round(scores.std(), 2))
# Each test is True (1) or False (0) per draw, so np.mean gives the share that pass
print(np.mean(np.abs(scores - 70) < 10))   # within 1 standard deviation
print(np.mean(np.abs(scores - 70) < 20))   # within 2 standard deviations
print(np.mean(scores > 90))                # above 90

69.99 10.0
0.68391
0.9546
0.02255

loc is NumPy's name for the mean and scale for the standard deviation.

Check yourself

  1. For a standard normal (mean 0, standard deviation 1), which range holds about 95% of draws?

    Answer

    From −2 to 2. (The exact value is ±1.96.)

  2. You double σ and keep μ fixed. What happens to the width and to the height of the peak? Why must the height change?

    Answer

    The bell becomes twice as wide and its peak half as tall. The total area must stay 1, so a wider bell has to be lower.

  3. A machine fills bags with a mean of 500 g and a standard deviation of 4 g. Is a bag of 488 g unusual?

    Answer

    Yes. 488 g is 12 g, or 3 standard deviations, below the mean. Only about 0.13% of bags are that light or lighter.

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