Normal Distribution Formula

The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

The Formula

f(x)=1σ2π e−(x−μ)22σ2

When to use: The normal distribution describes data that clusters symmetrically around the mean with a characteristic bell shape — most values are near the mean, and extreme values become rapidly less likely.

Quick Example

Height, test scores, measurement errors all tend to be normally distributed.

Notation

X∼N(μ,σ2) reads 'X follows a normal distribution with mean μ and variance σ2'

What This Formula Means

The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

The normal distribution describes data that clusters symmetrically around the mean with a characteristic bell shape — most values are near the mean, and extreme values become rapidly less likely.

Formal View

f(x)=1σ2πe−(x−μ)22σ2 for x∈(−∞,∞), with E(X)=μ and Var(X)=σ2

Worked Examples

Example 1

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Scores on a test are normally distributed with mean μ=75 and standard deviation σ=10. What percentage of students scored between 65 and 85?

Answer

≈68%

First step

1
Identify the mean μ=75 and standard deviation σ=10. Check whether 65 and 85 are within one standard deviation.

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Example 2

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Heights of adult women are normally distributed with μ=164 cm and σ=6 cm. What percentage of women are taller than 176 cm?

Example 3

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Cholesterol levels in adults are approximately normal with μ=200 mg/dL and σ=25 mg/dL. What percent of adults have cholesterol between 150 and 250 mg/dL?

Common Mistakes

  • Applying the 68-95-99.7 rule to non-normal data — the rule only holds for the symmetric bell.
  • Assuming any single-peaked data is normal — check for symmetry; a long tail means skew, not normal.
  • Confusing the curve's height with probability — for a continuous curve, probability is area under it over an interval, not the height at a point.

Why This Formula Matters

The normal distribution is the most important model in statistics: the central limit theorem makes sample means normal, and the 68-95-99.7 rule turns a mean and SD into instant probability estimates. It is the bridge from z-scores to real-world percentages. Recognizing it by "Is the data single-peaked, symmetric, and described by just a mean and a standard deviation?" — rather than by familiar numbers — is what lets a student tell it apart from skewed distribution and uniform distribution and standard normal in a mixed problem set.

Frequently Asked Questions

What is the Normal Distribution formula?

The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

How do you use the Normal Distribution formula?

The normal distribution describes data that clusters symmetrically around the mean with a characteristic bell shape — most values are near the mean, and extreme values become rapidly less likely.

What do the symbols mean in the Normal Distribution formula?

X∼N(μ,σ2) reads 'X follows a normal distribution with mean μ and variance σ2'

Why is the Normal Distribution formula important in Math?

The normal distribution is the most important model in statistics: the central limit theorem makes sample means normal, and the 68-95-99.7 rule turns a mean and SD into instant probability estimates. It is the bridge from z-scores to real-world percentages. Recognizing it by "Is the data single-peaked, symmetric, and described by just a mean and a standard deviation?" — rather than by familiar numbers — is what lets a student tell it apart from skewed distribution and uniform distribution and standard normal in a mixed problem set.

What do students get wrong about Normal Distribution?

The procedure for normal distribution is the easy part; the trap is applying the 68-95-99.7 rule to non-normal data. Asking "Is the data single-peaked, symmetric, and described by just a mean and a standard deviation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Normal Distribution formula?

Before studying the Normal Distribution formula, you should understand: mean, standard deviation.

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