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.

The Formula

f(x)=1σ2π e−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}}

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)X \sim N(\mu, \sigma^2) reads 'XX follows a normal distribution with mean μ\mu and variance σ2\sigma^2'

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)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} for x∈(−∞,∞)x \in (-\infty, \infty), with E(X)=μE(X) = \mu and Var(X)=σ2\text{Var}(X) = \sigma^2

Worked Examples

Example 1

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Scores on a test are normally distributed with mean μ=75\mu = 75 and standard deviation σ=10\sigma = 10. What percentage of students scored between 6565 and 8585?

Answer

≈68%\approx 68\%

First step

1
Identify the mean μ=75\mu = 75 and standard deviation σ=10\sigma = 10. Check whether 6565 and 8585 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\mu = 164 cm and σ=6\sigma = 6 cm. What percentage of women are taller than 176176 cm?

Example 3

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Cholesterol levels in adults are approximately normal with μ=200\mu = 200 mg/dL and σ=25\sigma = 25 mg/dL. What percent of adults have cholesterol between 150150 and 250250 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)X \sim N(\mu, \sigma^2) reads 'XX follows a normal distribution with mean μ\mu and variance σ2\sigma^2'

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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