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)=1ฯƒ2ฯ€โ€‰eโˆ’(xโˆ’ฮผ)22ฯƒ2f(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 โ€” 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

XโˆผN(ฮผ,ฯƒ2)X \sim N(\mu, \sigma^2) reads 'XX follows a normal distribution with mean ฮผ\mu and variance ฯƒ2\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 โ€” most values are near the mean, and extreme values become rapidly less likely.

Formal View

f(x)=1ฯƒ2ฯ€eโˆ’(xโˆ’ฮผ)22ฯƒ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} for xโˆˆ(โˆ’โˆž,โˆž)x \in (-\infty, \infty), with E(X)=ฮผE(X) = \mu and Var(X)=ฯƒ2\text{Var}(X) = \sigma^2

Worked Examples

Example 1

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

Answer

โ‰ˆ68%\approx 68\%

First step

1
Identify the mean ฮผ=75\mu = 75 and standard deviation ฯƒ=10\sigma = 10. Check whether 6565 and 8585 are within one standard deviation.

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Example 2

medium
Heights of adult women are normally distributed with ฮผ=164\mu = 164 cm and ฯƒ=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 ฮผ=200\mu = 200 mg/dL and ฯƒ=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 โ€” the rule only holds for the symmetric bell.
  • Assuming any single-peaked data is normal โ€” check for symmetry; a long tail means skew, not normal.
  • Confusing the curve's height with probability โ€” 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?" โ€” rather than by familiar numbers โ€” 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 โ€” most values are near the mean, and extreme values become rapidly less likely.

What do the symbols mean in the Normal Distribution formula?

XโˆผN(ฮผ,ฯƒ2)X \sim N(\mu, \sigma^2) reads 'XX follows a normal distribution with mean ฮผ\mu and variance ฯƒ2\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?" โ€” rather than by familiar numbers โ€” 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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