Normal Distribution Formula

The Formula

f(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 \sim N(\mu, \sigma^2) reads 'X follows a normal distribution with mean \mu and variance \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) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} for x \in (-\infty, \infty), with E(X) = \mu and \text{Var}(X) = \sigma^2

Worked Examples

Example 1

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

Solution

  1. 1
    Identify the mean \mu = 75 and standard deviation \sigma = 10. Check whether 65 and 85 are within one standard deviation.
  2. 2
    Verify: 65 = 75 - 10 = \mu - \sigma and 85 = 75 + 10 = \mu + \sigma, so the interval [65, 85] is exactly \mu \pm \sigma.
  3. 3
    By the empirical rule (68-95-99.7 rule), approximately 68\% of data in a normal distribution falls within one standard deviation of the mean.

Answer

\approx 68\%
The empirical rule provides quick approximations for normal distributions: about 68\% within 1\sigma, 95\% within 2\sigma, and 99.7\% within 3\sigma of the mean.

Example 2

medium
Heights of adult women are normally distributed with \mu = 164 cm and \sigma = 6 cm. What percentage of women are taller than 176 cm?

Common Mistakes

  • Assuming all data sets are normally distributed — income, wait times, and many real data sets are skewed
  • Applying the 68-95-99.7 rule to distributions that are not approximately normal
  • Confusing the standard normal (\mu = 0, \sigma = 1) with a general normal distribution

Why This Formula Matters

The normal distribution underpins most of classical statistics, from hypothesis testing to confidence intervals, because of the Central Limit Theorem. It models SAT scores, measurement errors, heights, and blood pressure, making it indispensable in medicine, engineering, and social science.

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

Why is the Normal Distribution formula important in Math?

The normal distribution underpins most of classical statistics, from hypothesis testing to confidence intervals, because of the Central Limit Theorem. It models SAT scores, measurement errors, heights, and blood pressure, making it indispensable in medicine, engineering, and social science.

What do students get wrong about Normal Distribution?

Not everything is normal—income and city sizes follow different distributions.

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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