Normal Distribution

Normal distribution is a way of describing how data spreads out symmetrically around a central value, forming a bell-shaped curve when graphed. Most values cluster near the mean, and fewer values appear as you move further away in either direction.

A normal distribution is a continuous probability distribution that is symmetric about its mean, with data concentrated near the center and tapering off toward the tails. It is fully described by two parameters: the mean μ, which sets the center, and the standard deviation σ, which controls the spread. The total area under the curve equals 1, representing all possible outcomes. The standard normal distribution is the special case where μ = 0 and σ = 1.

Key Formula

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

Where:

$f(x)$ = the probability density at value x
$μ$ = the mean (center of the distribution)
$σ$ = the standard deviation (controls the spread)
$e$ = Euler's number, approximately 2.718
$π$ = pi, approximately 3.14159

Worked Example

Problem: Scores on a standardized test are normally distributed with a mean of 500 and a standard deviation of 100. What percentage of students score between 400 and 600?

Step 1: Identify the mean and standard deviation.

\mu = 500, \quad \sigma = 100

Step 2: Find how many standard deviations each boundary is from the mean. This is called the z-score.

z = \frac{x - \mu}{\sigma} = \frac{400 - 500}{100} = -1 \quad \text{and} \quad \frac{600 - 500}{100} = +1

Step 3: Apply the empirical rule (68-95-99.7 rule): approximately 68% of data falls within 1 standard deviation of the mean.

P(400 < X < 600) = P(-1 < z < 1) \approx 68\%

Answer: Approximately 68% of students score between 400 and 600.

Visualization

Why It Matters

The normal distribution appears naturally in many real-world measurements — heights, exam scores, and measurement errors all tend to follow this pattern. In AP Statistics, it underpins hypothesis testing, confidence intervals, and the Central Limit Theorem, which explains why sample means become normally distributed even when the original population is not.

Common Mistakes

Mistake: Assuming any symmetric or mound-shaped dataset is perfectly normal.

Correction: Real data only approximately follows a normal distribution. Always check using a histogram or a normal probability plot before assuming normality.

Mistake: Confusing the mean and standard deviation of a normal distribution with those of a sample.

Correction: μ and σ are population parameters that define the theoretical distribution. Sample statistics x̄ and s are estimates from data, and the two should not be used interchangeably.

Related Terms

Mean — Sets the center of the normal distribution
Standard Deviation — Controls the width and spread of the bell curve
Probability — Areas under the curve represent probabilities