Mean (μ)

The center of the distribution - where the peak occurs. The mean determines the location of the bell curve on the number line. In a normal distribution, the mean, median, and mode are all equal and located at the center.

Standard Deviation (σ)

Measures the spread - how far values deviate from the mean. A larger standard deviation means the bell curve is wider and flatter (more spread out), while a smaller standard deviation means it's narrower and taller (more concentrated around the mean).

Bell Shape

The symmetric curve where most values cluster near the mean. About 68% of values fall within one standard deviation, 95% within two, and 99.7% within three. This consistent pattern makes the normal distribution incredibly useful for statistical analysis.

Also Known as the Empirical Rule

This rule describes how data is distributed in a normal distribution. It's one of the most important concepts in statistics.

The Three Ranges

• 68% of data falls within ±1σ of the mean
• 95% of data falls within ±2σ of the mean
• 99.7% of data falls within ±3σ of the mean

This means that values more than 3 standard deviations from the mean are extremely rare (less than 0.3% chance).

Z-Score (Standard Score)

Formula: z = (x - μ) / σ
The z-score tells you how many standard deviations a value is from the mean. Positive z-scores are above the mean, negative below. A z-score of 2 means the value is 2 standard deviations above the mean.

Test Scores

Standardized test scores (SAT, IQ tests) typically follow a normal distribution with a set mean and standard deviation. For example, IQ tests are designed with μ = 100 and σ = 15.

Heights and Weights

Adult heights and weights in a population tend to follow a normal distribution pattern. Most people are near the average, with fewer people at the extremes.

Measurement Errors

Random errors in scientific measurements often follow a normal distribution around the true value. This is why scientists take multiple measurements and average them.

Financial Returns

Daily stock returns and many financial metrics approximate a normal distribution, which is why it's fundamental to financial modeling and risk assessment.