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A normal distribution, denoted Ν (μ, σ²) is a symmetrical, bell-shaped distribution. It’s widely used in business and statistics because many real-life phenomena fit a bell-curve shape like heights of people, blood pressure readings, or standardized test scores like the SAT.

The empirical rule, depicted above, tells you what percentage of normally distributed data falls within x standard deviations (σ) from the mean (μ):
• 68% of data falls within 1σ.
• 95% of data falls within 2 σ.
• 99.7% of data falls within 3 σ

Standard deviation (σ) controls the spread of the distribution.

• Small standard deviations result in a tall, thin bell curve with data that is tightly clustered around μ.
• Larger standard deviations result in flatter, wider curves with data more widely spread out around μ.

Note that the notation Ν (μ, σ²) contains the variance, which is the standard deviation squared.

Standard Normal Distribution

A standard normal distribution (shown in the image above) has the following properties:

• μ = 0
• σ = 1

The standard normal distribution is also called the unit normal distribution.

Normal Distribution Properties

• Mean = mode = median = μ.
• Support (range): x ∈ ℝ (i.e., x is real valued).
• Symmetry around μ:  half of values are to the left of μ and half are to the right.
• Like all probability distributions, the total area under the curve is 1.
• Skewness = 0.
• Kurtosis = 3 (standard normal distribution).

General Probability density function (PDF) [2]

For the standard normal distribution, the equation becomes

The cumulative distribution function (which must be computed numerically) is the integral

See also:

• Lineo- normal distribution
• Radico- normal distribution

References

[1] Standard deviations from the mean image. D Wells, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Normal distribution. Engineering Statistics Handbook. Online: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm