Normal Distribution


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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.

Image: D Wells [1]

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:

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


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