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List the basic properties of the normal distribution
in Data Science & Statistics by Platinum (109k points) | 42 views

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Basic Properties of the Normal Distribution

The normal distribution, also known as the Gaussian distribution, has several key properties:

Symmetry: The distribution is symmetric about its mean, \( \mu \), meaning the left side is a mirror image of the right side.

Mean, Median, and Mode: For a normal distribution, the mean, median, and mode are all equal and located at the center.

Bell-shaped Curve: It has a bell shape with the peak at the mean, \( \mu \), and it approaches the x-axis asymptotically.

Defined by Two Parameters: Its shape is determined by the mean (\( \mu \)) and standard deviation (\( \sigma \)), where \( \mu \) affects the location and \( \sigma \) the spread.

Area under the Curve: The total area under the curve equals 1, a property of all probability distributions.

Empirical Rule: About 68\% of data falls within \( \pm 1\sigma \), 95\% within \( \pm 2\sigma \), and 99.7\% within \( \pm 3\sigma \).

Percentiles: Percentiles can be calculated, which predict the probability of a variable falling within certain ranges.

Inflection Points: There are inflection points at \( \mu - \sigma \) and \( \mu + \sigma \), where the graph's curvature changes.

Asymptotic: The tails of the curve are asymptotic to the x-axis, meaning they get closer indefinitely but never touch it.

Standard Normal Distribution: A special case with \( \mu = 0 \) and \( \sigma = 1 \), where any normal distribution can be standardized using the z-score formula. 

by Platinum (109k points)

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