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Chapter 12: Problem 76

Describe the normal distribution and discuss some of its properties.

Short Answer

Expert verified
Normal Distribution refers to a family of continuous probability distributions described by the normal equation, which involves parameters: mean and standard deviation. The graph of normal distribution is bell-shaped and symmetrical about the mean. Properties of the normal distribution include: symmetry around the mean, the mean=median=mode, and approximately 68% of data falls within 1 standard deviation, 95% falls within 2, and 99.7% falls within 3 standard deviations of the mean. It's widely used in various fields.

Step by step solution

01

Definition of Normal Distribution

The normal distribution is a type of continuous probability distribution for real-valued random variables. The general form of its probability density function is; \( f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ - \frac{(x - \mu)^2}{2\sigma^2} } \). Here, \(\mu\) is the mean or expectation of the distribution while \(\sigma\) is the standard deviation. The variance \(\sigma^2\) is the square of the standard deviation. The curve of the normal distribution is symmetric, centered about its mean, with its spread dictated by its standard deviation showing that data near the mean are more frequent in occurrence than data far from the mean.
02

Properties of Normal Distribution

The following points provide details about the properties of normal distribution: 1. The graph of the normal distribution depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. 2. Normal distributions are symmetric around their mean. 3. The mean, mode, and median are all equal in a normal distribution and are located at the center of the distribution. 4. Approximately 68% of the data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations from the mean.
03

Applications of Normal Distribution

Normal distribution has several applications in real life including the physical and social sciences, in business, and in other disciplines. It often is used in statistics and data science, where the data tend to cluster around the mean. Examples include heights of people, test scores, or a series of many dice rolls.
04

Normal Distribution Properties Summary

In conclusion, a normal distribution has important properties such as its shape being symmetric, mean = median = mode, and follows the 68-95-99.7 rule. Its applications are widespread and used in fields from business to social sciences.

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