Normal distribution

Definition

A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped, where most of the observations cluster around the central peak. It is characterized by its mean ($\mu$) and standard deviation ($\sigma$).

5 Must Know Facts For Your Next Test

1. The properties of a normal distribution include symmetry around the mean, a total area under the curve equal to 1, and asymptotic tails.
2. In small sample cases when the population standard deviation is unknown, the t-distribution is often used instead of the normal distribution.
3. The empirical rule states that approximately 68% of data falls within one standard deviation from the mean, about 95% within two standard deviations, and about 99.7% within three.
4. Normal distributions are used to construct confidence intervals for estimating population parameters.
5. When constructing confidence intervals with small samples (n < 30), it’s important to verify that the sample data approximates a normal distribution.

Review Questions

• What are the key properties of a normal distribution?
• How does the t-distribution relate to the normal distribution in small sample cases?
• What percentage of data falls within two standard deviations from the mean in a normal distribution?

Related terms

t-Distribution: A probability distribution used in place of the normal distribution when dealing with small sample sizes or an unknown population standard deviation.

Confidence Interval: A range of values derived from sample statistics that is likely to contain the value of an unknown population parameter.

$\mu$ (Mean): The average or central value in a set of data, which serves as a measure of central tendency in statistics.