Normal Distribution

The

**normal distribution**refers to a family of continuous probability distributions described by the normal equation.The Normal Equation

The normal distribution is defined by the following equation:

**Normal equation.**The value of the random variable

*Y*is:

Y = [ 1/σ * sqrt(2π) ] * e

^{-(x - μ)2/2σ2}where

*X*is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and*e*is approximately 2.71828.The random variable

*X*in the normal equation is called the**normal random variable**. The normal equation is the probability density function for the normal distribution.The Normal Curve

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. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions look like a symmetric, bell-shaped curve, as shown below.

The curve on the left is shorter and wider than the curve on the right, because the curve on the left has a bigger standard deviation.

Probability and the Normal Curve

The normal distribution is a continuous probability distribution. This has several implications for probability.

- The total area under the normal curve is equal to 1.
- The probability that a normal random variable
*X*equals any particular value is 0. - The probability that
*X*is greater than*a*equals the area under the normal curve bounded by*a*and plus infinity (as indicated by the*non-shaded*area in the figure below). - The probability that
*X*is less than*a*equals the area under the normal curve bounded by*a*and minus infinity (as indicated by the*shaded*area in the figure below).

Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following "rule".

- About 68% of the area under the curve falls within 1 standard deviation of the mean.
- About 95% of the area under the curve falls within 2 standard deviations of the mean.
- About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

Collectively, these points are known as the

**empirical rule**or the**68-95-99.7 rule**. Clearly, given a normal distribution, most outcomes will be within 3 standard deviations of the mean.Statistics Tutorial: Standard Normal Distribution

Standard Normal Distribution

The

**standard normal distribution**is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.The normal random variable of a standard normal distribution is called a

**standard score**or a**z-score**. Every normal random variable*X*can be transformed into a*z*score via the following equation:*z*= (

*X*- μ) / σ

where

*X*is a normal random variable, μ is the mean mean of*X*, and σ is the standard deviation of*X*.The Normal Distribution as a Model for Measurements

Often, phenomena in the real world follow a normal (or near-normal) distribution. This allows researchers to use the normal distribution as a model for assessing probabilities associated with real-world phenomena. Typically, the analysis involves two steps.

- Transform raw data. Usually, the raw data are not in the form of z-scores. They need to be transformed into z-scores, using the transformation equation presented earlier:
*z*= (*X*- μ) / σ. - Find probability. Once the data have been transformed into z-scores, you can use standard normal distribution tables, online calculators (e.g., Stat Trek's free normal distribution calculator), or handheld graphing calculators to find probabilities associated with the z-scores.

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