Properties of Normal Distributions

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Chapter: Biostatistics for the Health Sciences: The Normal Distribution

The normal distribution has three main characteristics. First, its probability density is bell-shaped, with a single mode at the center.


The normal distribution has three main characteristics. First, its probability density is bell-shaped, with a single mode at the center. As the tails of the normal distribu-tion extend to ±`, the distribution decreases in height and remains positive. It is symmetric in shape about μ, which is both its mean and mode. As detailed as this description may sound, it does not completely characterize the normal distribution. There are other probability distributions that are symmetric and bell-shaped as well. The normal density function is distinguished by the rate at which it drops to zero. Another parameter, σ, along with the mean, completes the characterization of the normal distribution.

The relationship between σ and the area under the normal curve provides the second main characteristic of the normal distribution. The parameter σ is the standard deviation of the distribution. Its square is the variance of the distribution.

For a normal distribution, 68.26% of the probability distribution falls in the interval from μσ to μ + σ. The wider interval from μ  – 2σ to μ + 2 σ contains 95.45% of the distribution. Finally, the interval from μ – 3σ to μ + 3σ contains 99.73% of the distribution, nearly 100% of the distribution. The fact that nearly all observations from a normal distribution fall within ±3σ of the mean explains why the three-sigma limits are used so often in practice.

Third, a complete mathematical description of the normal distribution can be found in the equation for its density. The probability density function f(x) for a nor-mal distribution is given by

One awkward fact about the normal distribution is that its cumulative distribution does not have a closed form. That means that we cannot write down an explicit formula for it. So to calculate probabilities, the density must be integrated numerically. That is why for many years statisticians and other practitioners of statistical methods relied heavily on tables that were generated for the normal distribution.

One important feature was very helpful in making those tables. Although to specify a particular normal distribution one has to provide the two parameters, the mean and the variance, a simple equation relates the general normal distribution to one particular normal distribution called the standard normal distribution.

For the general normal distribution, we will use the notation N(μ, σ2). This ex-pression denotes a normal distribution with mean μ and variance σ2. The standard normal distribution has mean 0 and variance 1. So N(0, 1) denotes the standard nor-mal distribution. Figure 6.1 presents a standard normal distribution with standard deviation units shown on the x-axis.

Figure 6.1. The standard normal distribution.

Suppose X is N(μ, σ 2); if we let Z = (Xμ)/σ, then Z is N(0, 1). The values for Z, an important distribution for statistical inference, are available in a table. From the table, we can find the probability P for any values a < b, such that P(a Z b). But, since Z = (X μ)/σ, this is just P(a (X μ)/σ b) = P(aσ (X μ) bσ) = P(aσ + μ X bσ + μ). Thus, to make inferences about X, all we need to do is to convert X to Z, a process known as standardization.

So, probability statements about Z can be translated into probability statements about X through this relationship. Therefore, a single table for Z suffices to tell us everything we need to know about X (assuming both μ and σ are specified).

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