Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 72

A normal distribution

 is a bell- or mound-shaped distribution.

 is completely characterized by its mean and standard deviation. The mean determines the center of the distribution and the standard deviation determines the spread about the mean.

 has probabilities and percentiles that are determined by the mean and standard deviation.

 is symmetric about the mean.

 has mean, median, and mode that are equal (i.e., μ=μ~=M).

 has probability density function given by

Example 2.33

The intelligence quotient (IQ) is based on a test of aptitude and is often used as a measure of an individual’s intelligence. The distribution of IQ scores is approximately normally distributed with mean 100 and standard deviation 15. The normal probability model for IQ scores is given in Figure 2.25.

Figure 2.25 The approximate distribution of IQ scores with µ = 100 and σ = 15.

The standard normal, which will be denoted by Z, is a normal distribution having mean 0 and standard deviation 1. The standard normal is used as the reference distribution from which the probabilities and percentiles associated with any normal distribution will be determined. The cumulative probabilities for a standard normal are given in Tables A.1 and A.2; because 99.95% of the standard normal distribution lies between the values −3.49 and 3.49, the standard normal values are only tabulated for z values between −3.49 and 3.49. Thus, when the value of a standard normal, say z, is between −3.49 and 3.49, the tabled value for z represents the cumulative probability of z, which is P(Z≤z) and will be denoted by Φ(z). For values of z below −3.50, Φ(z) will be taken to be 0 and for values of z above 3.50, Φ(z) will be taken to be 1. Tables A.1 and A.2 can be used to compute all of the probabilities associated with a standard normal.

The values of z are referenced in Tables A.1 and A.2 by writing z=a.bc as z=a.b+0.0c. To locate a value of z in Table A.1 and A.2, first look up the value a.b in the left-most column of the table and then locate 0.0 c in the first row of the table. The value cross-referenced by a.b and 0.c in Tables A.1 and A.2 is Φ(z)=P(Z≤z). The rules for computing the probabilities for a standard normal are given below.