Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 76

The result of converting a non-standard normal value, a raw value, to a Z-value is a Z score. A Z score is a measure of the relative position a value has within its distribution. In particular, a Z score simply measures how many standard deviations a point is above or below the mean. When a Z score is negative the raw value lies below the mean of its distribution, and when a Z score is positive the raw value lies above the mean. Z scores are unitless measures of relative standing and provide a meaningful measure of relative standing only for mound-shaped distributions. Furthermore, Z scores can be used to compare the relative standing of individuals in two mound-shaped distributions.

Example 2.41

The weights of men and women both follow mound-shaped distributions with different means and standard deviations. In fact, the weight of a male adult in the United States is approximately normal with mean µ = 180 and standard deviation σ = 30, and the weight of a female adult in the United States is approximately normal with mean µ = 145 and standard deviation σ = 15. Given a male weighing 215 lb and a female weighing 170 lb, which individual weighs more relative to their respective population?

The answer to this question can be found by computing the Z scores associated with each of these weights to measure their relative standing. In this case,

and

Since the female’s weight is 1.67 standard deviations from the mean weight of a female and the male’s weight is 1.17 standard deviations from the mean weight of a male, relative to their respective populations a female weighing 170 lb is heavier than a male weighing 215 lb.