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Values are in g/cm2.

Values are in g/cm2.

measurements may be different because the individual measurements used to calculate the average may be slightly different. It is useful, therefore, to know what the range of average values would be if repeated sets of three measurements each were performed on Mrs. B. an infinite number of times. This range is called the confidence interval and is calculated by finding the standard error of the sample mean. The standard error is different from, but related to, the standard deviation. The formula for the standard error (SE), is as follows:

4n where SD is the standard deviation and n is the number of measurements. The SD for the first set of three measurements on Mrs. B. was previously calculated as being 0.01 g/cm2. The SE, therefore, for that first set of three measurements is:

The value of 0.006 g/cm2 is the SE of the sample mean. The sample refers to the set of three measurements. The mean (or average) for this sample has already been calculated and found to be 1.021 g/cm2. The SE and sample mean are used to calculate the 95% confidence interval (CI) for the sample. The 95% CI is bounded by the mean plus or minus two times the SE.2 The formula is written as follows:

The 95% confidence interval based on the first set of three measurements on Mrs. B. is:

The interpretation of the 95% CI is that 95% of the means that would be obtained from an infinite number of series of three scans each will fall within the range of 1.009 g/cm2 to 1.033 g/cm2.

There are two characteristics of the SE that become apparent on reviewing the formula for its calculation. First, the SE will always be smaller than the SD. Second, the greater the number of measurements, or n, which make up the sample, the smaller the SE will be.

1 The actual value by which the SE is multiplied depends on the sample size. For samples with an n of greater than 20, the value is very close to 2. For smaller samples, the value will be slightly larger. The formula shown here is a practical characterization of the calculation of the 95% CI.

Fig. 3-7. Accuracy and precision. Target A illustrates accuracy without precision. Target B illustrates precision without accuracy. In target C, a high degree of both accuracy and precision is illustrated.

The smaller the SE, the more narrow the CI. The more narrow the CI, the greater the likelihood that the average value from the limited sample of scans is representative of the average that would be obtained if Mrs. B. was tested an infinite number of times.

Another example of a CI comes from Cummings et al. (5) who estimated a woman's lifetime risk of having a hip fracture. Using population-based data, it was calculated that the lifetime risk of hip fracture for a 50-year-old white woman was 15.6%. The 95% CI for this risk was 14.8% to 16.4%. This very narrow CI gives increasing credibility to the risk estimate of 15.6%.

CIs and statistical significance are closely related but CIs tend to provide more useful clinical information. Many medical journals now require that confidence intervals be presented in addition to assessments of statistical significance for reported data. In densitometry an understanding of CIs is imperative in interpreting the significance of changes in the BMD over time. This is discussed in the following section on precision and in greater detail in Chapter 11.

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