Relatively small values for the ran

Relatively small values for the range or the standard deviation imply that the observations are clustered near the mean.These sample statistics have their own distribution, which we call a sampling distribution. For example, in the lab analysis process, an important performance variable is the time it takes to get results to the critical care unit. Suppose that management wants results available in an average of 25 minutes. That is, it wants the process distribution to have a mean of 25 minutes. An inspector periodically taking a sample of five analyses and calculating the sample mean could use it to determine how well the process is doing. Suppose that the process is actually producing the analyses with a mean of 25 minutes. Plotting a large number of these sample means would show that they have their own sampling distribution with a mean centered on 25 minutes, as does the process distribution mean, but with much less variability. The reason is that the sample means offset the highs and lows of the individual times in each sample. Figure 3.5 shows the relationship between the sampling distribution of sample means and the process distribution for the analysis times.Some sampling distributions (e.g., for means with sample sizes of four or more and proportions with sample sizes of 20 or more) can be approximated by the normal distribution, allowing the use of the normal tables (see Appendix 1, “Normal Distribution”). For example, suppose you wanted to determine the probability that a sample mean will be more than 2.0 standard deviations higher than the process mean. Go to Appendix 1 and note that the entry in the table for z = 2.0 standard deviations is 0.9772. Consequently, the probability is 1.0000 - 0.9772 = 0.0228, or 2.28 percent. The probability that the sample mean will be more than 2.0 standard deviations lower than the process mean is also 2.28 percent because the normal distribution is symmetric to the mean. The ability to assign probabilities to sample results is important for the construction and use of control charts.