## Question 739:

1## Answer:

No answer provided yet.As with most sample statistics the larger the sample, the better the estimate the sample statistic provides of the unknown population parameter. Since the z-score is based on the sample mean and sample standard deviation of a sample there will be fluctuation. When you sample gets above around 30 things get more stable than say when your sample is around 10.With that said, much of this depends on the variability of the data you're working with and the need for precision. My recommendation is to generate a confidence interval around your z-scores. To do this just compute a confidence interval around your sample mean using the t-statistic if your data are quantitative. If you're using a proportion then use the adjusted wald confidence interval for the lower and upper bounds of your interval. Then when you have this interval you can convert the raw data to a z-score.

For example. Lets say you have a sample mean of 100 seconds and standard deviation of 30 seconds. Lets say you have a 1-sided specification limit of 120 seconds (meaning we want this process to take less than 120 seconds). The z-score around the mean for this value is (100-120)/30 = .6667 (reverse the sign b/c lower is better here). Now we can do nothing with the process and sample another 10 and get a different value do to chance fluctuations alone. To account for this compute a 95% 2-sided t confidence interval. It is computed as the mean +/- a margin of error. The margin of error is approximately 2 standard errors (2.2622 for a sample of this size see Inverse t-Distribution Calculator ). The standard error is the standard deviation divided by the square root of the sample size or 30/SQRT(10). Our 95% interval is then between 78.539 and 121.461 seconds. Now use these to compute 2 more z-scores = (78.539-120)/30 = -1.38203 (again reverse sign b/c lower is better for task times). The other z-score is (121.461-120)/30 = .0487 again reverse.

So the 95% confidence interval around the z-scores (or process sigma) suggest a z-scores between -.0487 and 1.38203.