Show
When you compare monthly QC data or perform initial method validation experiments, you do a lot of mean comparison. Dr. Madelon F. Zady, Ph.D., talks about the means of means and other important statistical calculations. EdD Assistant ProfessorClinical Laboratory Science Program University of LouisvilleLouisville, KentuckyJune 1999 Mean or averageThe previous lesson described the calculation of the mean, SD, and CV and illustrated how these statistics can be used to describe the distribution of measurements expected from a laboratory method. A common application of these statistics is the calculation of control limits to establish the range of values expected when the performance of the laboratory method is stable. Changes in the method performance may cause the mean to shift the range of expected values, or cause the SD to expand the range of expected values. In either case, individual control values should exceed the calculated control limits (expected range of values) and signal that something is wrong with the method. The calculation of a mean is linked to the central location or correctness of a laboratory test or method (accuracy, inaccuracy, bias, systematic error, trueness) and the calculation of an SD is often related to the dispersion or distribution of results (precision, imprecision, random error, uncertainty). In estimating the central location of a group of test results, one could attempt to measure the entire population or to estimate the population parameters from a smaller sample. The values calculated from the entire population are called parameters (mu for the mean, sigma for the standard deviation), whereas the values calculated from a smaller sample are called statistics (Xbar for the mean, SD for the standard deviation). A simulated experimentConsider the situation where there are 2000 patients available and you want to estimate the mean for that population. Blood specimens could be drawn from all 2000 patients and analyzed for glucose, for example. This would be a lot of work, but the whole population could be tested and the true mean calculated, which would then be represented by the Greek symbol mu (µ). Assume that the mean (µ) for the whole population is 100 mg/dl. How close would you be if you only analyzed 100 specimens? This situation can be demonstrated or simulated by recording the 2000 values on separate slips of paper and placing them in a large container. You then draw out a sample of 100 slips of paper, calculate the mean for this sample of 100, record that mean on a piece of paper, and place it in a second smaller container. The 100 slips of paper are then put back into the large container with the other 1900 (a process called with sampling with replacement) and the container shuffled and mixed. You then draw another sample of 100 slips from the large container, calculate the mean, record the mean on paper, place that slip of paper in the small container, return the 100 slips of paper to the large container, and shuffle and mix. If you repeat this process ten more times, the small container now has 12 possible estimates of the "sample of 100" means from the population of 2000. Calculation of the mean of a sample (and related statistical terminology)We will begin by calculating the mean and standard deviation for a single sample of 100 patients. The mean and standard deviation are calculated as in the previous lesson, but we will expand the statistical terminology in this discussion. The table below shows the first 9 of these values, where X is an individual value or score, Xbar is the mean, and X minus Xbar is called the deviation score or delta ( ).
It's important to recognize again that it is the sum of squares that leads to variance which in turn leads to standard deviation. This is an important general concept or theme that will be used again and again in statistics. The variance of a quantity is related to the average sum of squares, which in turn represents sum of the squared deviations or differences from the mean. Calculation of the mean of the means of samples (the standard error of the mean)Now let's consider the values for the twelve means in the small container. Let's calculate the mean for these twelve "mean of 100" samples, treating them mathematically much the same as the prior example that illustrated the calculation of an individual mean of 100 patient values.
Why are the standard error and the sampling distribution of the mean important?Important statistical properties. Conclusions about the performance of a test or method are often based on the calculation of means and the assumed normality of the sampling distribution of means. If enough experiments could be performed and the means of all possible samples could be calculated and plotted in a frequency polygon, the graph would show a normal distribution. However, in most applications, the sampling distribution can not be physically generated (too much work, time, effort, cost), so instead it is derived theoretically. Fortunately, the derived theoretical distribution will have important common properties associated with the sampling distribution.
In short, sampling distributions and their theorems help to assure that we are working with normal distributions and that we can use all the familiar "gates." Important laboratory applications. These properties are important in common applications of statistics in the laboratory. Consider the problems encountered when a new test, method, or instrument is being implemented. The laboratory must make sure that the new one performs as well as the old one. Statistical procedures should be employed to compare the performance of the two.
Self-assessment questions
About the author: Madelon F. ZadyMadelon F. Zady is an Assistant Professor at the University of Louisville, School of Allied Health Sciences Clinical Laboratory Science program and has over 30 years experience in teaching. She holds BS, MAT and EdD degrees from the University of Louisville, has taken other advanced course work from the School of Medicine and School of Education, and also advanced courses in statistics. She is a registered MT(ASCP) and a credentialed CLS(NCA) and has worked part-time as a bench technologist for 14 years. She is a member of the: American Society for Clinical Laboratory Science, Kentucky State Society for Clinical Laboratory Science, American Educational Research Association, and the National Science Teachers Association. Her teaching areas are clinical chemistry and statistics. Her research areas are metacognition and learning theory. |