Standard deviation Information and Courses from MediaLab, Inc.

In the last section, we saw several methods of determining the approximate "center" of a set of data. Another important characteristic of a set of data is how closely those data group around the center. Take a look at the following graphs:Figure 8Frequency Distribution of Test ScoresFigure 9Frequency Distribution of Test ScoresFigure 10Frequency Distribution of Test Scores

Now we will do an example calculation of the standard deviation of a set of data. Here are the data we will use:Table VII Urea Nitrogen Concentration in Five EmployeesConcentration (mg/dL)97111310

Coefficient of variation is commonly used as a means of measuring the variability of an instrument. The data are gathered by recording the values for the normal and abnormal controls for each test run. At the end of the month, the standard deviation, mean, and coefficient of variation are calculated. The testing data for a particular instrument might look like this:Normal ControlAbnormal ControlsCVsCVJanuary100.92.432.41209.54.412.11February103.12.992.90211.64.001.89March102.02.212.17206.83.951.91The coefficient of variation stays fairly constant from month to month. If there is a sudden increase, there might be a problem with the method or the equipment.In the clinical laboratory, the use of CV as a measure of relative variability should not be confused with the use of the standard deviation as a measure of absolute variability. For example, support physicians agreed that for accurate patient treatment, the inherent variability in a glucose method should be less than 5 mg/dL. In this case, neither the hexokinase nor the orthotoluidine method is acceptable. It does not matter which is more precise if neither is precise enough to result in adequate patient care.

As stated before, many of the measurements you make will be approximately normally distributed. If you plot your data, and they fall roughly in the bell curve shape, then you can make the assumption that your underlying population distribution is normal. Using this assumption, you can make several inferences about your population based on the sample data. First, approximate the population mean µ with the sample mean, , and the population standard deviation σ with the sample standard deviation, s. Then you can say that 68% of all data from the population will be within 1s of , 95% within 2s, and 99% within 3s. An example will illustrate. Suppose you have urea nitrogen data with a sample mean of 15 mg/dL, and a sample standard deviation of 5 mg/dL. Then the following is true: approximately 68% of healthy people will have urea nitrogen in the range ± 1s = 15 ± 5 mg/dL = 10-20 mg/dL.approximately 95% of healthy people will have urea nitrogen in the range ± 2s = 15 ± 10 mg/dL = 5-25 mg/dL.approximately 99% of healthy people will have urea nitrogen in the range ± 3s = 15 ± 15 mg/dL = 0-30 mg/dL. These data can be used to set standards for healthy urea nitrogen levels. Most labs set the 95% window as reference ranges for all tests performed

External quality assessment or proficiency testing (PT) is performed to ensure the reliability of test results by comparing results to other laboratories that use the same method system and/or to an assigned value.The CLIA standards for handling proficiency testing specimens are as follows:PT samples must be tested with the laboratory's regular patient load. PT samples must be tested the same number of times that patients' samples are tested routinely. Laboratories participating in PT programs must not engage in interlaboratory comparison of PT sample results. Laboratories may not send PT samples to another laboratory for analysis. Laboratories must document all steps of processing for PT samples. PT is required for only the primary method used for testing of analytes in patients' samples during the period covered by the PT event.In return for their participation, the laboratory will receive the following information:Results for each analyte sample Mean result for each analyte Standard deviation of results by the comparative method Number of laboratories using the same method Standard deviation index (SDI) Lower and upper limits of acceptability of resultsPT results that are between the lower and upper limits of acceptability are considered satisfactory.External quality control serves several purposes, including: Evaluates the internal quality control programDetects errors in a lab's methods, including technical errorsProvides a comparison of testing methods, which is useful in selecting new methods

Many physical and biological processes are well-modeled by a distribution having a roughly bell-like shape. This curve is called the gaussian, bell curve, or normal distribution. The normal distribution has the following characteristics: 68.3% of the area lies between the mean minus one standard deviation (1 SD) and x¯ plus 1 SD95.5% of the area lies between x¯ - 2 SD and x¯ + 2 SD99.7% of the area lies between x¯ - 3 SD and x¯ + 3 SD For example, if a certain control gives a mean test result of 5.6 with an SD of 0.8, then 95.5% of future control test values will be in the range of 4.0 - 7.2, which is 2 standard deviations of the mean. Values that fall outside of these limits may not be acceptable, depending on the laboratory's QC rules, and could indicate a problem with the measuring system.

Quality control charts are examined to see if there are problems in the measuring system. The Westgard multi-rule approach can help to determine whether there is a problem, and whether that problem is due to random or systematic error. If two controls are used, the Westgard Rules that may be considered for rejecting an analytical run are: 13s: This rule applies when a control result falls outside of the 3s limit, either above or below the mean. The run should be rejected. Usually, this indicates that a random error has occurred. 22s rule: This rule applies when two consecutive results exceed the +2 or the -2 standard deviation limit. The controls could be normal or abnormal (across runs), or one of each (within a run and both outside the same 2SD). A violation of this rule usually indicates a systematic error. The run is rejected. R4s rule: This rule applies when the difference between the highest and lowest result of a run exceeds 4 standard deviations. This rule detects random errors and only applies within a run (ie, not across runs) The run is rejected. 41s rule: This rule applies when four consecutive control samples all exceed the +1 or the -1 limit. The controls could be normal, abnormal, or a combination of the two. This rule detects systematic errors. The run is rejected. 8xrule: This rule applies when 8 consecutive controls all fall on the same side of the mean, either above or below. The rule could also apply if 4 consecutive controls fall on the same side of the mean with both controls. This rule detects a systematic error. The run is rejected.

Reference ranges can help show when a test result is drastically out-of-line with expectations by providing a range of most likely values for any given analyte. Reference ranges should reflect the mean value in the population and a certain level of variation (usually 2 standard deviations). 95% of all normal patients will fall inside the reference range of an analyte. Values outside the reference ranges could indicate not only an abnormality in the patient, but also a problem with the test results. For example, should 20% of results suddenly begin to exceed a given reference range, there is most likely a testing error.

A smaller SD represents data where the results are very close in value to the mean. The larger the SD the more variance in the results. Data points in a normal distribution are more likely to fall closer to the mean. In fact, 68% of all data points will be within ±1SD from the mean, 95% of all data points will be within + 2SD from the mean, and 99% of all data points will be within ±3SD. Statisticians have determined that values no greater than plus or minus 2 SD represent measurements that are more closely near the true value than those that fall in the area greater than ± 2SD. Thus, most QC programs call for action should data routinely fall outside of the ±2SD range.

Westgard rule 22s states that if two consecutive control measurements across runs exceed the same mean -2 standard deviations (SD) or exceed the same mean +2SD, or, within a run, if two consecutive control values are outside the same 2 SD, the run must be rejected. If this circumstance occurs, a systematic error is likely. The top chart represents the day's "normal" control, while the bottom chart shows the day's "elevated" control. The L-J plots on the 13th day for both the normal and elevated controls show greater than +2SD. Troubleshooting must be performed before testing can continue. Had only one of the controls been greater than +2SD, the run would have been accepted as "in control," but would have been rejected on the next QC run if the same control was again out +2SD.

Like the Westgard Rules, the Cumulative Summation Limit or Rule (CUSUM for short) has different approaches. The CUSUM type used on the following pages is more sensitive to systematic than random error. Nevertheless, it does provide an easy means to detect impending problems. CUSUM is calculated on worksheets like the one below. Basically CUSUM works in the following manner: a decision limit is predetermined (See section E. on the right side of the chart, where CUSUM limit is defined as SD x 2.7), and when the CUSUM of control observations exceed this limit, one must look for error in the testing process. The right side of the worksheet is used to determine the mean, standard deviation (SD), and CUSUM limit.

In the late 1950's, Dr. William Youden (1900-1971) developed what has now become known as the Youden Plots. This statistical technique involves both normal and abnormal controls and graphically helps to differentiate between systematic and random errors. The inner square of the plot (yellow) represents one standard deviation (1SD). The next larger square (green) represents 2SD, and the outer square (blue) represents 3SD. A horizontal median line is drawn parallel to the X-axis and a second median line is drawn parallel to the Y-axis. The intersection of the two median lines is called the Manhattan Median. One or two 45-degree lines are drawn through the Manhattan Median. The results of at least two different levels of controls (e.g. Level 1/Level 2 or Normal/Abnormal) are then plotted on the chart as X-axis versus Y-axis.