|Identify the underlined phrase:The research team at the hospital selected 16 employees at random, and tested their BUN levels, and found an average of 16 mg/dL with a standard deviation of 6.5 mg/dL. They used this data to construct a range of normal values for the whole healthy population.||View Page|
|Table IXCreatinine (mg/dL)x-(x-)2.87.981.04.86.901.051.08.84.9188.8.131.521.021.01.93.184.108.40.2061.04.93 What is the standard deviation, s, of these data?||View Page|
|Statistics and Parameters|
Raw information collected from an experiment or test is called data. However, it would be very impractical and difficult to list all of one's data in a report, and the reader would have a hard time making sense of it anyway. Therefore, researchers commonly report just a few numbers, called statistics, which attempt to capture all of the essential information about the whole data set.Common statistics include the mean, the standard deviation, the median, the maximum, and the minimum. The statistics you use will depend on the kind of data being studiedA parameter is a property of the population, and since you cannot measure all of the members of a population, you cannot measure a parameter directly. You can however make inferences about a parameter, based on your statistics.
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
|Standard Deviation (continued)|
All of these distributions have the same mean (62%) but you can see that they differ greatly. The difference lies in their spread. Set A is the most spread out, while C is the most clustered. One way of quantifying this spread is with the standard deviation, which is denoted with s. To calculate the standard deviation, first find (x - ) for each point, square the results, add them, divide by n-1, and finally take the square root. The formula is: For computerized calculations and for estimating the cumulative standard deviation, the equation that is frequently used is:
|Standard Deviation Example|
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
|Standard Deviation Example (continued)|
The first step in calculating the standard deviation is to calculate the mean, . In this case, = 10.Now, subtract that mean from each of the data values, and then square those results: Table VIIUrea Nitrogen Concentration in 5 Employees (mg/dL) Concentration (mg/dL)x-(x-)29-117-39111113391000Total20Use this total to calculate the standard deviation:The standard deviation is about 2.23.
| Using the standard deviation formula,what is the standard deviation of the following data? You may find it helpful to duplicate this chart to perform your calculations: Table VII Urea Nitrogen Concentration in 9 Employees (mg/dL) Concentration (mg/dL)x-(x-)210111113951579Total||View Page|
|A Measure of Relative Variability|
Since standard deviation, mean, median, and mode are all absolute data on statistical samples, they do not permit a direct comparison of variation between samples with different means or different units of measurement.One way to obtain a measure of variation that has no units is to divide the standard deviation (s) by the mean (), and multiply by 100 to give a percent. This quantity is called the coefficient of variation (CV), and can be used to compare methods that give different units. For example, the coefficient of variation for two different glucose methods would be calculated as shown below after the mean and standard deviation for each method has been established. The hexokinase method has = 99 mg/dL, and s = 8.0 mg/dL. The orthotoluidine method has = 105 mg/dL, and s = 12.5 mg/dL. From these CV's we would conclude that the hexokinase method is relatively more precise because it has a lower CV.
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, the standard deviation is a measure of the spread of a distribution. Here are several normal curves with the same center, but different population standard deviations: As you can see, as the spread of the distribution increases, so does σ. Therefore, the area of the curve lying within a certain number of standard deviations from the mean is fixed, over all normal distributions. Specifically: 68% of the area of the curve is within the range of μ ± 1σ 95% of the area of the curve is within the range of μ ± 2σ 99% of the area of the curve is within the range of μ ± 3σ Commit these numbers to memory: 68-95-99!
|Inferences from Sample Data|
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
|Introduction to the Normal Distribution|
Many of the data sets you will study will follow a similar distribution, with a peak around a certain value, and a few data points that lie outside the central cluster. This curve is called the normal distribution, the bell curve or the Gaussian distribution. A typical normal distribution and its formula are shown below: As you can see, the population mean μ and population standard deviation σ appear explicitly in the formula for the normal curve. In this example, μ = 0 and σ = 1. The normal curve appears in many areas of science, due to a mathematical result called the Central Limit Theorem. This theorem states that when many distributions are added together, the sum will look like a normal distribution, no matter what the original distributions were. So if the quantity you are measuring is the result of many factors, as most biological processes are, that quantity will often be normally distributed.
|Suppose you measured the plasma BUN levels in a sample of several healthy people. You found that the mean (average) was 19.6 mg/dL and the standard deviation was 6.1 mg/dL. The histogram of the data showed roughly the bell curve shape. What percent of the whole population of healthy people has plasma BUN levels between 13.5 and 25.7 mg/dL?||View Page|
|What is the standard deviation of the following data set, rounded to the nearest whole number? 15, 16, 18, 16, 14, 10, 20, 12, 14 ||View Page|
|Ideally, what testing should be done prior to starting a new lot of control material in order to establish its mean and standard deviation?||View Page|
|External Quality Assessment|
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
|Mean and Standard Deviation|
Mean can be defined as the average of the data points. Standard deviation (SD) is a measure of imprecision. It indicates the variability or dispersion around the mean. Together, mean and SD determine acceptable ranges for a lot of control material. New control values must be calculated and acceptable ranges established for each new lot of control materials. Ideally, at least 20 samples should be tested over time for good statistical data.The mean is calculated by adding all of the values, and dividing by the number of values. The formula is: For example, suppose you wanted to find the mean of the values 4, 6, 2, 8, and 5. The mean is: The standard deviation (abbreviated s or SD) is calculated according to the following formula: That is, calculate the deviation from the mean for each point, square those results, sum them, divide by the number of points minus one, and finally take the square root. For example, the deviations from the mean in the above example are -1, 1, -3, 3, and 0. The squared deviations are 1, 1, 9, 9, and 0. The standard deviation is therefore:
|Calculating Acceptable Ranges|
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.
|Choose whether each statement refers to the mean, standard deviation, or coefficient of variation.||View Page|
|For a certain method, a control has a mean result of 12 with a standard deviation (SD) of 2. What is the acceptable 95% range for this control?||View Page|
|Levey-Jennings Quality Control Charts|
Quality control charts are used to record the results of measurements on control samples, to determine if there are systematic or random errors in the method being used. The most common type of chart is the Levey-Jennings chart.There should be a separate control chart for each method being monitored, and separate charts for normal and abnormal controls. The mean and standard deviation of the control being used should be noted on the chart. These should be determined based on at least 20 measurements over 20 days. Here is an example of a Levey-Jennings chart. Each time the control is tested, the result is marked on the chart at the appropriate standard deviation level. For instance, if the mean for a control is 15 and the standard deviation 5, if you test a control, and get a value of 22.5, the chart is marked at +1.5 SD for that day.
|Westgard Multi-Rule Approach|
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.
|Responding to Out-of-Control Results|
If any QC rule required by your laboratory's QC program is violated, do not report patient results until the unacceptable result has been investigated and resolved. Once the problem has been resolved, it may be necessary to retest patient samples from previous runs, especially if the error proved to be a systematic error.It is important to document all steps that were taken to resolve the QC error.
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.
In the illustration, you'll note that the curve is divided into eight equal sections. Each of these sections is one standard deviation or SD. In the middle of these eight sections is a line that represents the mean. The illustrated Gaussian curve shows a normal distribution, which means that most of the data are close to the mean with very few of the data points being at one extreme or the other.
|Acceptable Standard Deviation (SD)|
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 13S|
Westgard rule 13s states that if a control is greater than ± 3 standard deviations from the mean, it should be rejected and rerun. This is because either a random error or a very large systematic error has occurred, as less than 1% of all test values exceed ± 3SD. In the accompanying example, the control for Day 13(noted by the arrow) is greater than +3SD from the mean. Consequently, the 13s rule applies and the run is rejected. Troubleshooting must be performed before further testing can be done.
|Westgard Rule 22S|
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.
|Westgard Rule 41s (2)|
The second part of Westgard Rule 41s says that a run must be rejected if control values have fallen on the same side of the mean for six consecutive testing days, regardless of standard deviation. Looking closely at the plots you will observe that there has been a movement of the values from one level of the control chart to another, as though the mean has changed. Both parts of this rule reveal systematic errors and troubleshooting must be done before testing can resume.
|What is a Cumulative Summation Limit?|
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.
|CUSUM Example: Plotting Control Data|
To illustrate the use of CUSUM in the laboratory, we'll use daily control values for glucose testing. In the example laboratory, testing is not performed on weekends, explaining the lack of data on days 1, 7, and 8.First, we'll list daily control values under "daily results." Then, we'll calculate mean by using formula A. Next, we can find the difference from the mean for each result, and square that result for the two relevant columns. Using all of the squared differences from the mean, we can find the standard deviation using formula B. Using the mean from formula A and the standard deviation calculations from formulas B and C, we can plot our data points on the Levey-Jennings chart. Formula D helps us calculate the coefficient of variation (CV), which expresses SD as a percentage of mean value and is more reliable for comparing precision at different concentration levels. The lower the CV the greater the precision.
|What is a Youden Plot?|
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.
|Using a Youden Plot|
Controls are run and plotted. Plots that lie near the 45-degree reference line and within the one and two standard deviation squares show acceptable results. Points that lie near the 45-degree reference lines but outside the 2SD square indicate a systematic error. Points that lie far from ether 45-degree reference line indicate a random error.
|Which of the following can be defined as the average of the data points, or the sum of all the data points divided by the number of points?||View Page|
|The mean for the normal hemoglobin control is 14.0 mg/dL. The standard deviation is 0.15 with an acceptable control range of +/- 2 standard deviations (SD). What are the acceptable limits of the control?||View Page|
|Which of the following actions would be incorrect when determining quality control (QC) limits for a new lot of control that is being tested in parallel with the old lot of control?||View Page|
|In this example glucose run. possible random errors occurred on days:||View Page|
|In a normal distribution, approximately what percent of data would be more than +/- 3 standard deviations (SD) from the mean?||View Page|
|On which days did the control data fall at least one standard deviation from the mean?||View Page|