1 Mean, Standard Deviation, and Coefficient of Variation
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CHAPTER 4:
Laboratory Statistics and Quality Control
4.11 BlandÀAltman
Plot ......................... 60
4.12 ReceiverÀ
Operator Curve..... 60
4.13 What is Six
Sigma? ................... 61
4.14 Errors
Associated with
Reference
Range .................... 62
4.15 Basic Statistical
Analysis: Student
t-Test and Related
Tests ...................... 63
Coefficient of variation is also a very important parameter because CV can be
easily expressed as a percent value; the lower the CV, the better the precision
for the measurement. The advantage of CV is that one number can be used
to express precision instead of stating both mean value and standard deviation. CV can be easily calculated with Equation 4.4:
CV 5 SD=Mean 3 100
ð4:4Þ
Sometimes standard error of mean is also calculated (Equation 4.5).
Standard error of mean 5 SD=On
ð4:5Þ
Here, n is the number of data points in the set.
Key Points ............. 63
References ............ 66
4.2 PRECISION AND ACCURACY
Precision is a measure of how reproducible values are in a series of measurements, while accuracy indicates how close a determined value is to
the target values. Accuracy can be determined for a particular test by analysis of an assayed control where the target value is known. This is typically provided by the manufacturer or made in-house by accurately
measuring a predetermined amount of analyte and then dissolving it in a
predetermined amount of a solvent matrix where the matrix is similar to
plasma. An ideal assay has both excellent precision and accuracy, but
good precision of an assay may not always guarantee good accuracy.
4.3 GAUSSIAN DISTRIBUTION AND
REFERENCE RANGE
Gaussian distribution (also known as normal distribution) is a bellshaped curve, and it is assumed that during any measurement values will
follow a normal distribution with an equal number of measurements
above and below the mean value. In order to understand normal distribution, it is important to know the definitions of “mean,” “median,” and
“mode.” The “mean” is the calculated average of all values, the “median”
is the value at the center point (mid-point) of the distribution, while the
“mode” is the value that was observed most frequently during the measurement. If a distribution is normal, then the values of the mean,
median, and mode are the same. However, the value of the mean,
median, and mode may be different if the distribution is skewed (not
4.3 Gaussian Distribution and Reference Range
Gaussian distribution). Other characteristics of Gaussian distributions are
as follows:
■
■
■
Mean 6 1 SD contain 68.2% of all values.
Mean 6 2 SD contain 95.5% of all values.
Mean 6 3 SD contain 99.7% of all values.
A Gaussian distribution is shown in Figure 4.1. Usually, reference range is
determined by measuring the value of an analyte in a large number of normal subjects (at least 100 normal healthy people, but preferably 200À300
healthy individuals). Then the mean and standard deviations are determined.
The reference range is the mean value 22 SD to the mean value 12 SD. This
incorporates 95% of all values. The rationale for reference range to be the
mean 6 2 SD is based on the fact that the lower end of abnormal values and
upper end of normal values may often overlap. Therefore, mean 6 2 SD is a
conservative estimate of the reference range based on measurement of the analytes in a healthy population. Important points for reference range include:
■
■
■
■
Reference range may be the same between males and females for many
analytes, but reference range may differ significantly between males and
females for certain analytes such as sex hormones.
Reference range of an analyte in an adult population may be different
from infants or elderly patients.
Although less common, reference range of certain analytes may be
different between different ethnic populations.
For certain analytes such as glucose, cholesterol, triglycerides, highdensity and low-density cholesterol, etc., there is no reference range but
FIGURE 4.1
A Gaussian distribution showing percentage of values within a certain standard deviation from the mean.
(Courtesy of Andres Quesda, M.D., Department of Pathology and Laboratory Medicine, University of
Texas-Houston Medical School.)
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CHAPTER 4:
Laboratory Statistics and Quality Control
there are desirable ranges which are based on the study of a large
population and risk factors associated with certain values of analytes
(e.g. various lipid parameters and risk of cardiovascular diseases).
Although many analytes in the normal population when measured follow normal distribution, not all analytes follow that pattern (e.g. cholesterol and triglycerides). In this case distribution is skewed and, as expected, mean, median, and
mode values are different.
4.4 SENSITIVITY, SPECIFICITY, AND PREDICTIVE
VALUE
An assay cannot be 100% sensitive or specific because there is some overlap
between values of a particular biochemical parameter observed in normal
individuals and patients with a particular disease (Figure 4.2). Therefore, during measurement of any analyte there is a gray area where few abnormal
values are generated from analysis of specimens from healthy people
(false positive) and few normal results are generated from patients (false
negative).
■
The gray area depends on the width of normal distribution as well as the
reference range of the analyte.
FIGURE 4.2
Distribution of values in normal and diseased states where TN: true negative values; TP: true positive
values; FN: false negative values; and FP: false positive values. (Courtesy of Andres Quesda, M.D.,
Department of Pathology and Laboratory Medicine, University of Texas-Houston Medical School.)
4.5 Random and Systematic Errors in Measurements
■
■
■
False positive results may mislead the clinician and lead to unnecessary
investigation and diagnostic tests as well as increased anxiety of the patient.
A false negative result is more dangerous than a false positive result
because diagnosis of a disease may be missed or delayed, which can
cause serious problems.
For a test, as clinical sensitivity increases, specificity decreases. For
calculating clinical sensitivity, specificity, and predictive value of a test,
the following formulas can be used:
■
TP 5 True positive (result correctly identifies a disease)
■
FP 5 False positive (result falsely identifies a disease)
■
TN 5 True negative (result correctly excludes a disease when the
disease is not present in an individual)
■
FN 5 False negative (result incorrectly excludes a disease when the
disease is present in an individual).
Therefore, when assay results are positive, results are a combination of TP
and FP, and when assay results are negative, results are combination of TN
and FN (Equations 4.6À4.8).
Sensitivity ðindividuals with disease who show positive test resultsÞ
TP
3 100
5
TP 1 FN
ð4:6Þ
Specificity (individuals without disease who show negative test results)
5
Positive predictive value 5
TN
3 100
TN 1 FP
TP
3 100
TP 1 FP
ð4:7Þ
ð4:8Þ
A positive predictive value is the proportion of individuals with disease who
showed a positive value compared to all individuals tested. Let us consider
an example where a particular analyte was measured in 100 normal individuals
and 100 individuals with disease. The following observations were made:
TP 5 95, FP 5 5, TN 5 95, and FN 5 5. Therefore, sensitivity 5 95/(95 1 5) 3
100 5 95%, and specificity 5 95/(95 1 5) 3 100 5 95%.
4.5 RANDOM AND SYSTEMATIC ERRORS
IN MEASUREMENTS
Random errors and systematic errors are important issues in the laboratory
quality control process. Random errors are unavoidable and occur due to
imprecision of an analytical method. On the other hand, systematic errors
have certain characteristics and are often due to errors in measurement using
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