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1 Mean, Standard Deviation, and Coefficient of Variation

1 Mean, Standard Deviation, and Coefficient of Variation

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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


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


Sometimes standard error of mean is also calculated (Equation 4.5).

Standard error of mean 5 SD=On


Here, n is the number of data points in the set.

Key Points ............. 63

References ............ 66


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.



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


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.)




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.



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


The gray area depends on the width of normal distribution as well as the

reference range of the analyte.


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


3 100




Specificity (individuals without disease who show negative test results)


Positive predictive value 5


3 100



3 100




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%.



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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