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measurement device specifications usually provide a statement as to the expected linearity of the
static calibration curve for the device. The relationship between y L (x) and measured value y(x) is a
measure of the nonlinear behavior of a system:
u L x
ð Þ ¼ y x
ð Þ À y L x
ð Þ
ð1:8Þ
where u L (x) is a measure of the linearity error that arises in describing the actual system behavior by
Equation 1.7. Such behavior is illustrated in Figure 1.13b in which a linear curve has been fit through
a calibration data set. For a measurement system that is essentially linear in behavior, the extent of
possible nonlinearity in a measurement device is often specified in terms of the maximum expected
linearity error as a percentage of full-scale output range, r o ,
%u L max ¼
u L max
r o
 100
ð1:9Þ
This is how the linearity error for the pressure transducer in Table 1.1 was estimated. Statistical
methods of quantifying data scatter about a line or curve fit are discussed in Chapter 4.
Sensitivity and Zero Errors
The scatter in the data measured during a calibration affects the precision in predicting the slope of
the calibration curve. As shown for the linear calibration curve in Figure 1.13c, in which the zero
intercept is fixed, the scatter in the data about the curve fit are random errors. The sensitivity error,
u K , is a statistical measure of the random error in the estimate of the slope of the calibration curve
(we discuss the statistical estimate further in Chapter 4). The static sensitivity of a device is also
temperature dependent, and this is often specified. In Table 1.1, the sensitivity error reflects
calibration results at a constant reference ambient temperature, whereas the thermal sensitivity error
was found by calibration at different temperatures.
If the zero intercept is not fixed but the sensitivity is constant, then a drift in the zero intercept
introduces a vertical shift of the calibration curve, as shown in Figure 1.13d. This shift is known as
the zero error with uncertainty, u z . Zero error can usually be reduced by periodically adjusting the
output from the measurement system under a zero input condition. However, some random variation
in the zero intercept is common, particularly with electronic and digital equipment subjected to
temperature variations (e.g., thermal zero drift in Table 1.1).
Instrument Repeatability
The ability of a measurement system to indicate the same value on repeated but independent
application of the same input provides a measure of the instrument repeatability. Specific claims of
repeatability are based on multiple calibration tests (replication) performed within a given lab on the
particular unit. Repeatability, as shown in Figure 1.13e, is based on a statistical measure (developed
in Chapter 4) called the standard deviation, s x , a measure of the variation in the output for a given
input. The value claimed is usually in terms of the maximum expected error as a percentage of fullscale output range:
%u R max ¼
2s x
r o
 100
ð1:10Þ
The instrument repeatability reflects only the variations found under controlled calibration
conditions.
22 Chapter 1 Basic Concepts of Measurement Methods
15:40:35 Page 22
measurement device specifications usually provide a statement as to the expected linearity of the
static calibration curve for the device. The relationship between y L (x) and measured value y(x) is a
measure of the nonlinear behavior of a system:
u L x
ð Þ ¼ y x
ð Þ À y L x
ð Þ
ð1:8Þ
where u L (x) is a measure of the linearity error that arises in describing the actual system behavior by
Equation 1.7. Such behavior is illustrated in Figure 1.13b in which a linear curve has been fit through
a calibration data set. For a measurement system that is essentially linear in behavior, the extent of
possible nonlinearity in a measurement device is often specified in terms of the maximum expected
linearity error as a percentage of full-scale output range, r o ,
%u L max ¼
u L max
r o
 100
ð1:9Þ
This is how the linearity error for the pressure transducer in Table 1.1 was estimated. Statistical
methods of quantifying data scatter about a line or curve fit are discussed in Chapter 4.
Sensitivity and Zero Errors
The scatter in the data measured during a calibration affects the precision in predicting the slope of
the calibration curve. As shown for the linear calibration curve in Figure 1.13c, in which the zero
intercept is fixed, the scatter in the data about the curve fit are random errors. The sensitivity error,
u K , is a statistical measure of the random error in the estimate of the slope of the calibration curve
(we discuss the statistical estimate further in Chapter 4). The static sensitivity of a device is also
temperature dependent, and this is often specified. In Table 1.1, the sensitivity error reflects
calibration results at a constant reference ambient temperature, whereas the thermal sensitivity error
was found by calibration at different temperatures.
If the zero intercept is not fixed but the sensitivity is constant, then a drift in the zero intercept
introduces a vertical shift of the calibration curve, as shown in Figure 1.13d. This shift is known as
the zero error with uncertainty, u z . Zero error can usually be reduced by periodically adjusting the
output from the measurement system under a zero input condition. However, some random variation
in the zero intercept is common, particularly with electronic and digital equipment subjected to
temperature variations (e.g., thermal zero drift in Table 1.1).
Instrument Repeatability
The ability of a measurement system to indicate the same value on repeated but independent
application of the same input provides a measure of the instrument repeatability. Specific claims of
repeatability are based on multiple calibration tests (replication) performed within a given lab on the
particular unit. Repeatability, as shown in Figure 1.13e, is based on a statistical measure (developed
in Chapter 4) called the standard deviation, s x , a measure of the variation in the output for a given
input. The value claimed is usually in terms of the maximum expected error as a percentage of fullscale output range:
%u R max ¼
2s x
r o
 100
ð1:10Þ
The instrument repeatability reflects only the variations found under controlled calibration
conditions.
22 Chapter 1 Basic Concepts of Measurement Methods