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identify outliers in a data set,
specify the number of measurements required to achieve a desired confidence interval, and
execute a Monte Carlo simulation that predicts the behavior expected in a result due to
variations in the variables involved in computing that result.
4.2 STATISTICAL MEASUREMENT THEORY
Sampling refers to obtaining a set of data through repeated measurements of a variable under fixed
operating conditions. This variable is known as the measured variable or, in statistical terms, the
measurand. By fixed operating conditions, we mean that the external conditions that control the
process are held at constant values. In actual engineering practice, truly fixed conditions are difficult
to attain and the term ‘‘fixed operating conditions’’ is used in a nominal sense. That is, we consider
the process conditions to be maintained as closely as possible and deviations from these conditions
will show up in the data set variations.
This chapter considers the effects of random errors and how to quantify them. Recall that random
errors are manifested through data scatter and by the statistical limitations of a finite sampling to
predict the behavior of a population. For now, we will assume that any systematic errors in the
measurement are negligible.
1 Recall from Chapter 1 that this is the case where the average error in a
data set is zero. We begin by considering the following measurement problem: estimate the true value,
x
0 , based on the information derived from the repeated measurement of variable x. In the absence of
systematic error, the true value of x is the mean value of all possible values of x. This is the value that we
want to estimate from the measurement. A sampling of the variable x under controlled, fixed operating
conditions renders a finite number of data points. We use these limited data to infer x
0 based on the
calculated sample mean value, x. We can imagine that if the number of data points, N, is very small,
then the estimation of x
0 from the data set could be heavily influenced by the value of any one data
point. If the data set were larger, then the influence of any one data point would be offset by the larger
influence of the other data. As N ! 1 or towards the total number in the population, all the possible
variations in x become included in the data set. From a practical view, finite-sized data sets are
common, in which case the measured data can provide only an estimate of the true value.
From a statistical analysis of the data set and an analysis of sources of error that influence these
data, we can estimate x
0 as
x
0
¼ x Æ u x ðP%Þ
ð 4:1Þ
where x represents the most probable estimate of x
0 based on the available data and Æu x represents
the uncertainty interval in that estimate at some probability level, P%. Uncertainties are numbers
that quantify the possible range of the effects of errors. The uncertainty interval is the range about x
within which we expect x
0 to lie. It combines the uncertainty estimates of the random error and of
systematic error in the measurement of x.
2 This chapter discusses ways to estimate the uncertainty
due to the effects of random error, called the random uncertainty. Chapter 5 discusses systematic
errors and how to find the uncertainty interval by combining the uncertainties of both random and
systematic errors.
1 Systematic error does not vary with repeated measurements and so does not affect the statistics of a measurement.
Systematic errors are discussed in Chapter 5.
2 Statistics texts that entirely ignore systematic error refer to this uncertainty interval as a ‘‘confidence interval.’’
4.2 Statistical Measurement Theory 119
14:7:39 Page 119
identify outliers in a data set,
specify the number of measurements required to achieve a desired confidence interval, and
execute a Monte Carlo simulation that predicts the behavior expected in a result due to
variations in the variables involved in computing that result.
4.2 STATISTICAL MEASUREMENT THEORY
Sampling refers to obtaining a set of data through repeated measurements of a variable under fixed
operating conditions. This variable is known as the measured variable or, in statistical terms, the
measurand. By fixed operating conditions, we mean that the external conditions that control the
process are held at constant values. In actual engineering practice, truly fixed conditions are difficult
to attain and the term ‘‘fixed operating conditions’’ is used in a nominal sense. That is, we consider
the process conditions to be maintained as closely as possible and deviations from these conditions
will show up in the data set variations.
This chapter considers the effects of random errors and how to quantify them. Recall that random
errors are manifested through data scatter and by the statistical limitations of a finite sampling to
predict the behavior of a population. For now, we will assume that any systematic errors in the
measurement are negligible.
1 Recall from Chapter 1 that this is the case where the average error in a
data set is zero. We begin by considering the following measurement problem: estimate the true value,
x
0 , based on the information derived from the repeated measurement of variable x. In the absence of
systematic error, the true value of x is the mean value of all possible values of x. This is the value that we
want to estimate from the measurement. A sampling of the variable x under controlled, fixed operating
conditions renders a finite number of data points. We use these limited data to infer x
0 based on the
calculated sample mean value, x. We can imagine that if the number of data points, N, is very small,
then the estimation of x
0 from the data set could be heavily influenced by the value of any one data
point. If the data set were larger, then the influence of any one data point would be offset by the larger
influence of the other data. As N ! 1 or towards the total number in the population, all the possible
variations in x become included in the data set. From a practical view, finite-sized data sets are
common, in which case the measured data can provide only an estimate of the true value.
From a statistical analysis of the data set and an analysis of sources of error that influence these
data, we can estimate x
0 as
x
0
¼ x Æ u x ðP%Þ
ð 4:1Þ
where x represents the most probable estimate of x
0 based on the available data and Æu x represents
the uncertainty interval in that estimate at some probability level, P%. Uncertainties are numbers
that quantify the possible range of the effects of errors. The uncertainty interval is the range about x
within which we expect x
0 to lie. It combines the uncertainty estimates of the random error and of
systematic error in the measurement of x.
2 This chapter discusses ways to estimate the uncertainty
due to the effects of random error, called the random uncertainty. Chapter 5 discusses systematic
errors and how to find the uncertainty interval by combining the uncertainties of both random and
systematic errors.
1 Systematic error does not vary with repeated measurements and so does not affect the statistics of a measurement.
Systematic errors are discussed in Chapter 5.
2 Statistics texts that entirely ignore systematic error refer to this uncertainty interval as a ‘‘confidence interval.’’
4.2 Statistical Measurement Theory 119