Engineering Instrumentation and Measurement Experiments

Experiment No. 2 Calibration of a Electronic Load Cell and a Bourdon Tube-Type Pressure Gage
2.1 Intercept Calculations
We sum the observations, the squares of the Ys and Xs and the products X × Y to obtain the
following quantities.
and Sy similarly.
and S + Y Y similarly
SX = x1 + x2 + ... + xn
SXX = x 2 1 + x 2 2 + ... + x 2 n
SXY = x1y1 + x2y2 + ... + xnyn
We use the summary statistics above to estimate the slope β.
β ≈ mSXY − SXSY
nSXX − SXSX
We use the estimate of β to estimate the intercept, α:
α ≈ SY − βSX
(2)
n
A consequence of this estimate is that the regression line will always pass through the center of the
data.
2.2 Standard error of the linear relationship
The following is excerpted from Applications, Basics, and Computing of Exploratory Data Analysis,
Velleman and Hoaglin, Wadsworth, Inc. 1981.
A fundamental step in most data analysis and in all exploratory analysis is the computation and
examination of residuals. [...] Most analysts propose a simple structure or model to begin describing
the patterns in the data. Such models differ widely in structure and purpose, but all attempt to
fit the data closely. We therefore refer to any such description of the data as a fit. The residuals
are, then, the differences at each point between the observed data value and the fitted value, or the
actual value and the predicted value: Residual = Actual value − Predicted value.
The median-median line provides one way to find a simple fit, and its residuals, r, are found for
each data value, (xi, yi) as
ri = yi − (axi + b)
A pessimist might view residuals as the failure of a fit to describe the data accurately. He might even
speak of them as errors, although a perfect fit, which leaves all residuals equal to zero, would arouse
suspicion. An optimist sees in residuals details of the data’s behavior previously hidden beneath
the dominant patterns of the fit. Both points of view are correct. The best fits leave small residual,
and systematically large residuals may indicate a poorly chosen model. Nevertheless, even a good
fit may do nothing more than describe the obvious.
Any method of fitting models must determine how much each point can be allowed to influence
the fit. Many statistical procedures try to keep the fit close to every data point. If the data include
an outlier, these procedures may permit it to have an undue influence on the fit. As always in
exploratory data analysis, we try to prevent outliers from distorting the analysis. Using medians in
fitting lines to data provide resistance to outliers, and thus the line fitting technique [...] is called
the resistant line [2].
2
(1)