Calibration curve uncertainty

Figure 8-11 Effect of calibration curve uncertainty. The dashed lines show confidence limits for concentrations determined by the regression line. Note that uncertainties increase at the extremities of the plot. Usually, we estimate the uncertainty in analyte concentration only from the standard deviation of the response. Calibration curve uncertainty can significantly increase the uncertainty in the analyte concentration from to. s, as shown.

Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regression analysis can provide an equation for the standardization. 

It is important to remember that sometimes, in spite of the excellent performances of an AMS measurement, the final uncertainty on the true calendar age of a sample is a function of the behaviour of the calibration curve in that time interval a small error on the radiocarbon age does not necessarily correspond to a small, or a unique, calendar span on the BC/AD axis. 

The final best fit calibration curve can be seen at Figure 2.9. Note the limitations on the use of the lower, extrapolated, portion of the curve. The line obtained will always pass through the centroid of all the points, i.e. the point on the graph corresponding to x y. A further aspect is the uncertainty associated with the values of a and b. These may be estimated in terms of the variability ofx and y. 

Preparation of a calibration curve has been described. From the fit of the least-squares line we can estimate the uncertainty of the results. Using similar equations we can determine the standard deviation of the calibration line (similar to the standard deviation of a group of replicate analyses) as... 

To gain some perspective on the problem, it is well to realize that assessment of the detection limit is subject to all of the assumptions and restrictions of the estimation process. That is, the functional and error structure of the calibration curve must be known or assumed, and the respective parameters and their uncertainties must be estimated. Although large... 

The procedure we use assumes that the errors in the y values are substantially greater than the errors in the x values.7 This condition is usually true in a calibration curve in which the experimental response (y values) is less certain than the quantity of analyte (x values). A second assumption is that uncertainties (standard deviations) in all the y values are similar. 

The method of least squares is used to determine the equation of the best straight line through experimental data points. Equations 4-16 to 4-18 and 4-20 to 4-22 provide the least-squares slope and intercept and their standard deviations. Equation 4-27 estimates the uncertainty in x from a measured value of y with a calibration curve. A spreadsheet greatly simplifies least-squares calculations. 

Confidence interval for calibration cu ve. To use a calibration curve based on n points, we measure a new value of y and calculate the corresponding value of x. The one-standard-deviation uncertainty in x, xx, is given by Equation 4-27. We express a confidence interval for x, using Student s r ... 

All obsidian samples were analyzed as unmodified samples they were washed in the field. Each sample was placed in the sample chamber with the flattest part of the surface facing the x-ray beam. All samples were at least 3 cm in length with varying widths and thicknesses. The width of the sample did not produce errors when comparing obsidian artifact to potential obsidian source. Accuracy errors result from inaccuracies of the regression model, statistical error of the calibration spectra, inaccuracy of the intensity of the calibration curve and the energy calibration. When the error is taken into account, the relative analytical uncertainty for this project is less than seven percent with this portable XRF unit (26) ... 

Error of prediction. McElwain (2004) used stomatal density counts to predict the known altitude of eleven modem Q. kelloggii trees across California from single leaves. This test revealed that, after using the correction for sea-level C02 differences between the calibration curve and the modern data, the average error in prediction was 300 m. This prediction error is one of the smallest in any currently available paleoaltimetry method, but some other sources of uncertainty may increase this error. 

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