In: Statistics and Probability
In what specific cases the graphical method of calibration curve generation would be more appropriate than the least-squares method and vice versa?
Calibration curves are used to understand the instrumental response to an analyte and predict the concentration in an unknown sample. Generally, a set of standard samples are made at various concentrations with a range than includes the unknown of interest and the instrumental response at each concentration is recorded. For more accuracy and to understand the error, the response at each concentration can be repeated so an error bar is obtained. The data are then fit with a function so that unknown concentrations can be predicted. Typically the response is linear, however, a curve can be made with other functions as long as the function is known. The calibration curve can be used to calculate the limit of detection and limit of quantitation.
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Calibration Curves
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OVERVIEW
Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Calibration curves are used to understand the instrumental response to an analyte and predict the concentration in an unknown sample. Generally, a set of standard samples are made at various concentrations with a range than includes the unknown of interest and the instrumental response at each concentration is recorded. For more accuracy and to understand the error, the response at each concentration can be repeated so an error bar is obtained. The data are then fit with a function so that unknown concentrations can be predicted. Typically the response is linear, however, a curve can be made with other functions as long as the function is known. The calibration curve can be used to calculate the limit of detection and limit of quantitation.
When making solutions for a calibration curve, each solution can be made separately. However, that can take a lot of starting material and be time consuming. Another method for making many different concentrations of a solution is to use serial dilutions. With serial dilutions, a concentrated sample is diluted down in a stepwise manner to make lower concentrations. The next sample is made from the previous dilution, and the dilution factor is often kept constant. The advantage is that only one initial solution is needed. The disadvantage is that any errors in solution making—pipetting, massing, etc.—get propagated as more solutions are made. Thus, care must be taken when making the initial solution.
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JoVE Science Education Database. Analytical Chemistry. Calibration Curves. JoVE, Cambridge, MA, (2019).
PRINCIPLES
Calibration curves can be used to predict the concentration of an unknown sample. To be completely accurate, the standard samples should be run in the same matrix as the unknown sample. A sample matrix is the components of the sample other than the analyte of interest, including the solvent and all salts, proteins, metal ions, etc. that might be present in the sample. In practice, running calibration samples in the same matrix as the unknown is sometimes difficult, as the unknown sample may be from a complex biological or environmental sample. Thus, many calibration curves are made in a sample matrix that closely approximates the real sample, such as artificial cerebral spinal fluid or artificial urine, but may not be exact. The range of concentrations of the calibration curve should bracket that in the expected unknown sample. Ideally a few concentrations above and below the expected concentration sample are measured.
Many calibration curves are linear and can be fit with the basic equation y=mx+b, where m is the slope and b is the y-intercept. However, not all curves are linear and sometimes to get a line, one or both set of axes will be on a logarithmic scale. Linear regression is typically performed using a computer program and the most common method is to use a least squares fitting. With a linear regression analysis, an R2 value, called the coefficient of determination, is given. For a simple single regression, R2 is the square of the correlation coefficient (r) and provides information about how far away the y values are from the predicted line. A perfect line would have an R2 value of 1, and most R2 values for calibration curves are over 0.95. When the calibration curve is linear, the slope is a measure of sensitivity: how much the signal changes for a change in concentration. A steeper line with a larger slope indicates a more sensitive measurement.
Hence graphical method of calibration curve generation would be more appropriate than the least-squares method and vice versa.