Straight-line calibration equation

Very commonly, as in the calibration curve example, we use the measured y-value to estimate an x-value. Once the slope and intercept of a straight-line calibration curve have been established, it is easy to calculate an x-value (e.g., a concentration) from a measured y-value. The estimation limits of the estimated x-value are given by the equation... 

Measurements were made in our laboratory with a Perkin-Elmer Model 125 instrument on 14C-labelIed copolymers, prepared with seven different catalytic systems and covering the entire range of compositions. These were averaged to give a calibration straight line having equation A1.692/A1.764 = 0.07 + 1.93 CP (CP molar fraction of propylene). 

Such assays are closely monitored for both accuracy and precision. Calibration graphs are prepared by adding known amounts of analyte to blank samples of the biofluid under study and extracting these. A plot of the ratio of the response of the analyte to the response of the standard against concentration normally yields a straight line. The equation of this line can be used to calculate the concentration of analyte in unknown samples. These unknown samples should also include reference samples of analyte in biofluid where the concentration of the analyte is known to a quality assurance monitor but not to the analyst. These steps are necessary to ensure the accuracy of the results of the unknown. 

How do we find the best estimate for the relationship between the measured signal and the concentration of analyte in a multiple-point standardization Figure 5.8 shows the data in Table 5.1 plotted as a normal calibration curve. Although the data appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of mathematically determining the best equation for the calibration curve is called regression. 

A calibration curve shows us the relationship between the measured signal and the analyte s concentration in a series of standards. The most useful calibration curve is a straight line since the method s sensitivity is the same for all concentrations of analyte. The equation for a linear calibration curve is... 

Deming and Pardue studied the kinetics for the hydrolysis of p-nitrophenyl phosphate by the enzyme alkaline phosphatase. The progress of the reaction was monitored by measuring the absorbance due to p-nitrophenol, which is one of the products of the reaction. A plot of the rate of the reaction (with units of pmol mL s ) versus the volume, V, (in milliliters) of a serum calibration standard containing the enzyme yielded a straight line with the following equation... 

Example 10. Calculate by the least squares method the equation of the best straight line for the calibration curve given in the previous example. 

Usually (e.g. 4, 2 the ratio equivalent to A254/A235 is plotted versus as shown in Figure 13. However, a plot of A235/A254 versus (1/W ) is also useful. From Equation (3) and (4) the deviation from the straight line derived from the pure homopolymers can then be used to reveal their adequacy of homopolymers for composition calibration and the presence of other variables. Figure 14 shows such a plot. 

MLR is based on classical least squares regression. Since known samples of things like wheat cannot be prepared, some changes, demanded by statistics, must be made. In a Beer s law plot, common in calibration of UV and other solution-based tests, the equation for a straight line... 

COMPARISON OF CAROTENOID COMPOSITION (juG/G) OF LEAFY VEGETABLES OBTAINED BY ONE-POINT CALIBRATION, STRAIGHT LINE EQUATION AND RESPONSE FACTORS... 

Carotenoid Sample number One-point calibration Straight line equation... 

Figure 2-7 A diagram for IR calibration (Equation 2-86). The data for each given H20t content is obtained by heating the sample to different temperatures to vary the species concentrations. For a perfect calibration, all the trends would lie on a single straight line. However, there is some scatter. Furthermore, the slope defined by data for one fixed H20t content does not equal that for another. Hence, the calibration results (8523 = 0.168 8452 = 0.166) shown in this diagram have a relative precision of only about 10%, whereas the relative precision of IR band intensity data is about 1%.

Linear equations of the type v = ct — C, where c and C are constants, relate kinematic viscosity to efflux time over limited time ranges. This is based on the fact that, for many viscometers, portions of the viscosity—time curves can be taken as straight lines over moderate time ranges. Linear equations, which are simpler to use in determining and applying correction factors after calibration, must be applied carefully as they do not represent the true viscosity—time relation. Linear equation constants have been given (158) and are used in ASTM D4212. 

Straight lines drawn through the calibration points could then be used to find the concentrations of theobromine and caffeine in an unknown. From the equation of the theobromine line in Figure 0-7, we can say that if the observed peak height of theobromine from an unknown solution is 15.0 cm, then the concentration is 76.y micrograms per gram of solution. 

Figure 4-11 Calibration curve for protein analysis in Table 4-7. The equation of the solid straight line fitting the 14 data points (open circles) from 0 to 20 y.g, derived by the method of least squares, is y = 0.016 3 ( 0.000 22)x t-0.004, (+0.0024). The standard deviation of y is sy = 0.0059. The equation of the dashed quadratic curve that fits all 17 data points from...

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. 

The linear response range of the glucose sensors can be estimated from a Michaelis-Menten analysis of the glucose calibration curves. The apparent Michaelis-Menten constant KMapp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation, i = i - KMapp(i/C), where i is the steady-state current, i is the maximum current, and C is the glucose concentration. A plot of i versus i/C (an electrochemical Eadie-Hofstee plot) produces a straight line, and provides both KMapp (-slope) and i (y-intercept). The apparent Michaelis-Menten constant characterizes the enzyme electrode, not the enzyme itself. It provides a measure of the substrate concentration range over which the electrode response is approximately linear. A summary of the KMapp values obtained from this analysis is shown in Table I. 

Application of the Hatch equations to a screen analysis is as follows A summation curve is plotted on log-probability grid. The percentage less than calibrated sieve-size is plotted instead of the usual percentage less than stated sieve-size. If the summation curve is a straight line, Eq (5-9) applies. The method is useful where precise information is desired on particle-size of a screened product. 

Distance L and accelerating potential Us are constant for a given spectrometer and, thus, the terms in parentheses can be replaced with the constant A. The relationship between m1/2 and t is linear. A constant B is added to produce a simple equation for a straight line. This constant B allows correction of the measured time zero that may not correspond exactly with the true time zero. Indeed, calibration must take into account other effects. The most basic effect is the uncertainty of the starting time of an ion. This effect can be caused by finite time delays in cables etc. Thus ... 

The method consists first of carrying out measurements on synthetic samples containing the same known quantity of the internal standard and increasing quantities of the compound to be measured. With these results a calibration curve is constructed. This allows a mathematical relationship to be obtained between the intensities of the signals corresponding to the compound to be analysed and the internal standard (/x//sti) and the quantity of compound present in the sample (Mx). As a reminder, maximum precision is obtained if the relationship corresponds to the equation of a straight line with a slope equal to one. 

This mathematical equation corresponds to the equation of a curve. The straight line represents only a special case. The nature of the calibration curve that is obtained depends on the normalized relative isotopic abundances and thus on the nature of the isotopes that are introduced, on the increase in the molecular weight and on how enriched the labelled compound is. 

The results should be checked to ensure that the calibration data do trnly lie on a straight line. Any spnrions data points shonld be discarded and the remainder nsed to formulate the calibration eqnation. The method of least-squares is one techniqne which can be nsed, where the following simultaneous equations ... 

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