# Worksheets for Analytical Calibration Curves

### Excel and OpenOffice Calc Versions (March, 2017)

These are fill-in-the-blanks spreadsheet templates for performing the calibration curve fitting and concentration calculations for analytical methods using the calibration curve method. All you have to do is to type in (or paste in) the concentrations of the standard solutions and their instrument readings (e.g. absorbances, or whatever method you are using) and the instrument readings of the unknowns. The spreadsheet automatically plots and fits the data to a straight line, quadratic or cubic curve, then uses the equation of that curve to convert the readings of the unknown samples into concentration.  You can add and delete calibration points at will, to correct errors or to remove outliers; the sheet re-plots and recalculates automatically.

Note: to run these spreadsheets, you must have a recent version of either Excel or OpenOffice Calc installed (download from OpenOffice).  The Calc versions of these models will also run on LibreOffice Calc, for example on the \$38 single-board Raspberry Pi 3.

Background

In analytical chemistry, the accurate quantitative measurement of the composition of samples, for example by various types of spectroscopy, usually requires that the method be calibrated using standard samples of known composition. This is most commonly, but not necessarily, done with solution samples and standards dissolved in a suitable solvent, because of the ease of preparing and diluting accurate and homogeneous mixtures of samples and standards in solution form. In the calibration curve method, a series of external standard solutions is prepared and measured. A line or curve is fit to the data and the resulting equation is used to convert readings of the unknown samples into concentration. An advantage of this method is that the random errors in preparing and reading the standard solutions are averaged over several standards. Moreover, non-linearity in the calibration curve can be detected and avoided (by diluting into the linear range) or compensated (by using non-linear curve fitting methods). There are worksheets here for several different calibration methods:

1. A first-order (straight line) fit of measured signal A (Y-axis) vs concentration C (X-axis). The model equation is A = slope * C + intercept. This is the most common and straightforward method, and it is the one to use if you know that your instrument response is linear. This fit is performed using the equations described and listed on http://terpconnect.umd.edu/~toh/spectrum/CurveFitting.html. You need a minimum of two points on the calibration curve. The concentration of unknown samples is given by (A - intercept) / slope where A is the measured signal and slope and intercept from the first-order fit. If you would like to use this method of calibration for your own data, download in Excel or OpenOffice Calc format.  View equations for linear least-squares.

2. Linear interpolation calibration. In the linear interpolation method (sometime called the bracket method), the spreadsheet performs a linear interpolation between the two standards that are just above and just below each unknown sample, rather than doing a least-squares fit over then entire calibration set. The concentration of the sample Cx is calculated by C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s), where S1x and S2s are the signal readings given by the two standards that are just above and just below the unknown sample, C1s and C2s are the concentrations of those two standard solutions, and Sx is the signal given by the sample solution. This method may be useful if none of the least-squares methods are capable of fitting the entire calibration range adequately (for instance, if it contains two linear segments with different slopes). It works well enough as long as the standards are spaced closely enough so that the actual signal response does not deviate significantly from linearity between the standards. However, this method does not deal well with random scatter in the calibration data due to random noise, because it does not compute a "best-fit" through multiple calibration points as the least-squares methods do. Download a template in Excel (.xls) format.

3. A quadratic fit of measured signal A (Y-axis) vs concentration C (X-axis). The model equation is A = aC2 + bC + c.  This method can compensate for non-linearity in the instrument response to concentration. This fit is performed using the equations described and listed on http://terpconnect.umd.edu/~toh/spectrum/CurveFitting.html.  You need a minimum of three points on the calibration curve. The concentration of unknown samples is calculated by solving this equation for C using the classical "quadratic formula", namely C = (-b+SQRT(b2-4*a*(c-A)))/(2*a), where A = measured signal, and a, b, and c are the three coefficients from the quadratic fit.  If you would like to use this method of calibration for your own data, download in Excel or OpenOffice Calc format. View equations for quadratic least-squares.  The alternative version CalibrationQuadraticB.xlsx computes the concentration standard deviation (column L) and percent relative standard deviation (column M) using the bootstrap method. You need at least 5 standards for the error calculation to work.  If you get a "#NUM!" or #DIV/0" in the columns L or M, just press the F9 key to re-calculate the spreadsheet.

4. Weighted fits. A weighted curve fit applies more weight (emphasis) to some points than others. There are weighted versions of the linear (CalibrationLinearWeighted.xls) and quadratic (CalibrationQuadraticWeighted.xls) templates. There is also a weighted version of the drift-corrected calibration template (CalibrationDriftingQuadraticWeighted.xls); see #7 below.. A weight (usually between zero and 1) for each point must be entered in Column A. There are pre-calculated weights for 1/X, 1/X2, 1/Y, and 1/Y2 weighting in columns Z to AC (in the linear template) or AK to AN (in the quadratic template); you can either Copy and Paste (numbers only) these into column A, or you can enter =Z6 or =AK6 into cell A6, then "drag copy down" that cell to the last data points in column A. (Alternatively, you can enter equations into column A that calculate weights in any way you wish). If you want to disregard (ignore) one or more data points, make their weights zero. To make the calibration unweighted, make all the weights 1.0.

5. A reversed cubic fit of concentration C (Y-axis) vs measured signal A (X-axis). The model equation is C = aA3 + bA2 + cA + d.  This method can compensate for more complex non-linearity that the quadratic fit, for example "S"-shaped curves. A "reversed fit" flips the usual order of axes, by fitting concentration as a function of measured signal. This is done in order to avoid the need to solve a cubic equation when the calibration equation is solved for C and used to convert the measured signals of the unknowns into concentration.  (This coordinate transformation is a short-cut, commonly done in least-squares curve fitting, at least by non-statisticians, to avoid mathematical messiness when the fitting equation is solved for concentration and used to convert the instrument readings into concentration values). This fit is performed using the LINEST function on Sheet3. You need a minimum of four points on the calibration curve. The concentration of unknown samples is calculated directly by aA3+bA2+c*A+d, where A is the measured signal, and a, b, c, and d are the four coefficients from the cubic fit.  If you would like to use this method of calibration for your own data, download in Excel or OpenOffice Calc format.

6. Log-log Calibration. In log-log calibration, the logarithm of the measured signal A (Y-axis) is plotted against the logarithm of concentration C (X-axis) and the calibration data are fit to a linear or quadratic model, as in #1 and #2 above. The concentration of unknown samples is obtained by taking the logarithm of the instrument readings, computing the corresponding logarithms of the concentrations from the calibration equation, then taking the anti-log to obtain the concentration. (These additional steps do not introduce any additional error, because the log and anti-log conversions can be made quickly and without significant error by the computer). Log-log calibration is well suited for data with very large range of values because it distributes the relative fitting error more evenly among the calibration points, preventing the larger calibration points to dominate and cause excessive errors in the low points. In some cases (e.g Power Law relationships) a nonlinear relationship between signal and concentration can be completely linearized by a log-log transformation. However, because of the use of logarithms, the data set can not contain any zero or negative values. To use this method of calibration for your own data, download the templates for log-log linear (Excel or Calc) or log-log quadratic (Excel or Calc).

7. Drift-corrected calibration. All of the above methods assume that the calibration of the instrument is stable with time and that the calibration (usually performed before the samples are measured) remains valid while the unknown samples are measured.  In some cases, however, instruments and sensors can drift, that is, the slope and/or intercept of their calibration curves can gradually change with time after the initial calibration. You can test for this drift by measuring the standards again after the samples are run, to determine how different the second calibration curve is from the first. If the difference is not too large, it's reasonable to assume that the drift is approximately linear with time, that is, that the calibration curve parameters (intercept, slope, and curvature) have changed linearly as a function of time between the two calibration runs.  It's then possible to correct for the drift if you record the time when each calibration is run and when each unknown sample is measured. The drift-correction spreadsheet (CalibrationDriftingQuadratic.ods) does the calculations: it computes a quadratic fit for the pre- and post-calibration curves, then uses linear interpolation to estimate the calibration curve parameters for each separate sample based on the time it was measured.  The method works perfectly only if the drift is linear with time (a reasonable assumption if the amount of drift is not too large), but in any case it is better than simply assuming that there is no drift at all. If you would like to use this method of calibration for your own data, download in Excel or OpenOffice Calc format. (See Instructions: #8). There is also a weighted version of this template (CalibrationDriftingQuadraticWeighted.xls) which assumes that the same weights apply to both pre- and post-calibrations.

8. Error calculations. In many cases it is important to calculate the likely error in the computed concentration values (column K) caused by imperfect calibration. This is discussed on "Reliability of  curve fitting results". The linear calibration spreadsheet (download in Excel or OpenOffice Calc format) performs a classical algebraic error-propagation calculation on the equation that calculates the concentration from the unknown signal and the slope and intercept of the calibration curve. The quadratic calibration spreadsheet (Download in Excel or OpenOffice Calc format) performs a bootstrap calculation. You must have a least 5 calibration points for these error calculations to be even minimally reliable; the more the better. That is because these methods need a representative sample of deviations from the ideal calibration line. If the calibration line fits the points exactly, then the computed error will be zero.

### Instructions:

1. Download and open the desired calibration worksheet from among those listed above. If you wish, you may change the "Instrument Reading" and "Concentration" labels on the graph axes to something more specifically related to your work; just click on the label, type, and press the Enter key.

2. Enter the concentrations of the standards and their instrument readings (e.g. absorbance) into the blue table on the left. Leave the rest of the table blank. You must have at least two points on the calibration curve (three points for the quadratic method or four points for the cubic method), including the blank (zero concentration standard). If you have multiple instrument readings for one standard, it's better to enter each as a separate standard with the same concentration, rather than entering the average. The spreadsheet automatically gives more weight to standards that have more than one reading. If you are using one of the weighted versions (see #4 above) , enter the weights for each point in column A (usually from 0 to 1) .

3. Enter the instrument readings (e.g. absorbance) of the unknowns into the yellow table on the right. You can have any number of unknowns up to 20. (If you have multiple instrument readings for one unknown, it's better to enter each as a separate unknown, rather than averaging them, so you can see how much variation in calculated concentration is produced by the variation in instrument reading).

4. The concentrations of the unknowns are automatically calculated and displayed column K. If you edit the calibration curve, by deleting, changing, or adding more calibration standards, the concentrations are automatically recalculated.

For the linear fit (CalibrationLinear.xls), if you have three or more calibration points, the estimated standard deviation of the slope and intercept will be calculated and displayed in cells G36 and G37, and the resulting standard deviation (SD) of each concentration will be displayed in rows L (absolute standard deviation) and M (percent relative standard deviation). These standard deviation calculations are estimates of the variability of slopes and intercepts you are likely to get if you repeated the calibration over and over multiple times under the same conditions, assuming that the deviations from the straight line are due to random variability and not systematic error caused by non-linearity. If the deviations are random, they will be slightly different from time to time, causing the slope and intercept to vary from measurement to measurement.. However, if the deviations are caused by systematic non-linearity, they will be the same from from measurement to measurement, in which case these predictions of standard deviation will not be relevant, and you would be better off using a polynomial fit such as a quadratic or cubic. The reliability of these standard deviation estimates also depends on the number of data points in the curve fit; they improve with the square root of the number of points.

5. You can remove any point from the curve fit by deleting the corresponding X and Y values in the table. To delete a value; right-click on the cell and click "Delete Contents" or "Clear Contents". The spreadsheet automatically re-calculates and the graph re-draws; if it does not, press F9 to recalculate. (Note: the cubic calibration spreadsheet must have contiguous calibration points with no blank or empty cells in the calibration range).

6. The linear calibration spreadsheet also calculates the coefficient of determination, R2, which is an indicator of the "goodness of fit", in cell C37.  R2 is 1.0000 when the fit is perfect but less than that when the fit is imperfect.  The closer to 1.0000 the better.

7. A "residuals plot" is displayed just below the calibration graph (except for the interpolation method).  This shows the difference between the best-fit calibration curve and the actual readings of the standards. The smaller these errors, the more closely the curve fits the calibration standards.  (The standard deviation of those errors is also calculated and displayed below the residuals plot; the lower this standard deviation, the better).

You can tell a lot by looking at the shape of the residual plot: if the points are scattered randomly above and below zero, it means that the curve fit is as good as it can be given the random noise in the data. But if the residual plot has a smooth shape, say, a U-shaped curve, then it means that there is a mismatch between the curve fit and the actual shape of the calibration curve; suggesting that the another curve fitting techniques might be tried (say, a quadratic or cubic fit rather than a linear one) or that the experimental conditions be modified to produce a less complex experimental calibration curve shape.

8. If you are using the spreadsheet for drift-corrected calibration, you must measure two calibration curves, one before and one after the samples are run, and record the date and time each calibration curve is measured.  Enter the concentrations of the standards into column B. Enter the instruments readings for the first (pre-) calibration into column C and the date/time of that calibration into cell C5; enter the instruments readings for the post-calibration into column D and the date/time of that calibration into cell D5.  The format for the date/time entry is Month-Day-Year Hours:Minutes:Seconds, for example 6-2-2011 13:30:00 for June 2, 2011, 1:30 PM (13:30 on the 24-hour clock.  (Note: if both calibrations are run on the same day, you can leave off the date and just enter the time).  In the graph, the pre-calibration curve is shown in green and the post-calibration curve is shown in red. Then, for each unknown sample measured, enter the date/time (in the same format) into column K and the instrument reading for that unknown into column L. The spreadsheet computes the drift-corrected sample concentrations in column M. Note: Version 2.1 of this spreadsheet (July, 2011) allows different sets of concentrations for the pre- and post-calibrations. Just list all he concentrations used in the "Concentration of standards" column (B) and put the corresponding instrument readings in columns C or D, or both. If you don't use a particular concentration for one of the calibrations, just leave that instrument reading blank.

Click to see larger figure

This figure shows an application of the drift-corrected quadratic calibration spreadsheet. In this demonstration, the calibrations and measurements were made over a period of several days. The pre-calibration (column C) was performed with six standards (column B) on 01/25/2011 at 1:00 PM. Eight unknown samples were measured over the following five days (columns L and M), and the post-calibration (column D) was performed after then last measurement on 01/30/2011 at 2:45 PM.  The graph in the center shows the pre-calibration curve in green and the post-calibration curve in red. As you can see, the sensor (or the instrument) had drifted over that time period, the sensitivity (slope of the calibration curve) becoming smaller and curve becoming noticeably more non-linear (concave down).  However, both the pre- and post-calibration curves fit the quadratic calibration equations very well, as indicated by the residuals plot and the coefficients of determination (R2) listed below the graphs. The eight "unknown" samples that were measured for this test (yellow table) were actually the same sample measured repeatedly - a standard of concentration 1.00 units - but you can see that the sample gave lower instrument readings (column L) each time it was measured (column K), due to the drift.  Finally, the drift-corrected concentrations calculated by the spreadsheet (column M on the right) are all very close to 1.00, showing that the drift correction works well, within the limits of the random noise in the instrument readings and subject to the assumption that the drift in the calibration curve parameters is linear with time between the pre- and post-calibrations. If you are using the weighted version of this spreadsheet (see #4 above), enter the weights for each point in column A (usually from 0 to 1). There are pre-calculated weights for 1/X, 1/X2, 1/Y, and 1/Y2 in columns AN through AQ that you can Copy and Paste (numbers only) into Column A.  Alternatively, you can enter equations into column A that calculate weights in any way you wish.

### Frequently Asked Questions (taken from actual search engine queries)

1. Question: What is the the purpose of calibration curve?
Answer:   Most analytical instruments generate an electrical output signal such as a current or a voltage.  A calibration curve establishes the relationship between the signal generated by a measurement instrument and the concentration of the substance being measured. Different chemical compounds and elements give different signals. When an unknown sample is measured, the signal from the unknown is converted into concentration using the calibration curve.

2. Question:  How do you make a calibration curve?
Answer:  You prepare a series of "standard solutions" of the substance that you intend to measure, measure the signal (e.g. absorbance, if you are doing absorption spectrophotometry), and plot the concentration on the x-axis and the measured signal for each standard on the y-axis. Draw a straight line as close as possible to the points on the calibration curve (or a smooth curve if a straight line won't fit), so that as many points as possible are right on or close to the curve.

3. Question: How do you use a calibration curve to predict the concentration of an unknown sample? How do you determine concentration from a non-linear calibration plot?
Answer:  This can be done in two ways, graphically and mathematically. Graphically, draw a horizontal line from the signal of the unknown on the y axis over to the calibration curve and then straight down to the concentration (x) axis to the concentration of the unknown. Mathematically, fit an equation to the calibration data, and solve the equation for concentration as a function of signal. Then, for each unknown, just plug its signal into this equation and calculate the concentration. For example, for a linear equation, the curve fit equation is Signal = slope * Concentration + intercept, where slope and intercept are determined by a linear (first order) least squares curve fit to the calibration data. Solving this equation for Concentration yields Concentration (Signal - intercept) / slope, where Signal is the signal reading (e.g. absorbance) of the unknown solution. (Click here for a fill-in-the-blank OpenOffice spreadsheet that does this for you.

4. Question:   How do I know when to use a straight-line curve fit and when to use a curved line fit like a quadratic or cubic?
Answer:  Fit a straight line to the calibration data and look at a plot of the "residuals" (the differences between the y values in the original data and the y values computed by the fit equation). Deviations from linearity will be much more evident in the residuals plot than in the calibration curve plot. (Click here for a fill-in-the-blank OpenOffice spreadsheet that does this for you. View screen shot). If the residuals are randomly scattered all along the best-fit line, then it means that the deviations are caused by random errors such as instrument noise or by random volumetric or procedural errors; in that case you can use a straight line (linear) fit. If the residuals have a smooth shape, like a "U" shape, this means that the calibration curve is curved, and you should use a non-linear curve fit, such as a quadratic or cubic fit. If the residual plot has a "S" shape, you should probably use a cubic fit. (If you are doing absorption spectrophotometry, see Comparison of Curve Fitting Methods in Absorption Spectroscopy).

5. Question:  What if my calibration curve is linear at low concentrations but curves off at the highest concentrations?
Answer:  You can't use a linear curve fit in that case, but if the curvature is not too severe, you might be able to get a good fit with a quadratic or cubic fit. If not, you could break the concentration range into two regions and fit a linear curve to the lower linear region and a quadratic or cubic curve to the higher non-linear region.

6. Question:  What is the difference between a calibration curve and a line of best fit? What is the difference between a linear fit and a calibration curve.
Answer:  The calibration curve is an experimentally measured relationship between concentration and signal. You don't ever really know the true calibration curve; you can only estimate it at a few points by measuring a series of standard solutions. Then draw a line or a smooth curve that goes as much as possible through the points, with some points being a little higher than the line and some points a little lower than the line. That's what we mean by that is a "best fit" to the data points.  The actual calibration curve might not be perfectly linear, so a linear fit is not always the best.  A quadratic or cubic fit might be better if the calibration curve shows a gradual smooth curvature.

7. Question: Why does the slope line not go through all points on a graph?
Answer:  That will only happen if you (1) are a perfect experimenter, (2) have a perfect instrument, and (3) choose the perfect curve-fit equation for your data. That's not going to happen. There are always little errors. The least-squares curve-fitting method yields a best fit, not a perfect fit, to the calibration data for a given curve shape (linear. quadratic, or cubic). Points that fall off the curve are assumed to do so because of random errors or because the actual calibration curve shape does not match the curve-fit equation.

Actually, there is one artificial way you can make the curve go through all the points, and that is to use too few calibration standards: for example, if you use only two points for a straight-line fit, then the best-fit line will go right through those two points no matter what. Similarly, if you use only three points for a quadratic fit, then the quadratic best-fit curve will go right through those three points, and if you use only four points for a cubic fit, then the cubic best-fit curve will go right through those four points.  But that's not really recommended, because if one of your calibration points is really off by a huge error, the curve fit will still look perfect, and you'll have no clue that something's wrong. You really have to use more standards that that so that you'll know when something has gone wrong.

8. Question:  What happens when the absorbance reading is higher than any of the standard solutions?
Answer:  If you're using a curve-fit equation, you'll still get a value of concentration calculated for any signal reading you put in, even above the highest standard. However, it's risky to do that, because you really don't know for sure what the shape of the calibration curve is above the highest standard. It could continue straight or it could curve off in some unexpected way - how would you know for sure?  It's best to add another standard at the high end of the calibration curve.

9. Question: What's the difference between using a single standard vs multiple standards and a graph?
Answer:  The single standard method is the simplest and quickest method, but it is accurate only if the calibration curve is known to be linear. Using multiple standards has the advantage that any non-linearity in the calibration curve can be detected and avoided (by diluting into the linear range) or compensated (by using non-linear curve fitting methods). Also, the random errors in preparing and reading the standard solutions are averaged over several standards, which is better than "putting all your eggs in one basket" with a single standard. On the other hand, an obvious disadvantage of the multiple standard method is that it requires much more time and uses more standard material than the single standard method.

10. Question: What's the relationship between sensitivity in analysis and the slope of standard curve?
Answer:  Sensitivity is defined as the slope of the standard (calibration) curve.

11. Question: How do you make a calibration curve in Excel or in OpenOffice?
Answer:  Put the concentration of the standards in one column and their signals (e.g. absorbances) in another column.  Then make an XY scatter graph, putting concentration on the X (horizontal) axis and signal on the Y (vertical) axis.  Plot the data points with symbols only, not lines between the points.  To compute a least-squares curve fit, you can either put in the least-squares equations into your spreadsheet, or you can use the built-in LINEST function in both Excel and OpenOffice Calc to compute polynomial and other curvilinear least-squares fits.  For examples of OpenOffice spreadsheets that graphs and fits calibration curves, see Worksheets for Analytical Calibration Curves.

12. Question:  What's the difference in using a calibration curve in absorption spectrometry vs other analytical methods such a fluorescence or emission spectroscopy?
Answer:  The only difference is the units of the signal. In absorption spectroscopy you use absorbance (because it's the most nearly linear with concentration) and in fluorescence (or emission) spectroscopy you use the fluorescence (or emission) intensity, which is usually linear with concentration (except sometimes at high concentrations). The methods of curve fitting and calculating the concentration are basically the same.

13. Question:  If the solution obeys Beer's Law, is it better to use a calibration curve rather than a single standard?
Answer:  It might not make much difference either way. If the solution is known from previous measurements to obey Beer's Law exactly on the same spectrophotometer and under the conditions in use, then a single standard can be used (although it's best if that standard gives a signal close to the maximum expected sample signal or to whatever signal gives the best signal-to-noise ratio - an absorbance near 1.0 in absorption spectroscopy). The only real advantage of multiple standards in this case is that the random errors in preparing and reading the standard solutions are averaged over several standards, but the same effect can be achieved more simply by making up multiple copies of the same single standard (to average out the random volumetric errors) and reading each separately (to average out the random signal reading errors). And if the signal reading errors are much smaller than the volumetric errors, then a single standard solution can be measured repeatedly to average out the random measurement errors.

14. Question:  What is the effect on concentration measurement if the monochromator is not perfect?
Answer:  If the wavelength calibration if off a little bit, it will have no significant effect as long as the monochromator setting is left untouched between measurement of standards and unknown sample; the slope of the calibration curve will be different, but the calculated concentrations will be OK. (But if  anything changes the wavelength between the time you measure the standards and the time you measure the samples, an error will result). If the wavelength has a poor stray light rating or if the resolution is poor (spectral bandpass is too big), the calibration curve may be effected adversely.  In absorption spectroscopy, stray light and poor resolution may result in non-linearity, which requires a non-linear curve fitting method. In emission spectroscopy, stray light and poor resolution may result in a spectral interferences which can result in significant analytical errors.

15. Question:  What does it mean if the intercept of my calibration curve fit is not zero?
Answer:  Ideally the y-axis intercept of the calibration curve (the signal at zero concentration) should be zero, but there are several reasons why this might not be so.  (1) If there is substantial random scatter in the calibration points above and below the best-fit line, then it's likely that the non-zero intercept is just due to random error. If you prepared another separate set of standards, that standard curve would have different intercept, either positive or negative. There is nothing that you can do about this, unless you can reduce the random error of the standards and samples. (2) If the shape of the calibration curve does not match the shape of the curve fit, then it's very likely that you'll get a non-zero intercept every time. For example, if the calibration curve bends down as concentration increases, and you use a straight-line (linear) curve fit, the intercept will be positive (that is, the curve fit line will have a positive y-axis intercept, even if the actual calibration curve goes through zero). This is an artifact of the poor curve fit selection; if you see that happen, try a different curve shape (quadratic or cubic). (3) If the instrument is not "zeroed" correctly, in other words, if the instrument gives a non-zero reading when the blank solution is measured. In that case you have three choices: you can zero the instrument (if that's possible); you can subtract the blank signal from all the standard and sample readings; or you can just let the curve fit subtract the intercept for you (if your curve fit procedure calculates the intercept and you keep it in the solution to that equation, e.g. Concentration (Signal - intercept) / slope).

16. Question:  How can I reduce the random scatter of calibration points above and below the best-fit line?
Answer:  Random errors like this could be due either to random volumetric errors (small errors in volumes used to prepare the standard solution by diluting from the stack solution or in adding reagents) or they may be due to random signal reading errors of the instrument, or to both. To reduce the volumetric error, use more precise volumetric equipment and practice your technique to perfect it (for example, use your technique to deliver pure water and weigh it on a precise analytical balance). To reduce the signal reading error, adjust the instrument conditions (e.g. wavelength, path length, slit width, etc) for best signal-to-noise ratio and average several readings of each sample or standard.

17. Question:  What are interferences? What effect do interferences have on the calibration curve and on the accuracy of concentration measurement?
Answer:  When an analytical method is applied to complex real-world samples, for example the determination of drugs in blood serum, measurement error can occur due to interferences. Interferences are measurement errors caused by chemical components in the samples that influence the measured signal, for example by contributing their own signals or by reducing or increasing the signal from the analyte. Even if the method is well calibrated and is capable of measuring solutions of pure analyte accurately, interference errors may occur when the method is applied to complex real-world samples. One way to correct for interferences is to use "matched-matrix standards", standard solution that are prepared to contain everything that the real samples contain, except that they have known concentrations of analyte. But this is very difficult and expensive to do exactly, so every effort is made to reduce or compensate for interferences in other ways. For more information on types of interferences and methods to compensate for them, see Comparison of Analytical Calibration Methods.

18. Question:  What are the sources of error in preparing a calibration curve?
Answer:  A calibration curve is a plot of analytical signal (e.g. absorbance, in absorption spectrophotometry) vs concentration of the standard solutions. Therefore, the main sources of error are the errors in the standard concentrations and the errors in their measured signals. Concentration errors depend mainly of the accuracy of the volumetric glassware (volumetric flasks, pipettes, solution delivery devices) and on the precision of their use by the persons preparing the solutions.  In general, the accuracy and precision of handling large volumes above 10 mL is greater than that at lower volumes below 1 mL. Volumetric glassware can be calibrated by weighing water on a precise analytical balance (you can look up the density of water at various temperatures and thus calculate the exact volume of water from its measured weight); this would allow you to label each of the flasks, etc, with their actual volume. But precision may still be a problem, especially a lower volumes, and it's very much operator-dependent. It takes practice to get good at handling small volumes. Signal measurement error depends hugely on the instrumental method used and on the concentration of the analyte; it can vary from near 0.1% under ideal conditions to 30% near the detection limit of the method.  Averaging repeat measurements can improve the precision with respect to random noise. To improve the signal-to-noise ratio at low concentrations, you may consider modifying the conditions, such as changing the slit width or the path length, or using another instrumental method (such as a graphite furnace atomizer rather than flame atomic absorption).

19. Question:  How can I find the error in a specific quantity using least square fitting method? How can I estimate the error in the calculated slope and intercept?
When using a simple straight-line (first order) least-squares fit, the best fit line is specified by only two quantities: the slope and the intercept. The random error in the slope and intercept (specifically, their standard deviation) can be estimated mathematically from the extent to which the calibration points deviate from the best-fit line. The equations for doing this are given here and are implemented in the "spreadsheet for linear calibration with error calculation".  It's important to realize that these error computations are only estimates, because they are based on the assumption that the calibration data set is representative of all the calibration sets that would be obtained if you repeated the calibration a large number of times - in other words, the assumption is that the random errors (volumetric and signal measurement errors) in your particular data set are typical. If your random errors happen to be small when you run your calibration curve, you'll get a deceptively good-looking calibration curve, but your estimates of the random error in the slope and intercept will be too low. If your random errors happen to be large, you'll get a deceptively bad-looking calibration curve, and your estimates of the random error in the slope and intercept will be too high. These error estimates can be particularly poor when the number of points in a calibration curve is small; the accuracy of the estimates increases if the number of data points increases, but of course preparing a large number of standard solutions is time consuming and expensive.  The bottom line is that you can only expect these error predictions from a single calibration curve to be very rough; they could easily be off by a factor of two or more, as demonstrated by the simulation "Calibration Curve Method with Linear Curve Fit". Download in Excel or Calc format.

20. Question:  How can I estimate the error in the calculated concentrations of the unknowns?
You can use the slope and  intercept from the least-squares fit to calculate the concentration of an unknown solution by measuring its signal and computing (Signal - intercept) / slope, where Signal is the signal reading (e.g. absorbance) of the unknown solution. The errors in this calculated concentration can then be estimated by the usual rules for the propagation of error: first, the error in (Signal - intercept) is computed by the rule for addition and subtraction; second, the error in (Signal - intercept) / slope is computed by the rule for multiplication and division. The equations for doing this are given here and are implemented in the "spreadsheet for linear calibration with error calculation".  It's important to realize that these error computations are only estimates, for the reason given in #19 above, especially if the number of points in a calibration curve is small, as demonstrated by the simulation "Calibration Curve Method with Linear Curve Fit". Download in Excel or Calc format.

21. Question:  What is the minimum acceptable value of the coefficient of determination (R2)?
It depends on the accuracy required. As a rough rule of thumb, if you need an accuracy of about 0.5%, you need an R2 of 0.9998; if a 1% error is good enough, an
R2 of 0.997 will do; and if a 5% error is acceptable, an R2 of 0.97 will do. The bottom line is that the R2 must be pretty darned close to 1.0 for quantitative results in analytical chemistry.

22.
Question:  Why put concentration on the x-axis and absorbance on the y-axis; can I do it in reverse?
Ordinarily, the correct way is to put the standard concentrations on the x axis and the absorbance values on the y axis, because the usual least-squares curve fitting computations make the assumption that the major uncertainty is the y -axis values (the dependent variable), which is usually the case. Then, when you read the unknown solutions, you must solve the calibration equation for concentration:
Concentration  (Absorbance- intercept) / slope). An example is shown in the figure below on the left.

If you do it the other way, and assume that absorbance is the independent variable, then the results of the
least-squares computation will be different. This is shown in the figure above on the right. The result is that the best-fit slope and intercept will be different (specifically, the slope will be the reciprocal of the slope computed the usual way). Then, when you read the unknown solutions, you won't need to solve the calibration equation for concentration, because Concentration  Absorbance* slope+intercept. The resulting calculated concentrations will be exactly the same if there were no scatter in the calibration curve and if all point fall exactly on the line. But if there is some scatter, then there will be a slight difference in the calculated concentrations (compare the values in the Calculated concentration column in the two figures above). Assuming that the scatter in the calibration points is due to errors in reading in absorbance, and not errors in preparing the standard solutions, then the correct method is to put the standard concentrations on the x axis and the absorbance values on the y axis, and to perform the least-squares computation as if the x axis contains the independent variable. But if the scatter in the calibration points is due to errors in preparing the standard solutions, and not reading absorbance, then it's better if the columns and axes are switched and the calculations assume that absorbance is the independent variable.

But there is another situation when you might want to exchange the x and y axes, and that is when you want to fit a non-linear calibration curve, using a quadratic (2nd order) or cubic (3rd order) least-squares fit.  In those cases,
solving the calibration equation for concentration is more troublesome, especially for the cubic fit. The math is much simpler if the columns and axes are switched and the calculations assume that absorbance is the independent variable, because in that case you won't need to solve the calibration equation for concentration. See #5, "reversed cubic fit".

(c) 1991, 2016. This page is part of Interactive Computer Models for Analytical Chemistry Instruction, and "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver , Professor Emeritus, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu.
Number of unique visits since May 17, 2008. Last updated Marcj, 2017