1080 Anal. Chem. 1989, 61, 1080-1083

Calibration of Pipets: A Statistical View

Lowell M. Schwartz

Department of Chemistry, University of Massachusetts, Boston, Massachusetts 02125

Fixed volume pipets are typically calibrated gravimetrically by weighing replicate delivered aliquots. When a single weighlng vessel is used for this purpose, the procedure must be corrected for evaporation of the liquid, and if the vessel Is not tared between deliveries, the data analysis must be modified for serlal correlatlon. Appropriate statistical methods are shown for the calculation of both the systematic volume calibration and the standard deviation of the delivery volumes. The methods are illustrated by experiments calibrating a microliter pipet.

This communication discusses statistical methodology re- quired for rigorous calibration of devices that are designed to deliver a fixed aliquot volume without having to read volume graduations lines. Specifically, these include transfer pipets, Ostwald-Folin pipets and dispensing micropipets but not graduated volumetric glassware such as measuring, se- rological, or syringe pipets or burets unless these are used as fixed-volume devices by ignoring all but a single graduation line. The pipet may be used either “to contain” (TC) or “to deliver” (TD) the aliquot. By calibration we refer to the conventional volumetric calibration procedure using gravi- metry, i.e. by transferring aliquots of known density from the pipet to a weighing vessel and measuring the masses trans- ferred. We are particularly interested in the statistical un- certainties of the delivered volumes as well as their accuracies. As in any calibration, the experimental conditions used in the calibration procedure should match as closely as possible the conditions of the experiment itself.

The calibration of volumetric devices by gravimetry is simple, effective, and well-known and under ordinary cir- cumstances this procedure and the associated data analysis are straightforward. The analyst preweighs or tares a series of weighing vessels, dispenses an aliquot into each, and weighs each vessel. Assuming that the balance is properly calibrated and that its random measurement error is negligible relative to the random error of the weights of the dispensed volumes, the ratio of the mean of the aliquot weights to the liquid density is a suitable measure of the pipet’s delivery volume and the ratio of the standard deviation of the weights to the density is a suitable measure of the statistical uncertainty of that volume. If the analyst wishes to perform a highly reliable calibration, he needs a large data set, which requires a large number of weighing vessels. But the use of multiple weighing vessels is optional. The alternative is to use a single weighing vessel and to transfer the series of aliquots from the pipet into this one vessel, weighing after each transfer. Also the vessel may or may not be tared prior to the delivery of each aliquot. Depending on the operation of the balance, the taring oper- ation may be somewhat time-consuming and troublesome. On the other hand, the data analysis is simpler if taring is done. For example, if the capacity of the volumetric device is small, the evaporation of liquid from the weighing vessel may in- troduce a significant error over the longer time span required to complete an untared series of aliquot deliveries. Although as will be shown below, it is possible to correct for this effect in an untared calibration, the taring operation effectively

eliminates the effect of evaporation if weight measurements are made quickly or if the weighing vessel is stoppered. An- other problem that arises in an untared calibration is that the statistical data analysis is very cumbersome if there is a sig- nificant random error introduced by the weighing operation. If the balance is tared before each delivery, however, the effect of weighing errors is more easily treated. In this paper we focus on gravimetric calibration of pipets with a single weighing vessel using both the tared and untared procedures.

THEORY The volumetric apparatus is designed by the manufacturer

to deliver a nominal fixed volume of liquid. We let the symbol u,, represent this volume. However, vo is not the actual volume delivered because of the inevitable occurrence of two types of errors, systematic and random. A systematic error, which we denote by Av, may partly reflect variation within the manufacturer’s tolerance and may partly result from differ- ences in dispensing technique between that assumed by the manufacturer and that used in practice by the analyst. A random error, which we denote by tv, is involved in the dis- pensing operation and which fluctuates with successive aliquot deliveries, the magnitude depending on the operation skill of the analyst. The true volume u of any given aliquot delivered to the weighing vessel is u = uo + Au + 6,. The variance of t,. is assumed to be a constant (r: and to have a mean (ex- pectation) value of zero. The objective of the calibration experiment is to determine both quantities Av and u:.

We assume that the balance is properly calibrated so that negligible systematic error is involved in its readings. How- ever, we allow the possibility that the balance readings are subject to random errors tW caused by undamped mechanical disturbances or electronic fluctuations. We assume that the variance of tw is a constant uw2 over the limited weight range of a given calibration experiment.

The liquid used for the calibration experiment has a known density p.

Calibration with Taring. Because it is not possible to determine both u: and uw2 simultaneously, we must perform a separate experiment to find uw2. This is easily done by repeated weighing of a dry weighing vessel or any substitute object having similar mass. The variance of the series of recorded weights is sw2, which is an estimate of uW2. In order to obtain a good estimate of sw2, a reasonably large statistical sample is required, perhaps 30 or more replications rather than only a few. A plot should be made of the sequence of weights vs the order of acquisition. If the weights do not scatter randomly about the mean, the drift of the balance or some other source of systematic error in the experiment must be corrected.

The calibration experiment simply involves a series of re- peated steps: tare the balance loaded with the weighing vessel, dispense an aliquot, stopper the weighing vessel if evaporation is significant, and record the weight. Again a reasonably large statistical sample of replications is required in order to de- termine the calibration parameters with satisfactory precision. It is important that the density of the liquid be known and constant. Therefore, the liquid must be equilibrated at am- bient temperature and the temperature must not change during the calibration experiment.

0003-2700/89/0361-1080$01.50/0 1989 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989 1081

cov (w1,w2) = (dE1/de,,)(dE2/dtV1) var tvl = p2a: If these experimental conditions are realized, any given recorded weight w is given by w = pu + t, = p(u0 + Au + 6,)

+ t,. The random variable w has expectation value p(uo + Au) and variance p2u: + uwz. Thus the mean of the recorded weights is the best estimate of p(uo + Au) and the systematic calibration correction Au is easily calculated from a knowledge of uo and p. The sample variance of the weights is p2s? + sW2 from which the estimate s: of the variance of the aliquot delivery error is again easily calculated by using p2 and sW2.

Calibration without Taring. We shall consider here only the case where evaporation of liquid may be significant but that the random weighing error t, is negligible relative to other sources of random fluctuation. In practice this means that the balance must be able to operate with minimal mechanical and electronic fluctuation and that any weighing repeatability error must be negligibly small. These conditions may be difficult to achieve when attempting to calibrate a micropipet having a capacity of a few microliters using an analytical balance which reads to a tenth of a milligram. In these cir- cumstances the calibration is best done with taring as de- scribed above.

The experiments begins a t time t = 0 when the first aliquot is delivered to a weighing vessel having unknown weight wo. After a number n aliquots have been dispensed, the true total volume delivered is

V, = = ~ ( u O + Au) + E,, (1) where we have denoted the random error in V, by E,, and which error is described in more detail below. When the balance has equilibrated after delivery of the nth aliquot, the observed weight w, is recorded, and the time t , is recorded. We shall assume that the random measurement error in the recorded time is negligible relative to other sources of fluc- tuation. The observed nth weight is the sum of the initial weight wo and the weights of n aliquots less the weight loss due to evaporation, i.e.

(2) where p is the known density of the liquid and a is the un- known evaporation rate, a negative quantity. The requirement that the evaporation rate be constant during experiment re- quires that the liquid used for the calibration be degassed as well as be at ambient temperature. After substituting for V, from eq 1, the observed weights are

(3) Our objective is to perform a multiple regression calculation, regressing w, on the aliquot index n and on the time t,, to calculate the three least-squares coefficients wo, p(uo + Au), and a, to calculate residuals between the regression and the w, data, to calculate the variance of these residuals, and from this variance to calculate the variance of the aliquot volumes.

The random error E,, in the total volume of n dispensed aliquots is the sum of the nth aliquot error and all previous aliquot errors, i.e.

w, = wo + p V , + at,

w, = wo + p(uo + Au)n + at, + pE,,

n

E,, = Ce,, (41 1=1

and we see from eq 3 and 4 that the random error component in w, for the nth weighing is p z : t v i . Thus any particular w, is statistically correlated with each previous weighing. This serial correlation is due to the fact that the total volume dispensed is the sum of all previous individual aliquots dis- pensed and a random fluctuation in any single aliquot con- tinues to contribute to the total volume error throughout the rest of the experiment. With this understanding, the variances of the recorded weighings are var w, = pznu; and the co- variances between weighings develop through the common dependencies on the aliquot volume errors as, for example

or

By generalizing from these results, we see that the covariance between weighing wi and weighing w, is cov (wi,wj) = min- (io?,ju?), meaning that cov (wi,wj) is iu? if i < j but is ju: if j < i.

The method of least squares can be used to fit w, vs n and t , data provided that suitable account is taken of the variances and covariances between the w, data. The appropriate methodology is given in matrix form by Meyer ( I ) and is adapted to our notation as follows:

If the total number of recorded data points in the calibration experiment is N , and if p = 3 unknown coefficients are fitt+d, the design matrix a corresponding to linear model eq 3 is

(5)

a (N X p ) matrix, one row for each data point. The first column is the unit function multiplying the unknown coef- ficient c1 = wo, the second column contains the indices of aliquot volumes multiplying the unknown coefficient c2 = p(uo + Au), and the third column contains the recorded times multiplying the unknown coefficient c3 = a. The requirement that each column be linearly independent dictates that the data not be recorded at equally spaced intervals of time, but there is no practical difficulty in meeting this requirement. However, if evaporation of the liquid is not significant, only p = 2 regressors are required and the third column in a is omitted.

The N component vector of measurements of the random variable w is y = (w,, w2, ..., wN). The p = 3 component vector of unknown coefficients is c = (cl, c2, c3). The ( N X N) covariance matrix [cov(y)] of the variables w, has diagonal elements which are the variances var w, and off-diagonal elements which are the covariances between pairs (wi, w,) as shown in eq 6, where the (N X N) symmetrical matrix of

weighting factors W is defined through its inverse in eq 6. After p-2 is factored out from W, the matrix elements on the diagonal are all 2 except for the Nth which is 1, the adjacent diagonal elements are -1, and all other elements are zero as follows:

r2 -1 . 07

With these matrices and vectors defined, the least-squares estimate of the unknown coefficients are the three elements of the vector 2 such that

b = (aTWa)-'(aTWy) ( 7 )

In particular the fitted least-squares coefficient c^2 is the best estimate of the aliquot mass p(uo + Au) and this value together

1082 ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

with p and uo determines the systematic volume correction Ac.

The residuals of the data points about the fitted regression are the N components of the column vector r = (y - ai?). The weighted sum of squares of these residuals is

p2(SSmin) = rTWr = yTWy - yTWai. (8)

which is the minimum sum of squares achieved by the least-squares procedure. This quantity is used to calculate an estimate sb2 of the pipet volumetric variance g: from

(9)

If standard error estimates are required for any of the fitted parameters, these may be calculated from the symmetric ( p x p ) variance-covariance matrix

sV2 = ss,,,/(N - P )

(VCV) = p2s,Z(aTWa)-' (10)

which has parameter variances on the diagonal and covari- ances off the diagonal.

In the event that evaporation of liquid is not significant, the equations reduce to particularly simple algebraic forms so that matrix manipulations need not be used to calculate the least-squares parameters from the experimental data. If the matrix operations of eq 5-10 are expanded algebraically with only the first two columns of the a matrix in place, the following formulas are obtained:

0.01

(11)

-- .. . . - +,c.THc.- -,..--22+.-,3, lu-JI;.r j .

(12)

1

--f_(2w,u;N - NwI2 - N w X 2 ) (13a) N - 1

p2SSmin = 2 C r i 2 - 2 C rir, + rN2 (13b) i = l li-jl=l

where the residual r, is r , = w', - E l - E 2 i

and where the index notation li - j ( = 1 means all ( i , j ) pairs differing by 1, i.e. (1, 2), (2, 3), (3, 4), ... (N- 1, If). Equation 13b is preferable to eq 13a in computation because of round-off error considerations. Also the parameter variances are

var E , = __ (14) p2s,2N A' - 1

and

P 2 S V 2 var S2 = - N - 1

I t is interesting to note that Mandel (2, 3) , recognizing the phenomenon of serial correlation in certain physicochemical systems. derived results that are equivalent to eq 12 and 15 using a different approach of taking differences between successive measurements. In ref 3 he discusses and defends the somewhat surprising fact that the least-squares parameters as given by his eq 12.81 or by our eq 11 and 12 above are independent of all the recorded measurements except the first and the last.

EXPERIMENTAL SECTION We demonstrate here calculational procedures for both tared

and untared gravimetric calibration of a microliter pipet using a single weighing vessel. In both experiments we used the same apparatus: a 1000 1L capacity Gilson micropipet set at 20.0%

0.034 B

7 !Y -0.01 4

0.03 t C I I

Anal. Chem. 1989, 61, 1083-1086 1083

eq 11 and 12) were = 0.1854 g and e2 = 0.1892 g. The residuals between the weight measurements and the model equation fitted with these two parameter values are shown in Figure 1A. The residual plot is nonrandom with clear evidence of substantial evaporative loss between aliquots 20 and 21. Using the same data set with the evaporation term in the model equation yielded SI = 0.1846 f 0.0034 g, tp = 0.19064 f 0.00058 g, and t3 = -6.3 X lo4 f 0.9 X g s-l, where the uncertainties quoted are standard error estimates calculated from eq 8 ,9 , and 10 and based on 37 degrees of freedom. From t2 = p(vo + Av) we calculate the sys- tematic volume correction Au = -0.0094 & 0.0006. A residual plot for this fit is shown in Figure 1B. The evaporation term appears to have largely compensated for the loss between aliquots 20 and 21 shown in Figure 1A. Although the nonrandom pattern of residuals in Figure 1B seems to indicate a lack of fit of the model equations to the data, Mandel (3) points out that this type of pattern is characteristic of serially correlated data even when properly fitted by a model. Because the random error in each data point includes the random errors of all previous data points, there are fewer zero-crossings of successive residuals than would be expected in a set of independently random residuals. The independent random component of any particular measurement can be recovered from its residual by subtracting the immediately preceding residual. Thus a plot of successive residual differences as shown in Figure 1C is a more appropriate monitor of lack of model fit for serially correlated data. The variance of residuals from eq 8 was 1.15 X g2, which leads to a volumetric standard deviation s, = 0.0034 mL based on 37 degrees of freedom. This

level of random error seems visually consistent with the residual plot in Figure 1C but not with the plot in Figure 1B.

Finally, we check the statistical consistencies of the results of the two experiments. Experiment 1 yields Av of -0.0084 f 0.0005 mL after correcting for evaporation while experiment 2 yields -0.0094 & 0.0006. The difference is 0.0010 mL and the standard error estimate of the difference is about 0.0008 mL. A t test for the hypothesis that the difference is within statistical scatter of zero leads to an acceptance of this hypothesis at any reasonable confidence level. Secondly, experiment 1 yields a volumetric variance estimate s,2 = 1.02 x mL2 based on 39 degrees of freedom while experiment 2 yields a corresponding value of 1.16 X mL based on 37 degrees of freedom. The ratio of these variances is 1.14, which is less than the tabulated F-statistic for any reasonable confidence level. Thus we may accept the null hypothesis that the two variances are from the same statistical population and conclude that the volumetric standard deviations of 0.0032 and 0.0034 mL are consistent.

LITERATURE CITED (1) Meyer, S. L. Data Analysis for Scientists and Engineers; Wiley: New

York, 1975; Chapter 33. (2) Mandel, J. J . Am. Statist. Assoc. 1957, 52, 552-566. (3) Mandel, J. The Statistical Analysis of Experimental Data; Wiley: New

York, 1964; Sections 12.7, 12.8.

RECEIVED for review September 15,1988. Accepted February 15, 1989.

Laser Desorption from a Probe in the Cavity of a Quadrupole Ion Storage Mass Spectrometer

David N. Heller,* Ihor Lys, and Robert J. Cotter Middle Atlantic Mass Spectrometry Facility, Johns Hopkins University School of Medicine, Baltimore, Maryland 21205

0. Manuel Uy

Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland 20707

I t is possible to trap and mass-analyze ions that have been laser desorbed from a probe withln the ion storage volume of a quadrupole ion storage mass spectrometer. A Flnnigan ion trap detector was modlfled to enable a C02 laser pulse to strike a probe tip that passed radially through the ring eiec- trode. Molecular and fragment Ions from a variety of organic and blochemlcai compounds have been obtained. These ions are typical of COP laser desorption (LD) processes, while the instrument itself retains tunlng characteristics typlcal of the orlglnal electron Impact (E1 ) lonlzation performance. Be- cause the E1 filament assembly was left Intact, it has been posslbie to operate In LD/EI mode and produce additlonai fragment Ions.

INTRODUCTION In recent years a new approach to the operation of the

quadrupole ion storage trap, or QUISTOR ( l ) , has been in- troduced which has greatly expanded its utility as an analytical mass spectrometer (2) . This new approach, commercialized

by the Finnigan Corp. (San Jose, CA), involves the use of the “mass selective instability” scanning mode, wherein trapped ions are successively allowed to become unstable and fall out of the trap to strike a detector. Furthermore, relatively high pressures of a light gas are used to damp the motion of higher mass ions, bringing them to the center of the trap and thereby extending the mass range of the instrument. The Finnigan ion trap detector (ITD) was originally designed for interfacing to a gas chromatograph, but since its introduction many further modifications have been carried out. These include (but are not limited to) the capability for chemical ionization (3) , negative ion detection ( 4 ) , photoionization (5), tandem mass spectrometry (6), and interfacing to a supercritical fluid chromatograph (7). However, all of these improvements have depended on formation of ions from sample molecules already in the gas phase.

It has been of interest to us to discover if a quistor can be used to store and analyze ions formed by desorption tech- niques, and we have modified an ITD to enable a laser pulse to impinge on a probe within the storage volume itself. A different approach to the problem of mating alternative ion- ization methods with the ITD makes use of a lens system to

0003-2700/89/0361-1083$01.50/0 0 1989 American Chemical Society