Anal. Chem. 1983, 55,1611-1614
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Inaccuracies in Compuiter Calculation of Standard Deviation Sir: In a recent issue, Wanek et al. (11 drew attention to the importance of round-off errors when computing the standard deviation on ellectronic calculators. They compared the following three formullas for the sample standard deviation:
so = (CCX,- X ) Z / ( n- 1 ) y
Table I. Standard Deviation (SD) Computed on an IBM 4341 by the Four Algorithms Described in the Texta algorithm
SO S I
S* S3
These three formulas are algebraic identities, but SI and S2 allow more efficient computation via calculator or computer. The authors found that the use of So provided the most accurate estimate. SI and S2were sensitive to round-off errors. I want to point out that this problem is equally important when using improperly made computer programs. The round-off errors may be even greater on1 a computer when compared with an electronic calculator, since a computer's real variable of four bytes has a precision of only six decimal digits. For example, the subroutine TALLY in the popular IBM Scientific Subroutine Package (2) calculates the standard deviation as SI. It will be shown below (Table I) that this algorithm may be greatly inaccurate. Many computer programs use the safe Soestimate, but this algorithm needs two program loops, one for the mean X and one for the sum off squared deviations SS =: C(X,- Xi2. With large sets of data, especially when they are too large to be kept in memory, a faster, but still accurate, method is to compute the provisional (or running) mean X&and sum of squared deviations SS,.
single precision SD error,b %
-
0.000105 0.007813
2.9 7560 c
C
0.000097
-4.9
i = 1, n
Xi= Xi-, + di/i
ssi = ssi-1-t.
0 0 0
*
This algorithm is used, for example, in the BMDP package of statistical programs (3). Table I shows results obtained with the algorithms described above. The test data were taken from Wanek's paper (1) and consist of five numbers of eight digits with variation only in the last three. As should be expected, all algorithms worked well with double precision (approximately 15 decimal digits).
LITERATURE CITED (1) Wanek, P. M.; Whipple, R. E.; Fickles, T. E.; Grant, P. M. Anal. Cbsm. 1982, 5 4 , 1877-1878. (2) "IBM System/360 Scientific Subroutine Package, Programmer's Manual H20-0205-3"; International Business Machines Corp.; White Plalns, NY, 1968; p 27. (3) Dixon, W. J., Ed. "BMDP Statistical Software 1981"; University of California Press: Berkeley, CA, 1981; p 662.
Helge Erik Solberg
sso = 0 x. - X ,-I.
0.000102 0.000102 0.000102 0.000102
a Data (1): Xi = (19.51408, 19.51383,19.51388, 19.51386,19.51385). The errors are relative to the double precision So estimate. Impossible due to a negative argument to the square root function.
xo= 0 d; =
double precision SD error,b %
Department of Clinical Chemistry Rikshospitalet Oslo 1, Norway
i = 1, n
- Xi)
i = 1, n RECEIVED for review March 9, 1983. Accepted May 3, 1983.
Comparison of On-Line and Off-Line Liquid Chromatography/Nuclear Magnetic Resonance Spectrometry for Analysis of Steroid Mixtures Sir: The combination of chromatography and spectrometry is becoming increasingly important (1,2). As far as nuclear magnetic resonance spectrometry is concerned, the combination seems promising in cases where multicomponent mixtures with components imposing stereochemical problems are to be analyzed. The off-line combination is routine for both the high-pressure liquid chromatography/nuclear magnetic resonance spectrometry (HPLC/NMR) and for the gas chromatography/nuclea.r magnetic resonance spectrometry (GC/NMR). The on-line combination has only recently been described: for HI'LC/N.MR by others (3-5) and ourselves (6,
7) and for GC/NMR by ourselves (8). No investigation has been made to compare the off-line and on-line techniques. Here, we describe the advantages and disadvantages of off-line and on-line HPLC/NMR. Mixtures of steroids are analyzed by using hydrogen-chloroform as eluting solvent. In a similar fashion to previous studies (7) special techniques are applied to prevent overflow in the computer that may occur due to the large solvent vs. solute ratio. In principle other hydrogen containing solvents may be taken circumventing thus the use of the extremely expensive deuterated solvents. (With the exception of chloro-
0003-2700/83/0355-1611$01.50/00 1983 American Chemical Society
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* ANALYTICAL
CHEMISTRY, VOL. 55, NO. 9, AUGUST 1983
Ti
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i _ _
(ppmiB
rnin
i 2 36 30
7
5
6
i - 1
3
4
2
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Flgure 1. Chromatogram of a steroid mixture: (I) progesterone, (11) dehydroepiandrosterone, (I I I) testosterone (10 mg of each, dissolved in 100 mg chloroform),(v) vacant peak (YO, IY): column, 8 mm i.d., length 25 cm, filled with Li Chrosorb Si 100, 10 prn (Merck,Darmstadt); eluent, chloroform with 0.8% (v/v) methanol-d, and 0.2% deuterium oxide; flow, 1.9 mL/min; pressure, 40 bar.
form typical deuterated solvents are 100 to 1000 times more oxpensive than undeuterated ones.)
EXPERIMENTAL SECTION The spectra were recorded with the pulse NMR spectrometer Jeol FX-100, the field frequency stability of which was maintained by an external lock. The computer linked with the spectrometer consisted of a 12-bit ADC and a 32K 16 bit memory. A mixture of the three steroids progesterone, dehydroepiandrosterone, and testosterone was separated on a silica geI filled HPLC column, which was conditioned prior t o the separation procedure with the eluting solvent which consisted of 99% (by volume) [ lH]chloroform (the stabilizer ethanol has been removed by filtration through basic alumina), 0.8% methanol-d,, and 0.2% D20. On-Line Technique. The equipment used for the analysis of flowing HPLC fractions by NMR has been described earlier (6). Each fraction required 75 s to pass between HPLC exit and NMR probe. The sample volume in the NMR probe is 0.3 mL. Data were recorded in the pulse Fourier transform mode. The beginning and the end o f the pulse sequence were timed to coincide with the chromatogram (Figure 1)obtained with a refractive index detector. Off-Line Technique. A 0.5-mL portion of each fraction was collected in a NMR glass tube of 5 mm diameter. The collections were performed, whon a maximum of concentration passed the exit of tho column.
RESULTS AND DISCUSSION Figures 2 and 3 represent the NMR spectra of the three fractions obt~inedunder on-line conditions. The spectra in Figure 2 were obtained by suppressing the solvent line through selective presaturation (7). A computer with standard 16 bit words was used. The solvent signal io reduced by a factor of about 400 (compare solvent line with its partially reduced 13C satellite signals). Signals in the vicinity of the suppressed solvent signal are also reduced as can be seen from the integration curve. The percentage of reduction is given in Figure 2 and refers to a decoupler power of 1 W. The spectra in
5
i
l>prnl E
O@
7
6
5
4
3
i
1
b 6
Flgure 2. 'H NMR spectra of I, 11, and I11 in chloroform obtained
under on-line conditions. The solvent llne is suppressed by selective presatwratlon. The number of scans was 50; pulse repetition tlme, 3 s; 1, CD,HOD; 2,CH,BrCI; 3, CH,CI,; 4, 5, CHC13 and its I3C satellites. Numbers In percent, reduction caused by decoupling. Standard reference is the signal of chloroform set at 6 7.23 ppm. Figure 3 were obtained without any peak suppression. In order to prevent overflow in the computer, its word length has been extended from standard 16 to 32 bit by a double precision technique (7). Figure 4 shows the NMR spectra of the same components obtained by the off-line technique. The spectra were recorded under routine conditions. Recently, Bayer et al. (9)also analyzed steroid mixtures by on-line HPLC/NMR, but in the stopped flow variant and with a deuterated solvent. To compare the two techniques the methyl signal at 6 0.85 ppm of dehydroepiandrosterone (compound 11) was chosen. The line width is 1.96 Hz (both in the presaturation and in the double precision variant) compared to 1.46 Hz (off-line technique) and the signal-to-noise ratio is 28 (presaturation
ANALYTICAL CHEMISTRY, VOL. 55, NO. 9, AUGUST 1983
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5 5
/’
I
I
AI
-Avt=I.95 H z
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i I1 I1
I11 111
ippml7
Flgure 3. ’H NMR spectral of I,11, and 111 111 chloroform obtalned under on-tine conditions wlth the double precision technlque. For conditions, standard reference, and meaning of 1 to 5, see Flgure 2.
variant) and 20 (double precision variant) compared to 110 (off-line technique). The higher quality of the spectra received by the off-line technique is mainly due to the spinning of the sample (yielding sharper signals) and to the below discussed higher concentration of the solution (yielding higher signalto-noise ratio). The signal-to-noiseratio can be improved by a larger number of accumulations. For reasons of comparison,
6
5
4
3
2
1
0 s
Flgure 4. ‘H NMFl spectra of I, 11, and I11 in chloroform obtained under off-line conditions wlth the double precislon technique. For condltions, standard reference, and meaning of 1 to 5, see Flgure 2. Number 6 represents a spinning side band: X represents Impurities.
this number was chosen to be roughly the same in the three techniques. However, additional signals occur (see signals X in Figure 4) due tx, handling while collecting the fractions. The signal at 6 1.85 ppm arises from contamination with water. The origin of the signal at 6 1.3 ppm is not quite clear, it could arise from the side chain protons of the plasticizer dialkyl phthalate. The off-line technique imposes no technical problems. It
Anal. Chem. 1983, 55, 1614-1615
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produces fractions of higher concentration, because collections are only taken, while the maximum concentration is passing the detector. The fractions can be registered by NMR at any convenient time and long enough to get a sufficient signalto-noise ratio. The on-line technique demands a special tube connecting HPLC and NMR apparatus. The accumulation time and thus the signal-to-noise ratio is limited by the relatively short chromatographic separation procedure. This technique may be advantageous in cases where mixtures sensitive to air and moisture (organometallic compounds, “living” oligomers etc.) are to be analyzed. There is one problem that exists for both techniques: the interference of the NMR signal of the eluting solvent. Above, we have offered two techniques to diminish these interferences. However, these techniques are not optimal. Signals in the vicinity of the solvent line or under the solvent line are difficult, or impossible to detect. Thus, more effective solvent elimination techniques are needed before HPLC fractions can be conveniently investigated by NMR.
Griffiths, P. R. Appl. Spectrosc. 1977, 3 1 , 497. Watanabe, N.; Niki, E. R o c . Jpn. Acad., Ser. 6 1978, 5 4 , 194. Bayer, E.; Albert, K.; Nleder, M.; Grom, E.; Keller, T. J. Chromatogr. 1979, 186, 497. Haw, J. F.; Glass, T. E.; Hausler, D. W.; Motell, E.; Dorn, H. C. Anal. Chem. 1980, 52, 1135. Buddrus, J.; Herzog, H. Org. Magn. Reson. 1980, 13, 153. Buddrus, J.; Herzog, H.; Cooper, J. W. J. Magn. Reson. 1981, 4 2 ,
453. (8) Buddrus, J.; Herzog, H. Org. Magn. Reson. 1981, 15, 211. (9) Bayer, E.; Albert, K.; Nieder, M.; Grom, E.; Zhu An Fresenius’ Z. Anal. Chem. 1980, 304, 111. (IO) Solms, D. J.; Smuts, T. W.; Pretorlus, V. J . Chromatogr. Sci. 1971, 9 , 600. (11) Slais, K.; Krejci, M. J . Chromatogr. 1974, 91, 1981.
Joachim Buddrus* Helmut Herzog Institut fur Spektrochemie und angewandte Spektroskopie Bunsen-Kirchhoff-Strasse 11 D-4600 Dortmund 1, Federal Republic of Germany
LITERATURE CITED (1) Scott, R. P. W. I n “Techniques of Organic Chemistry”; Welssberger, A,, Ed.; Wiley: New York, 1982;Vol. X I , Chapter V I .
RECEIVED for review July 19,1982. Accepted April 22,1983.
AIDS FOR ANALYTICAL CHEMISTS Device for Continual Monitorlng of Flow Rates in Gas Chromatography
‘
Richard S. duvef, dr., * dean-Pierre Qllvo, and Georges Gulochon Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, Route de Saclay, Palaiseau, France
In a study of the precise quantitative resolution of strongly overlapping gas chromatographic peaks, to be published elsewhere, it is necessary to have continual, highly accurate, computer monitoring of the carrier gas flow rate. The device described allows the monitoring and computer storage of carrier gas flow rates with an accuracy of 0.1% and a measurement frequency of once every 30 s or longer.
EXPERIMENTAL SECTION This device makes use of a Gow-Mac (Bound Brook, NJ) Model 10-952-6 microthermal conductivity (T/C) cell. This T/C cell has a rapid response time owing to its flow-through geometry and small internal volume. A 31.8 cm length of stainlesssteel capillary tubing with approximately 1.6 mm inside diameter is attached to the micro T / C detector as indicated in Figure 1. The thermal conductivity cell is wired as a Wheatstone bridge using the four cell filaments as the arms of the bridge, and an Analog Devices Model 610L Instrumental Amplifier with Model 904 power supply amplifies the bridge output (10 V maximum) for transmittal to a Hewlett-Packard 21MX computer for data processing and, optionally, to a strip-chart recorder. The flow monitor is applied in conjunction with a gas chromatograph which uses a flame ionization detector for detection of the organic components being analyzed. Nitrogen is employed as carrier gas. Periodically,under computer control, an injection of ca. 50 KLof helium is made into the carrier gas at a position upstream from the point of sample introduction. The helium injected is detected by the microthermal conductivity cell but Present address: Department of Chemistry, Arizona State University, Tempe, AZ 85287.
causes no interference with the flame ionization detector (FID) used for measurement of the organic samples separated on the chromatographic column owing to the insensitivity of the FID to helium. Alternatively, nitrogen can be injected into a stream of helium carrier gas. Two Clippard (Cincinnati, OH) Model EVO-3 computer-operated pressure values are used to control a Carlo Erba eight-port pneumatic value for introduction of the helium. The helium introduced to the system passes through the first pair of thermal conductivity cells, through the empty capillary tubing t o the second pair of thermal conductivity cells, and thence to the sample introduction valve, the chromatographic column, and the FID detector.
RESULTS AND DISCUSSION An output from the T/C cell shown in Figure 2 is the result. Owing to the differences in thermal conductivity of the helium and the nitrogen present in the arms of the Wheatstone bridge, a negative and positive-going curve is produced. The time required for the slug of helium to pass from the first T/C cell to the second T/C cell is a function of the carrier gas flow rate. In actual practice there is no need to record this output. Two points on the output may be chosen (e.g., the peak minimum and maximum) and the time measured accurately between these two points by the computer. This difference in time, which is also a function of the carrier gas flow rate, may then be stored by the computer or, using calibration data, converted directly to flow rate and stored for later use. Since the time difference is commonly of the order of 10 s and since points are measured along this curve every 10 ms, it is possible to achieve accuracies of about 1 ppt in the time difference 0 1983 American Chemical Society
