NIST Technical Note 1715

Certified Transmittance 

Density Uncertainties for 

Standard Reference Materials 

using a Transfer 

Spectrophotometer

J. C. Travis (ret.)

M. V. Smith

S. J. Choquette

Hung-kung Liu

NIST Technical Note 1715
i

Certified Transmittance 

Density Uncertainties for 

Standard Reference Materials 

using a Transfer 

Spectrophotometer

J. C. Travis (ret.)

M. V. Smith

S. J. Choquette

Biochemical Science Division

First Operating Unit

Hung-kung Liu

Statistical Engineering Division

Second Operating Unit

November 2011

U.S. Department of Commerce

John Bryson, Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Director
ii

Certain commercial entities, equipment, or materials may be identified in this 

document in order to describe an experimental procedure or concept adequately. 

Such identification is not intended to imply recommendation or endorsement by the 

National Institute of Standards and Technology, nor is it intended to imply that the 

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1715

Natl. Inst. Stand. Technol. Tech. Note 1715, 13 Pages (November 2011)

CODEN: NTNOEF
iii

Abstract

Overall uncertainties are evaluated for the certification of transmittance density (absorbance 

referred to air) and regular spectral transmittance for solid neutral density filter Standard 

Reference Materials by means of a transfer spectrophotometer. Traceability is asserted by means 

of comparison measurements to a recognized reference spectrophotometer. The uncertainties are 

evaluated without bias correction, using the combination of the legacy combined standard 

uncertainty values with uncertainty components for the measured bias, the standard uncertainty 

of this measured bias, and the standard uncertainty characteristic of simple replication for a 

single measurement using the transfer spectrophotometer. Numerical results are given, and are 

anticipated to be the values initially quoted on certificates and in reports of recertification. 

However, these values are subject to change upon future adjustment of estimated uncertainty 

components for the filters or changes in the observed bias with future determinations.

Key Words

Absorbance, bias, certification, spectrophotometer, traceability, transmittance, transmittance 

density, uncertainty
iv

Table of Contents

1. Introduction ............................................................................................................................. 1

2. Certification Measurements with the Transfer Spectrophotometer......................................... 1

3. Traceability Measurements...................................................................................................... 2

3.1 Replication Uncertainty.................................................................................................... 4

3.2 Bias Determination........................................................................................................... 4

3.3 Transfer Spectrophotometer Uncertainties....................................................................... 5

4. Conclusion............................................................................................................................... 6

5. Acknowledgement................................................................................................................... 7

6. References ............................................................................................................................... 8
v

Glossary

Absorbance, A: Defined from the internal transmittance, Tint, as A = -log(Tint )= log(1/Tint).

Bias, : The difference in the reported value of a measurement and the best available estimate of 

the true value. In the present case, the bias reported is the difference in the transmittance density 

determined with the transfer spectrophotometer and the reference spectrophotometer.

Combined standard uncertainty, uc: The result of propagating all of the standard uncertainty 

components of the input quantities for a measurement to yield an uncertainty estimate for a 

measurement which is the equivalent of a standard deviation, or, approximately, the half width of

a 67% confidence interval.

Expanded uncertainty, U95: Multiplication of the combined standard uncertainty by an 

“expansion factor” k≥2 yields an uncertainty estimate corresponding to, approximately, the half 

width of a 95% confidence interval for a measurement. The value k = 2 is used in the present 

instance, reflecting the reasonably large effective degrees of freedom for the evaluation.

Internal transmittance, Tint: The portion of the overall transmittance determined by the 

absorption of light in the interior of the sample, between the reflective entry and exit faces of a 

cuvette or solid filter.

Optical density, OD: Defined from the transmittance, T, as OD = -log(T) = log (1/T).

Regular Spectral Transmittance, T(λ), referred to throughout as transmittance: The ratio of the 

exit intensity to the entry intensity for a collimated beam of nominally monochromatic light 

centered at a wavelength λ and normal to both the entry and exit faces of a sample with plane 

parallel faces. 

Standard uncertainty, u: Normally used with appropriate subscripts to describe uncertainty of 

the result of a measurement expressed as a standard deviation. 

Transmittance density, TD: Although it is entirely equivalent to optical density, this 

expression has been consistently used by the NIST optical filters SRM program in certificates 

and reports since 1970. These documents have carried the assurance that instruments designed 

to report absorbance will be reporting transmittance density when the reference sample (or 

chemical blank) is air (an empty filter holder is provided for this purpose with NIST SRM 

filters).
1

1. Introduction

Transmittance and transmittance density traceability has been offered to the chemical, 

pharmaceutical, and biochemical sectors of the US and global economy by the National Bureau 

of Standards/National Institute of Standards and Technology (NBS/NIST) since 1970. 

Certification in the ultraviolet and visible spectral regions (UV/vis) has been supported at 

NBS/NIST by a focused beam “High Accuracy Spectrophotometer” [1] (HAS, 1970-1999) and a 

second-generation optical clone of that instrument with modernized automation (HAS II, 2000 -

present). During this time, the concept of traceability [2] has been refined and the transmittance 

community has responded with the concept of the “transfer spectrophotometer” [3] as a link in 

the traceability chain. Such an instrument may be a commercial spectrophotometer linked to the 

primary transmittance scale by means of optical filter reference materials certified on a 

recognized national reference spectrophotometer.

The longstanding NBS/NIST program supporting the production and recertification of 

UV/visible Standard Reference Material (SRM) neutral density filters [4] in cuvette-simulation 

holders has qualified such a transfer spectrophotometer to assume the bulk of the certification 

and recertification work load and extend the useful life of the HAS II. This document details the 

proposed transmittance density uncertainty budget (from which transmittance uncertainties may 

be computed) for SRM certification with the Biochemical Science Division Transfer 

Spectrophotometer (BSDTS). The actual uncertainty budget is subject to change as needed, and 

is noted on certificates and reports that accompany certified values. 

It is anticipated that the program will not construct a third generation reference 

spectrophotometer at the end of the HAS II life cycle, but will migrate the traceability of the 

BSDTS to a national reference spectrophotometer participating in international Consultative 

Committee for Photometry and Radiometry (CCPR) key comparisons1. [5] 

2. Certification Measurements with the Transfer Spectrophotometer

The BSDTS is a Varian Cary 6000i2 UV/visible/NIR spectrophotometer with specifications well 

documented in the commercial literature [6]. The instrument is equipped with an optional 

temperature-controlled sample changer with positions for six cuvettes in the sample beam and 

six cuvettes in the reference beam. It is operated under the NIST quality system [7], with an 

initial qualification followed by quarterly performance validation and annual preventive 

maintenance (including lamp changes) by the manufacturer. 

The application software furnished with the instrument is written in an “Application 

Development Language” (ADL) that is also available to end users. The operational routines 

used for certified transmittance density and transmittance measurements have been custom 

 

1 This document includes links to websites that may have information of interest to our users. NIST does not 

necessarily endorse the views expressed or the facts presented on these sites. Further, NIST does not endorse 

any commercial products that may be advertised or available on these sites.

2 Certain commercial equipment, instruments, or materials are identified in this document. Such identification does 

not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it 

imply that the products identified are necessarily the best available for the purpose.
2

written in ADL to conform to the greatest degree possible with the measurement sequencing and 

timing of the HAS II. Measurements are made and recorded in the database in transmittance. 

Results are reported both in transmittance (T) and in “transmittance density” (TD), given as 

log(1/T) and equivalent to “optical density.” The terminology is employed to emphasize that the 

measurement is not blank-corrected for surface reflection. However, the certified values for 

transmittance density should conform to absorbance measurements on customer instruments with 

the empty holder furnished with the SRM serving as the blank.

All measurements are an average of three individual readings with an averaging time of 1 second 

each and are made at a spectral bandwidth setting of 0.8 nm. The “reduced height” slit image is 

centered 15 mm above the base of the holder, to best approximate the illumination of the HAS II 

beam. SRM filter sets each include three filters of different nominal transmittance level. A 

single filter set is measured during each run, with the filters placed in positions 2, 4, and 6 in the 

order of increasing transmittance. Position 1 is used to determine the transmittance density 

baseline with the reference position empty and an empty filter holder in the sample position. 

Position 1 is used again at the end of the run as a control to record the quality of the zero of 

transmittance density. 

The “native” measurement position of the Cary beam is 20 mm above the base, and a “sample 

lifter” is built into the sample carriage to enable easy extraction of cuvettes. This device is used 

to raise the samples 5 mm to the certification position. The empty sample holder in Position 1 

has had the top milled off to be flush with the surface of the sample carriage when the vertical 

position of the samples centers the optical beam as described above. The size and position of 

the beam is matched as closely as possible to that of the HAS II to minimize the confounded 

effects of beam geometry and filter heterogeneity in establishing traceability.

Data acquisition begins by taking the transmittance baseline in position 1 for all 5 wavelengths 

(SRM 930, SRM 1930, and SRM 2930) or 10 wavelengths (SRM 2031). The measurement 

proceeds by advancing to position 2, recording the transmittance at each of the 5 or 10 

wavelengths, and repeating the process twice, for a total of three one-second measurements at 

each wavelength. This process is repeated for the other two filter-containing positions and for 

the empty control position. The average values of the three transmittance readings for each filter 

and wavelength are transmitted to the master database. The complete raw data set is retained, 

identified by set number and date, in addition to the reduced information transferred to the 

database.

The central certified values reported on new material certificates or on Reports of Recertification 

for returned materials represent the averages of the three transmittance measurements and the 

transmittance densities calculated from these average transmittances. The uncertainties reported 

for each filter and wavelength represent modifications of the legacy uncertainties given for HAS 

II measurements to include additional uncertainty resulting from the traceability of the transfer 

instrument to the reference instrument. 

3. Traceability Measurements

A sequence of measurements has been made to evaluate uncertainty components resulting from 

the traceability of the transfer instrument based on NIST guidelines [8]. Traceability is here 

chosen to be provided by including uncorrected bias in the measurement uncertainty rather than 
3

by bias correction [9]. The expanded uncertainty (U95) for a certified value at a given wavelength 

for a given nominal transmittance density using the BSDTS is given by

ref U ux u u

2 2 2

95 |  | 2   

(1)

where ux is the standard uncertainty of a single transfer instrument measurement determined by 

simple replication, uref is the combined standard uncertainty for a reference standard used to 

compare the transfer instrument to the stated reference spectrophotometer, and  is the bias of the 

transfer instrument as determined by the average of n determinations of the measured value 

minus the certified value with a sample standard deviation of S and a standard uncertainty of

u S n/

2

 .

(2)

A standard uncertainty is a good approximation to a standard deviation and is expanded by a 

factor which is approximately 2 due to the large number of effective degrees of freedom to yield 

the half-width of a 95 % confidence interval. [8] The standard uncertainty in Eq. 2 is so 

expressed as to estimate the standard deviation of the mean of the n measurements used to 

determine the average value of . The standard uncertainty ux is expressed as ux = Sx although 

replicate measurements have gone into the determination of Sx , as discussed below. This is

because a single measurement will be used by the instrument to determine each certified value

(where a “measurement” is taken as the average of three readings). 

In Eq. 1, uref is given as half of the original certified U95 value for the given wavelength and filter 

(transmittance density), and represents the combined standard uncertainty of the certified value. 

The original combined standard uncertainty is thus further combined by the root-sum-of-squares 

with standard uncertainties for the bias determination and for a single measurement with the 

transfer instrument. After expansion by the factor of 2, the addition of the absolute value of the 

measured bias provides a symmetric expanded uncertainty for transfer measurements made 

without bias correction. The new U95 for the transfer instrument approaches the original U95 for 

the reference instrument in the limit as , ux, and S approach zero. 

The largest contribution to the right-hand side of Eq. 1 is uref, the combined standard uncertainty 

of the certified value of a given filter (i.e. nominal transmittance) at a given wavelength using the 

HAS II. This uncertainty is dominated by components relating to properties of the reference 

materials such as heterogeneity and temporal drift, discussed elsewhere [10]. 

For practical and traditional reasons, all uncertainty evaluations are conducted in transmittance 

density and are converted to transmittance as needed. Also for traditional reasons, a single 

uncertainty (in transmittance density) is associated with each level, applying to all certified 

wavelengths. To do this, the relevant uncertainty components are computed for each wavelength 

and the maximum is then applied to measurements at all wavelengths as a conservative estimate 

for each nominal transmittance.
4

3.1 Replication Uncertainty

One set of each of the SRM types shown in the first column of Table 1 below was measured four 

times on each of three days for a total of 12 measurements. The values shown in the ux column 

acquired at measurement wavelength λx represent the maximum computed sample standard 

deviations of these 12 measurements over all certification wavelengths for each filter. This 

maximum value for each filter is conservatively used for all certification wavelengths. 

Table 1. Standard uncertainty of the transfer

spectrophotometer for simple replication.

The SRM samples were measured in sequence four times on each day, so that removal and 

replacement of the samples and time separation of the four measurements was assured for each 

sample set. Use of the three days was intended to confound any “day” effect into the 

determination.

3.2 Bias Determination

Three independent measurements were made on each of two control sets for each of the four 

types of SRM on the BSDTS and on the HAS II. Pairwise data sets were then used for six 

determinations of bias, and the average and sample standard deviation were computed for each 

nominal filter and wavelength. Figure 1 shows a representative result for the nominal 10 % 

transmitting (TD ≈ 1) filter of SRM 930. Curiously, a statistically significant wavelength 

dependence to the bias is observed, and the wavelength of maximum bias is reproduced in 

quarterly traceability validations and for the other filters measured. As before, the maximum 

bias over all of the wavelengths for a given filter is conservatively assigned as the bias () for all 

wavelengths. Table 2 gives the measured values for the bias with the corresponding sample 

standard deviation (S ) along with the wavelength (λ) for which the maximum bias was 

observed. The first row of data in Table 2 represents the 635 nm data point of Fig. 1, and the 

wavelength dependence consistency across filters is supported by the preponderance of long 

wavelengths in the table. Since the average of six determinations is used for the bias 

determination, n = 6 in Eq. 2.

SRM Filter TDnom λ x

u x

10 % 1 440 nm 0.00004

20 % 0.7 440 nm 0.00006

30 % 0.5 440 nm 0.00004

1 % 2 465 nm 0.00010

3% 1.5 440 nm 0.00010

50 % 0.3 590 nm 0.00003

0.1 % 3 440 nm 0.00040

0.3 % 2.5 440 nm 0.00015

90 % 0.04 440 nm 0.00006

10 % 1 340 nm 0.00006

30 % 0.5 360 nm 0.00005

90 % 0.04 340 nm 0.00016

2031

930

1930

2930
5

Figure 1. Transmittance density bias as a function of wavelength for the nominal 10 % 

transmitting filter of SRM 930.

3.3 Transfer Spectrophotometer Uncertainties

Table 3 combines the uncertainty components of Eqs. 1 and 2 from Tables 1 and 2 with the uref

value characteristic of certification on the HAS II and shows the computed value of the expanded 

uncertainty for transmittance density for SRMs certified on the BSDTS. NIST certificates and 

reports for these SRMs include tables of certified values for both transmittance density and for 

regular spectral transmittance. The final transmittance density uncertainties are used to compute 

the final uncertainties for certified transmittances through the relationship derived by 

differentiating the equation relating transmittance density to transmittance, TD = -log(T).

(3)

The uncertainties for transmittance are individually computed for each wavelength of 

certification using the actual measured transmittance rather than the nominal transmittance. 

Thus the certified transmittance tables in NIST certificates and reports shows the uncertainties 

varying with transmittance over the certified wavelengths whereas the transmittance density 

tables quote the same uncertainty for each wavelength for a given filter.

-0.0005

0.0000

0.0005

0.0010

0.0015

400 450 500 550 600 650 Transmittance Density Bias

Wavelength/nm
6

Table 2. The bias determinations for the BSDTS

with respect to the HAS II.

Table 3. Computation of the expanded uncertainty for the certification of transmittance

density of NIST SRMs with the BSDTS using Eq. 1.

4. Conclusion

The expanded uncertainties in transmittance density for the certification of four SRMs using the 

Biochemical Science Division Transfer Spectrophotometer have been calculated using 12 

replicate measurements of control standards to evaluate the replication uncertainty of the transfer 

instrument and 6 independent measurements of the bias for each filter and wavelength. The 

conservative nature of the results is assured by using maximum values over the various 

wavelengths for each transmittance density and by choosing to evaluate the symmetric 

uncertainty interval for uncorrected transmittance density measurements. The numerical 

uncertainties of Table 3 are illustrative of the process, but are subject to change as a result of 

future experimental results. Transmittance uncertainties are computed from the transmittance 

density uncertainties using the known relationship between the scales.

SRM Filter TDnom λ   S 

10 % 1 635 0.00082 0.00058

20 % 0.7 590 0.00053 0.00018

30 % 0.5 635 0.00041 0.00021

1 % 2 635 0.00196 0.00119

3% 1.5 635 0.00121 0.00053

50 % 0.3 590 0.00035 0.00007

0.1 % 3 635 0.00238 0.00152

0.3 % 2.5 635 0.00215 0.00083

90 % 0.04 546 0.00033 0.00015

10 % 1 500 0.00179 0.00046

30 % 0.5 590 0.00091 0.00007

90 % 0.04 635 0.00023 0.00009

2031

930

1930

2930

SRM Filter TDnom u X u  u ref  U 95

10 % 1.0 0.00004 0.00024 0.00120 0.0008 0.0033

20 % 0.7 0.00006 0.00007 0.00115 0.0005 0.0028

30 % 0.5 0.00004 0.00009 0.00115 0.0004 0.0027

1 % 2.0 0.00010 0.00049 0.00275 0.0020 0.0076

3% 1.5 0.00010 0.00022 0.00270 0.0012 0.0066

50 % 0.3 0.00003 0.00003 0.00100 0.0003 0.0023

0.1 % 3.0 0.00040 0.00062 0.00905 0.0024 0.0205

0.3 % 2.5 0.00015 0.00034 0.00450 0.0021 0.0112

90 % 0.04 0.00006 0.00006 0.00100 0.0003 0.0023

10 % 1.0 0.00006 0.00019 0.00210 0.0018 0.0060

30 % 0.5 0.00005 0.00003 0.00195 0.0009 0.0048

90 % 0.04 0.00016 0.00004 0.00095 0.0002 0.0022

930

1930

2930

2031
7

The inflation of the final uncertainties by the use of the transfer instrument is expected to be 

acceptable for the bulk of applications in the user community. Certification using the HAS II is 

available at specified times and for a cost commensurate with the loss of economy of scale.

5. Acknowledgement

The authors are grateful to David L. Duewer of the Analytical Chemistry Division for many 

helpful discussions and for careful review of the manuscript.
8

6. References

[1] R. Mavrodineanu, An Accurate Spectrophotometer for Measuring the Transmittance of Solid and Liquid 

Materials, J. Res. NBS 76A, 405 (1972). 

[2] See http://www.nist.gov/traceability/nist_traceability_policy_external.cfm with links and references therein.

[3] See, for instance, “Technical Specifications for Certification of Spectrophotometric NTRMs,” NIST Special 

Publication 260-140 (2000), pages 7 – 9. Can be downloaded from: 

http://ts.nist.gov/MeasurementServices/ReferenceMaterials/upload/SP260-140.PDF

[4] Find descriptions of SRM 930, SRM 1930, SRM 2930, and SRM 2031 at 

http://www.nist.gov/mml/biochemical/bioassay/optical_filters_program.cfm

[5] Gaël Obein and Jean Bastie, Report on the CCPR Key Comparison K6: Spectral regular transmittance, 

Metrologia 46, 02002 (2009). Can be downloaded from: http://iopscience.iop.org/0026-1394/46/1A/02002 .

[6] Available on line at: http://www.chem.agilent.com/Library/specifications/Public/si-0188.pdf .

[7] NIST Quality System, QM-I (http://www.nist.gov/qualitysystem/upload/NIST-QM-I-V6-NonControlled.pdf ); 

Biochemical Sciences Division Quality Manual, QM-II; Bioassay Methods Group Technical Procedure 

631.02.09

[8] Barry N. Taylor and Chris E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST 

Measurement Results, NIST Technical Note 1297, U.S. Government Printing Office, Washington (1993).

[9] Bertil Magnusson & Stephen L. R. Ellison, Treatment of uncorrected measurement bias in uncertainty estimation 

for chemical measurements, Anal. Bioanal. Chem. 390, 201 (2008). 

[10] See the discussion of uncertainty estimation for SRM 930d in: R. Mavrodineanu, R.W. Burke, J.R. Baldwin, 

M.V. Smith, J.D. Messman, J.C. Travis, and J.C. Colbert, Glass Filters as a Standard Reference Material for 

Spectrophotometry – Selection, Preparation, Certification, and Use of SRM 930 and SRM 1930, NIST Special 

Publication 260-116, U.S. Government Printing Office, Washington (1994), pages 60 – 62.