Section: 2 | Expression of Uncertainty of Measurements |
Help Manual

Page of 1
Type a page number and hit Enter.
/1
  Back to Search Results
Type a page number and hit Enter.
Additional Information
Summary of table differences
No records found.
How to Cite this Reference
The recommended form of citation is:
John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

EXPRESSION OF UNCERTAINTY OF MEASUREMENTS

In general, the result of a measurement is only an approximation or estimate of the true value of the quantity subject to measurement, and thus the result is of limited value unless accompanied by a statement of its uncertainty. Much (but not all) of the scientific data appearing in the literature does include some indication of the uncertainty, but this may be stated in many different ways and is often explained poorly. In an effort to encourage consistency in uncertainty statements, the International Committee for Weights and Measures (CIPM) of the Bureau International des Poids et Mesures initiated a project, in collaboration with several other international organizations, to prepare a set of guidelines expressing international consensus on the recommended method of stating uncertainties. This project resulted in the publication of the Guide to the Expression of Uncertainty in Measurement (Refs. 1 and 2), which is often referred to as GUM. The recommendations of GUM have been summarized by the National Institute of Standards and Technology in NIST Technical Note 1297, Guidelines for Evaluating the Uncertainty of NIST Measurement Results (Ref. 3).

In the notation of GUM, we are concerned with the measurand, i.e., the quantity that is being measured. In physics and chemistry this is usually called a physical quantity and represents some inherent characteristic of a material, system, or process that can be expressed in numerical terms — specifically as the product of a number and a reference, commonly called a unit. Thus, the density of water at room temperature is (approximately) 0.998 g/mL (grams per milliliter) or, alternatively 998 kg m–3 (kilograms per meter cubed). This statement gives the most likely value of the measurand, to this level of precision, but gives no information on how much the stated value might differ from the true value. A more detailed discussion of measurement terminology is given in the International Vocabulary of Metrology (VIM) (Ref. 4).

It is important to differentiate between the terms error and uncertainty. The error in a measurement is the difference between the measured value and the true value; the error can be stated if the true value is known (to some level of accuracy). The uncertainty is an estimate of the maximum reasonable extent to which the measured value is believed to deviate from the true value, in a situation where the true value is not known (most often the case). The result of a measurement can unknowably be very close to the true value, and thus have negligible error, even though its uncertainty is large.

The uncertainty of the result of a measurement generally consists of several components, which may be grouped in two types according to the method used to estimate their numerical values:

The terms “random uncertainty” and “systematic uncertainty” are often used, but these terms do not always correspond in a simple way to the A and B categories. This is because the nature of an uncertainty component is conditioned by how the quantity appears in the mathematical model that describes the current measurement process. An uncertainty component arising from a systematic effect may in some cases be evaluated by methods of Type A while in other cases by methods of Type B.

In the GUM formulation, each component of uncertainty, whether in the A or B category, is represented by an estimated standard deviation, termed standard uncertainty, symbol ui , and equal to the positive square root of the estimated variance ui2.

For an uncertainty component of Type A, ui = si , where si is the statistically estimated standard deviation, as determined from a series of observations by appropriate statistical analysis. Any valid statistical method may be used. Examples are calculating the standard deviation of the mean of a series of independent observations; using the method of least squares to fit a curve to data in order to estimate parameters of the curve and their standard deviations; and carrying out an analysis of variance (ANOVA) in order to identify and quantify random effects in certain types of measurements. Details of statistical analysis are given in Refs. 5–9 and many other places.

In a similar manner, each uncertainty component of Type B is represented by a quantity uj , which is obtained from an assumed probability distribution based on all the available information about the measurement process. Because uj is treated like a standard deviation,the standard uncertainty in each Type B component is simply uj . The evaluation of uj is usually based on scientific judgment using all the relevant information available, which may include

The specific approach to evaluating the standard uncertainty uj of a Type B uncertainty will depend on the detailed model of the measurement process. The following are examples of steps that may be used:

  1. Convert a quoted uncertainty (for example, in a calibration factor) that is a stated multiple of an estimated standard deviation to a standard uncertainty by dividing the quoted uncertainty by the multiplier.
  2. Convert a quoted uncertainty that defines a “confidence interval” having a stated level of confidence, such as 95% or 99%, to a standard uncertainty by treating the quoted uncertainty as if a normal distribution had been used to calculate it (unless otherwise indicated) and dividing it by the appropriate factor for such a distribution. These factors are 1.960 and 2.576 for the two levels of confidence given.
  3. Model knowledge of the quantity in question by a normal distribution and estimate lower and upper limits a and a+ such that the best estimated value of the quantity is (a+ + a)/2 (i.e., the midpoint of the limits) and there is 1 chance out of 2 (i.e., a 50% probability) that the value of the quantity lies in the interval a to a+. Then uj ≈ 1.48 a, where a = (a+a)/2 is the half-width of the interval.
  4. Model knowledge of the quantity in question by a normal distribution and estimate lower and upper limits a and a+ such that the best estimated value of the quantity is (a+ + a)/2 and there is about a 2 out of 3 chance (i.e., a 67% probability) that the value of the quantity lies in the interval a to a+. Then uj a, where a = (a+a )/2.
  5. Estimate lower and upper limits a and a+ for the value of the quantity in question such that the probability that the value lies in the interval a to a+ is, for all practical purposes, 100%. Provided that there is no contradictory information, treat the quantity as if it is equally probable for its value to lie anywhere within the interval a to a+; that is, model it by a uniform or rectangular probability distribution. The best estimate of the value of the quantity is then (a+ + a)/2 with u= a/√3 where a = (a+a)/2. If the distribution used to model the quantity is triangular rather than rectangular, then uj = a/√6. The rectangular distribution is a reasonable default model in the absence of any other information. But if it is known that values of the quantity in question near the center of the limits are more likely than values close to the limits, a triangular or a normal distribution may be a better model.

When all the standard uncertainties of Type A and Type B have been determined in this way, they should be combined to produce the combined standard uncertainty (suggested symbol uc), which may be regarded as the estimated standard deviation of the measurement result. This process, often called the law of propagation of uncertainty or “root-sum-of-squares,” involves taking the square root of the sum of the squares of all the ui . In many practical measurement situations, the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y) is approximately normal (Gaussian). When this is the case, uc(y) defines an interval y – uc(y) to y + uc(y) about the measurement result y within which the value of the measurand Y estimated by y is believed to lie with a level of confidence of approximately 68%. That is, it is believed with an approximate level of confidence of 68% that y – uc(y) ≤ Y ≤ y + uc(y), which is commonly written as Y = y ± uc(y).

In fundamental metrological research (involving physical constants, calibration standards, and the like) the combined standard uncertainty uc is normally used as the statement of uncertainty in a measurement. In most cases, however, it is desirable to use a measure of uncertainty that defines an interval about the measurement result y within which the value of the measurand Y is confidently believed to lie. The measure of uncertainty intended to meet this requirement is termed expanded uncertainty, suggested symbol U, and is obtained by multiplying uc (y) by a coverage factor, suggested symbol k. Thus U = kuc(y) and it is believed with high confidence that y – UYy + U, which is commonly written as Y = y ± U. The value of the coverage factor k is chosen on the basis of the desired level of confidence to be associated with the interval defined by U = kuc. Typically, k is in the range 2 to 3. When the normal distribution applies, U = 2uc (i.e., k = 2) defines an interval having a level of confidence of approximately 95%, and U = 3uc defines an interval having a confidence level greater than 99%. In current international practice it is most common to use k = 2, corresponding to about 95% confidence, but the value of k should be stated in each case to avoid confusion. See Refs. 1 and 3 for methods of calculating k when a value other than k = 2 is needed for a specific requirement.

It should be noted that the International Union of Pure and Applied Chemistry (IUPAC) is reviewing recommendations on metrological and quality concepts in analytical chemistry (Ref. 10). 

Summary of Key Steps

References

  1. Evaluation of Measurement Data - Guide to the Uncertainty in Measurement, JCGM 100:2008, BIPM, Sevres, 2008, <bipm.org/utils/common/documents/jcgm/JCGM_1002008_E.pdf>
  2. ISO, Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland, 1993. Several supplements have been published; see Bich, W., Cox, M. C., and Harris, P. M., “Evolution of the Guide to the Expression of Uncertainty in Measurement,Metrologia 43, S161, 2006. [https://doi.org/10.1088/0026-1394/43/4/S01]
  3. Taylor, B. N., and Kuyatt, C. E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, National Institute of Standards and Technology, Gaithersburg, MD, 1994; available for free download at <physics.nist.gov/cuu/Uncertainty/bibliography.html>.
  4. International Vocabulary of Metrology - Basic and General Concepts and Associated Terms, JCGM 200:2012, Third Edition, BIPM, Sevres, 2012.
  5. Bell, S., A Beginner’s Guide to Uncertainty of Measurement, National Physical Laboratory, Teddington, Middlesex, UK, 2001; available on the Internet at <www.npl.co.uk/server.php?show=ConWebDoc.1785>.
  6. Eisenhart, C., “Realistic Evaluation of the Precision and Accuracy of Instrument Calibration Systems,” J. Res. Natl. Bur. Stand. (U.S.) 67C, 161, 1963. [https://doi.org/10.6028/jres.067C.015]
  7. Mandel, J., The Statistical Analysis of Experimental Data, Dover Publishers, New York, 1984.
  8. Nantrella, M. G., Experimental Statistics, NBS Handbook 91, U.S. Government Printing Office, Washington, DC, 1966.
  9. Box, G. E. P., Hunter, J. S., and Hunter, W. G., Statistics for Experimenters: Design, Innovation, and Discovery, Second Edition, John Wiley & Sons, Hoboken, NJ, 2005.
  10. https://iupac.org/recommendation/metrological-and-quality-concepts-in-analytical-chemistry/>
Page 1 of 1
1/1

Entry Display
This is where the entry will be displayed

Log In - Individual User
You are not within the network of a subscribing institution.
Please sign in with an Individual User account to continue.
Note that Workspace accounts are not valid.

Confirm Log Out
Are you sure?
Log In to Your Workspace
Your personal workspace allows you to save and access your searches and bookmarks.
Username
Password
Remember Me
This will save a cookie on your browser



If you do not have a workspace Log In click here to create one.
Forgotten your workspace password? Click here for an e-mail reminder.
Log Out From Your Workspace
Are you sure?
Create your personal workspace
Title
First Name (Given)
Last Name (Family)
Email address
Username
Password
Confirm Password


Incorrect login details
You have entered your Workspace sign in credentials instead of Individual User sign in credentials.
You must be authenticated within your organisation's network IP range in order to access your Workspace account.
Click the help icon for more information on the differences between these two accounts.
Incorrect login details
You have entered your Individual User account sign in credentials instead of Workspace credentials.
While using this network, a personal workspace account can be created to save your bookmarks and search preferences for later use.
Click the help icon for more information on the differences between Individual User accounts and Workspace accounts.
My Account

Change Your Workspace Password
Username
Current Password

New Password
Confirm New Password


Update your Personal Workspace Details
Username
Title
First Name (Given)
Last Name (Family)
Email address


Workspace Log In Reminder
Please enter your username and/or your e-mail address:

Username
Email Address


Searching for Chemicals and Properties

The CRC Handbook of Chemistry and Physics (HBCP) contains over 700 tables in over 450 documents which may be divided into several pages, all categorised into 17 major subject areas. The search on this page works by searching the content of each page individually, much like any web search. This provides a challenge if you want to search for multiple terms and those terms exist on different pages, or if you use a synonym/abbreviation that does not exist in the document.

We use metadata to avoid some of these issues by including certain keywords invisibly behind each table. Whilst this approach works well in many situations, like any web search it relies in the terms you have entered existing in the document with the same spelling, abbreviation etc.

Since chemical compounds and their properties are immutable, a single centralised database has been created from all chemical compounds throughout HBCP. This database contains every chemical compound and over 20 of the most common physical properties collated from each of the >700 tables. What's more, the properties can be searched numerically, including range searching, and you can even search by drawing a chemical structure. A complete list of every document table in which the compound occurs is listed, and are hyperlinked to the relevant document table.

The 'Search Chemicals' page can be found by clicking the flask icon in the navigation bar at the top of this page. For more detailed information on how to use the chemical search, including adding properties, saving searches, exporting search results and more, click the help icon in to top right of this page, next to the welcome login message.

Below is an example of a chemical entry, showing its structure, physical properties and document tables in which it appears.

image of an example chemical entry
We use cookies to improve your website experience. To learn about our use of cookies and how you can manage your cookie settings, please see our Cookie Policy. By continuing to use the website, you consent to our use of cookies.
Cookie Policy

Cookie Policy

We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.

The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.

Please note that our use of any personal information we collect about you is subject to our Privacy Policy.

What are Cookies?

Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

Cookies can be ‘Session Cookies’ or ‘Persistent Cookies’. Session Cookies allow a website to link a series of your actions during one browser session, for example to remember the items you have added to a shopping basket. Session Cookies expire after a browser session and are therefore not stored on your computer or device afterwards. Persistent Cookies are stored on your computer or device between browser sessions and can be used when you make subsequent visits to the website, for example to remember your website preferences, such as language or font size.

Cookies We Use and Their Purpose

We use three types of Cookies - ‘Strictly Necessary’ Cookies, ‘Performance’ Cookies and ‘Functionality’ Cookies. Each type of Cookie and the purposes for which we use them are described in this section. To learn about the specific Cookies we use, please see our List of Cookies.

1. Strictly Necessary Cookies

‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.

Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.

2. Performance Cookies

‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.

3. Functionality Cookies

‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.

You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.

First and Third Party Cookies

The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.

The Cookies placed on your computer or device may also include ‘Third Party’ Cookies, meaning Cookies that are placed there by third parties. These Cookies may include third party advertisers who display adverts on our Website and/or social network providers who provide ‘like’ or ‘share’ capabilities (see the above section on Targeting or Advertising Cookies). They may also include third parties who provide video content which is embedded on our Website (such as YouTube). Please see the website terms and policies of these third parties for further information on their use of Cookies.

To learn about the specific First Party and Third Party Cookies used by our, please see our List of Cookies.

Managing Cookies

You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.

You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.

You can learn more about how to manage Cookies on the following websites: www.allaboutcookies.org and www.youronlinechoices.com.

Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.

Changes to Cookie Policy

In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.

Questions or Concerns

If you have any questions or concerns about this Cookie Policy or our use of Cookies on the Website, please contact us by email to [email protected]

You can also contact the Privacy Officer for the Informa PLC group at [email protected].


Our Cookies

Here is a list of cookies we have defined as 'Strictly Necessary':

Taylor and Francis 'First Party' Cookies

JSESSIONID

TandF.ACCT.CNB.cookieId

TandF.WS.CNB.cookieId

TandF.SU.CNB.cookieId

TandF.PORTAL.cookiesAgreed

TandF.LOGIN.cookiesAgreed

TandF.HBCP.cookiesAgreed

TandF.CCD.cookiesAgreed

TandF.DNP.cookiesAgreed

TandF.DOC.cookiesAgreed

TandF.DOD.cookiesAgreed

TandF.DIOC.cookiesAgreed

TandF.POLY.cookiesAgreed

TandF.DFC.cookiesAgreed

TandF.DMNP.cookiesAgreed

TandF.DCCC.cookiesAgreed

TandF.POC.cookiesAgreed

Here is a list of the cookies we have defined as 'Performance'.

'Third Party' Cookies

Google Analytics:

_ga

_gid

_gat

Accessibility

The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.

The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.

Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.

Click here for more information about how to use this web application using the keyboard.


This is replaced with text from the script
This is replaced with text from the script
Top Notification Bar Dialog Header
Your Session is about to Expire!
Your session will expire in seconds

Please move your cursor to continue.