## Webinars

Every year, SPEX CertiPrep produces several FREE, informative webinars on a variety of topics including Clean Laboratory Techniques, CRM's: Beyond the Basics, and Uncertainty of Analytical Data.

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# Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis

## Abstract

As technology continues to improve, new analytical instrumentation is capable of quantifying concentrations in the PPT and even PPQ levels. However, there is always an uncertainty associated with the reliability of the result. Uncertainty calculations must be taken into account for each step of the analytical process, and estimating these values increases the confidence in the accuracy of a measurement result. Join SPEX CertiPrep as we explore the nature of uncertainty and quantify it to denote a measure of confidence that can be placed on an analytical result. We will explain the process of identifying and quantifying the components of uncertainty in the analytical process, using the instrumental analysis of a gold standard as an example.## Transcription

Hello everyone! I’d like to welcome you all to Spex CertiPrep’s webinar on Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis. I’m Peter Eskow, the Marketing and Communications Manager here at Spex CertiPrep and I’ll be moderating today.

Before we begin, I’d like to get a few housekeeping items out of the way. First off, everyone here today will be sent a copy of the presentation files. We’ll send an email later today with a link where you can download them. The webinar is being recorded and we will post it on our YouTube channel so that you can watch it again at your convenience. It will be uploaded in about a week or so and you’ll receive a follow-up email with the link when it’s available. We do have a Q&A session at the end of today’s event so just type the question into the question box as you think of them, and we’ll get to it as many as we can. With that out of the way, let’s get on with today’s presentation.

I would like to introduce Vanaja Sivakumar. Vanaja received her PhD from the Department of Inorganic and Physical Chemistry from the Indian Institute of Science in Bengaluru, India. She worked with Weaver Brothers India as a research chemist for 9 years before coming to Spex CertiPrep 20 years ago as the QA regulatory manager. She is now our Vice President of Manufacturing. Vanaja, I’ll pass it up to you.

Thank you, Peter. Good afternoon everyone. Today’s topic, as Peter mentioned, is Identifying and Quantifying the Uncertainty Associated with Instrumental Analysis. Calculating uncertainty is a mystery to all of us. I hope by end of this presentation you’ll get more familiarity with this presentation. So what are we going to cover in today’s presentation? In the past, chemists focus on positional results from any given method. However, these methods are just not enough to have a precise method. We have to establish the quality, our accuracy of results by stating a measure of confidence. One useful measure is measurement of uncertainty. Since precision, accuracy, confidence interval and errors are fundamental in analytical chemistry, we will be covering them here when we briefly.

Our main goal for today’s presentation is to explain about uncertainty in practical terms. We will be showing one of the ways to calculate uncertainty taking into account the analysis of gold by ICP. We will calculate uncertainty associated with the measurement as applied to the process we have chosen.

Two value for the quantity measured depends on method, instruments and operators. Any of these could introduce errors in measurement. You can be confident about the results from a validated method and well-maintained instrument, highly trained operators, applying his or her mind to the task at hand and through experience can gauge the quality of the results. Inspite of all these, an experienced chemist, a precise instrument and a validated method, it’s very hard to get the true value of measurement accurately. There are always errors and uncertainty associated with measurement. But what are these errors? Typically there are three types of errors: Determinate Errors, Indeterminate Errors as well as Spurious Errors.

Determinate errors can be avoided if they are recognized. Unpredictable variations in repeated observations from the measured value is an example. You can reduce these errors by increasing the number of observation. Indeterminate errors, on the other hand, exist by the very nature of measurement data. The analyst does not know their magnitude. These indeterminate errors affect the precision and accuracy of all chemical work. One perfect example for this is the error due to inadequate control of experimental condition. This can give the systematic errors that are not constant. These errors cannot be eliminated totally. However, we should attempt to contain them in as narrow zone as possible. The other possible errors that come to these errors, these are mainly due to human errors or instrument failures. Whatever be the type of errors, we have to eliminate or minimize them. One way to minimize these errors is to make a series of measurements on the same object and record the average.

So what is an average? When we do an experiment, we allocate measurements of few number of replicates and then calculate the average value from all these measurements. It is the sum of all individual measurements divided by the number of measurements. As you all know, average represents the central tendency wherein lies the true value. It is not an absolute value; it's only a point estimate of the true value. How can we estimate the true value? One way to do is through standard deviation usually represented as “s” or sigma. Standard deviation is given by this equation that X is the measurement value, x-bar is the average value and n is the total number of measurements. I said earlier, an average gives the estimate of true value. This is called point average. By using standard deviation, you can pinpoint the range within which the true value lies at the certain confidence level. The term confidence level, confidence interval, confidence limit and interval estimate are used interchangeably. So you can express true value at x-bar plus or minus 1.96 at 95% confidence or x-bar plus or minus 2.94s at 99% confidence. Usually, this 1.96s is rounded to 2s and 2.94s is rounded to 3s in practical terms.

So we discussed about confidence interval and so what is confidence interval? Confidence interval is a likely range of true value and is useful in expressing true value. There is another important concept in confidence limit is it is also an estimate for precision. The wider the confidence interval, the less is the precision. We usually express the accuracy of result at 90% confidence, 95% confidence or 99% confidence interval. Earlier I said, you take an average of measurement, by taking average of measurement is one way to get an estimate of accurate true value.

What is accuracy? Accuracy represents the closeness of the measured value to the true value. In this graph, the red line is intended true value. The green graph indicates the values obtained by the measurement. As you can see, the measurements are close to the true value and hence accurate. In the second graph, you have, the red line is again, red line is a value that is obtained after the measurement. This is about 7.39 and it is not very accurate. Although, down below you see that graph, it shows a fairly good precise value obtained in that.

So, now what is precision? Precision is the reproducibility of the method. Here, you can see the red line again is the true value that they got and it has a very good precision because the measurement values are closely related to the true value. Here, they got a fairly good true value but as you see the point measurements are scattered all over. There is a quite dispersion of values from this. This dispersion or scatter is measured by average deviation, variance and standard deviation. What is average deviation? It’s given, represented by the formula here x as the individual value and x-bar is the average value, and n is the total number of replicates. The smaller the AD, average deviation, the more precise the measurements are. However, it is not an accurate measure of precision. The scientists use, more commonly, variant and standard deviation. Standard deviation is the square root of variance and these all indicated state of measurement around the data. And they do offer a better tool to measure the precision.

Now, these are the terms that we introduced here which you all know already and these, we are going to use these to calculate uncertainty.

So what is uncertainty? All the parameters like standard deviation or multiples of standard deviation shows a dispersion of results around the true value. This dispersion is the uncertainty around the true value. In general, the word “uncertainty” relates to the concept of doubt. The doubt is certainly not about the validity of measurement. In fact, the knowledge of this uncertainty increases the confidence in your measurement result. So, it is, at this point, it is important to distinguish between error and uncertainty. Two slides ago I had mentioned to you about various errors. Those errors you can combine them to single value we cannot know exactly what the value is. However, we can estimate the uncertainty.

Uncertainty estimation is simple in principle. As a first step, you should have a clear idea on what is to be measured. Then, you will write down all various process involved in your measurements. You will be identifying forces of uncertainty within this process... You write down the process: “What is the sample dilution?”, “What are all the environmental conditions?”, “Is there any uncertainty associated with the equipment?”. You have to ponder over all these details and write it down in order to calculate the uncertainty. Once you know these components, you should quantify uncertainty from each of these within the process. Now, the information obtained in step 4 will consist of a number of quantified contributions to overall uncertainty, whether they are associated with the individual sources or with the combined effects of several sources. This contribution is expressed as standard deviation and should be combined according to appropriate rules to give combined standard uncertainty. Once you have an idea about the combined standard uncertainty for the overall process, you would expand it by applying your appropriate coverage factor.

There are two types of uncertainty: Type A is associated with repeated measurements like when you do replicates in titration or an average value when reading instruments. The Type A Uncertainty is expressed as standard deviation or standard error of the mean. Type B, on the other hand, is based on scientific judgment using all the relevant information available including previous measurement data, experience, manufacturer specification, and data provided in the calibration report.

There are three types of Type B Uncertainty. They are the rectangular, triangular and normal. To calculate the standard uncertainty from this listed uncertainty, we have to apply a normalizing factor for each of these. We will review these factors for each type.

Now the first type is the rectangular distribution. Rectangular distribution is used when uncertainty is stated without specifying level of confidence. That is, when there is no confidence interval stated, there is no reason to expect extreme values. So you convert the listed uncertainty for this by dividing by square root of three which is the normalization factor here.

The second type is the triangular distribution. Here, the measured value lies very close to the target value. We use this distribution when we take into account uncertainty or tolerance given for volumetric glassware. For example, if you have a 500ml flask, it is a Class A and usually a Class A has a tolerance of 0.2ml for 500ml flask. Now the normalizing factor for this is the square root of six. Hence(?), to convert this listed uncertainty or tolerance to standard uncertainty, you would divide that by square root of 6.

The third type is the normal distribution type. When we use this when we make an estimate from repeated observation and express the results at certain confidence interval, it follows normal distribution. Example of this would be the certificate of reference materials that states uncertainty with certain level of confidence. Here to convert the listed uncertainty, the standard uncertainty, you use the coverage factor K if it is given or if they just gave you 95%, you would use the normalization factor 1.96 and if it is stated at 99% confidence, you would use 2.94.

So what we did, so far, we determined the process, we outlined the process, we identified the sources, we calculated the uncertainty for the individual sources, and now we are going to proceed to combine this uncertainty.

There are two types of uncertainty we have here. One model is called the interim uncertainty. The interim uncertainty is used for a particular component. Say for an instance, if you have a flask, and if you have a 500ml flask, you have two components of uncertainty here, the tolerance of the flask, the volumetric flask as well as the temperature coefficient. Now you would be pulling these two uncertainties to give the overall uncertainty for the flask then you will be proceeding further to determine for the flasks that you have used, the pipette you have used, for the balance that you have used so you will be writing all the uncertainty for the entire process. Now to combine all of the individual uncertainty of the interim uncertainty to overall uncertainty for the process, you would use this equation.

And here we have used the term standard, multi-standard uncertainty. Why do we use this? It is observed in chemical measurements, dominant contributions to the overall uncertainty, very improportioned to the level of analytes. Hence, it is sensible to use working standard deviations then we combine each source. Then once you have combined the uncertainty for the overall process, now you have to expand it. Although the combined standard uncertainty is used to express the uncertainty of many measurement results, few regulating bodies require to define an interval around the result. The measure of the uncertainty intended to meet this requirement is termed as expanded uncertainty. This is usually done by multiplying by coverage factor K. Coverage factor depends on the confidence interval and number of observations. K=2 for observations greater than 7 at 95%. K=3 at 99% confidence. So finally you would represent the true value at x equals average plus or minus expanded uncertainty which is due.

Now, these are the additional terms that we used while in the preceding factors combined uncertainty and expanded uncertainty. These are all the important concepts in calculating the uncertainty for the process that I am going to describe.

So far we outlined the theory behind the calculation now how do we take a practical approach to this. So what we did we took gold pellets and dissolved it into solution with appropriate acid to make up volume. We analyzed it on our ICP and the result was 10,004 mg per liter concentration. Now how sure are we about this concentration? So we got through all the steps I have said earlier, identify the uncertainty from each step, combine them, and expand them.

So what are the steps I would use here? First, I have to determined what is to be measured. I’m measuring the concentration of the gold sample that I have prepared. Then I will be outlining my process which is going to come later. This process is dissolving gold in a sample solution and analyzing with ICP by comparing with NIST SRM. So there are various components of sources of uncertainties for this particular process so I will be identifying the sources. Now once I have identified the sources, I will estimate the uncertainty for each source then I will combine all the components and expand in.

So what is the process outlined for the analysis of gold? Now, in order to analyze my sample, I have to use the appropriate NIST SRM so I prepared the NIST SRM and the preparation of the NIST SRM is given by the symbol standard P. And there are some components of uncertainty associated with mass separation. I weighed, I made up the volume and then the recent SRM given, there is an uncertainty given in the certificate of SRM so I would have to determine uncertainty from each of this component in order to find the total uncertainty for standard P. The next step I have is the dilution of my sample. This I represented as CRM F, now this has two components uncertainty because I took a certain volume in a thermal pipette and made up the volume in a flask so there are two components of uncertainty rising from... for my CRM F. The third is I measured the concentration of SRM in ICP and there is an uncertainty component associated with my measurement. The similar approach was taken for my CRM and I measured the concentration of my sample and that is the uncertainty associated with the CRM concentration here.

So first and foremost, I have to determine my concentration. That’s my, one of the goals. So what is the equation I used? The concentration of gold in the sample is the concentration I measured from the ICP, the standard concentration I prepared from the NIST SRM, and the dilution factor that I used divided by the measurement of the standard by my ICP, which is the ICP measured concentration. These values are plotted here, given here and when you apply this here, you will be arriving at 10,004 mg per liter. How we arrived at these values is shown here. I will be explaining a bit later in my slide.

Now the next goal for me is to determine the uncertainty. I have determined my concentration. Now I have to determine how sure I am about this concentration so I have to calculate the uncertainty associated with the process as a whole. So what are the components of uncertainty here? This gives me the one component that is preparation of the standard represented by standard P. Here I weighed it so there is a mass component and I made up a certain volume so there is a tolerance component for the flask and temperature component arising from a portion expansion of the liquid in the flask. Then, the SRM certificate can weigh the uncertainty already so I have to scatter that into this. Then next step is sample dilution. I prepared the sample in order to analyze my, in my ICP instrument. And there are two components here. One is the pipette uncertainty and that has two components, tolerance of the pipette and temperature due to the coefficient expansion of the liquid. And then I have the flask component which is again just a tolerance and temperature component as the pipette. So combining all these, I will get the uncertainty for my CRM F sample dilution. Then I measured the concentration by ICP and this represents the uncertainty due to ICP measurement for SRM. Okay, similar process was taken for the ICP measurement for the sample. That component has to be controlled. So now you have four major components and very many minor components to each of this four major components. So usually it is better to represent all the components by means of this fishbowl diagram. This gives you a better idea on how to proceed.

Now first we are going to calculate the uncertainty associated with the standard. We used NIST SRM 3121. We prepared about 5.067 grams in a balance and diluted to 500 ml flask. So as I said in my fishbowl diagram, there are three components of uncertainty… weighing on the balance, making up to volume in 500ml flask and the certification that SRM has given me also has some uncertainty. So I have to factor all of these to calculate the total uncertainty for this process.

So let us take first weighing on balance. In my balance certificate, when I examined, the calibration was given but it had not stated any confidence interval so I would use a rectangular distribution in such a scenario. So the value was 0.0001 grams and I used the normalization factor root 3 and arrived at 5.774 x105 as my uncertainty. Now, I have used a... weight and then used the gross weight in order to measure this 5.067 grams so I have uncertainty twice the amount. I listed it, I pulled it, and that value is given here. So what did we have now. In this table, we have the uncertainty, the value, which is 5.067g, the uncertainty that I combined and I formed for this balance. Now I have to calculate, relative uncertainty which is combined uncertainty divided by the value. This is what I’m going to use later to calculate my overall uncertainty. So we did the calculation for weighing on balance. The next step is making up to volume in 500ml flask. There are two uncertainties associated with dilution. One is the tolerance given for the flask which was .2 and since it a triangular distribution as I have mentioned earlier for volumetric glassware you use a triangular distribution so you normalize it and to calculate the standard uncertainty from the listed uncertainty, you would use the tolerance divided by the normalizing factor and you got about 0.08165 The next contribution is from the liquid that is in that usually it is water. So normal coefficient of water external expansion is 2.1 ten to the power minus 4 degrees celsius per ml. ...radiation of three deg celsius in our lab, and then the volume coefficient for this is 500ml. So when you factor all of these and divided by the normalizing factor root three for rectangular distribution, you get a value of 0.1819. So, where do we, how do we pull the uncertainty for the 500ml flask that is you would do to volume and you would do to temperature so then you pull this, you get this value. Now, this is the interim uncertainty. Now we have to find the multi-standard uncertainty to use in our overall process. So for a 500 ml value, I will combine the uncertainty of .19935 so the relative uncertainty was, you see by P and that is the value given here as 0.0003987.

So we did…for this standard P to calculate the uncertainty for standard P, we did two components already the weighing on balance and the volume components due to 500ml flask. Now, there is a third component, SRM certification. SRM certification had the certified value for gold at 9.89 mg per gram. It had an extended uncertainty of 0.02. A coverage factor stated there was 2 so you have to convert that to standard uncertainty. You divide the uncertainty given with the certificate by 2 so you arrived at the value of 0.01. Now you calculate the relative uncertainty for this approach, and you arrive at a value 1.011 x 10 -3. Now, putting it all together, the uncertainty components from the balance, from the flask, and SRM certification, putting it all together you got a total value of 1.1816 x 10-6

The next uncertainty component, major component, was our CRM dilution. So what did we do here? We took… and diluted it to 500ml. So there are two components of uncertainty here, one is from the pipet and the other is from the flask. Just as similar to the earlier calculation, the listed U for the pipet was 0.01 and the triangular distribution will be normalized by root of 6. We arrived at the value 0.004082. Now again it has a temperature, the liquid coefficient expansion of water is to be used so for that, the same expression we used, our the temperature variation was three degrees and the volume that we use in the pipettor is 5ml and then we normalize it by root of three, you arrived at this value. So now we have to pull the uncertainty for the pipet. What, so that is U tolerance plus U temperature. When you pull it, you get a value 0.004469 or you find the validity (??) of uncertainty by dividing the value of five and you arrive at this value 0089... We finished the calculation for uncertainty for the pipet the next task is to calculate for the flask. For the flask, we used a 500ml…so we listed uncertainty of .2ml. The normalization factor, this is triangular, so we used root 6. The value when you convert to standard uncertainty of a U tolerance is 0.8165. Now again, this also has the liquid coefficient expansion of the liquid component.. so similar equations used, the volume is 500ml, temperature variation of 3 degrees, so you normalize the whole thing by root of 3 and arrive at a value 0.1819. So now, our task is to pull the uncertainty for the flask from their individual components like U tolerance and U temperature. So this is the value you get for the flask. Now, relative uncertainty for the flask is given here as 0.003987. So what do we have now for overall uncertainty for CRM F. Pipet uncertainty where relative uncertainty is calculated to be 0.0008938. Flask uncertainty, the relative uncertainty is calculated to be 0.0003987 and the total uncertainty is given here. This is the value we are going to use later in pulling the overall uncertainty for the method.

Now the third component that you saw in the fish bowl diagram was calculation of the measurement uncertainty for my ICP. We took 9 different samples, analyzed them, and we found an average of 100.985 by the ICP measurement. The standard deviation for the whole process was 0.29397. So this is a Type A Uncertainty and so you would apply this equation and you arrived at the standard relative standard uncertainty by dividing the combined uncertainty by the value 100.985. So this also will be used later on in our process. Now, similar to what we did for SRM, we did for CRM, we took 9 samples, made analysis on that by ICP, now the average is given here is x-bar 100.797, the standard deviation is 0.3065. Now, this again is a Type A Uncertainty. So the value of the uncertainty calculate relative to be 0.102165. This is the standard deviation divided by the number of replicates and we arrive at this 0.102165. So you calculate the relative standard uncertainty by dividing the uncertainty by the value here. So now what do we have? This represents the uncertainty from each component as you had seen uncertainty for SRM prep had its own component we pulled uncertainty from each of those individual components and finally we calculated S for Standard P, the relative standard uncertainty. This took a similar approach for SamplePrep and there were quite a few individual components to this and we pulled that and we got this value. Now, for the uncertainty for SRM measurements is pretty straight forward and same thing for uncertainty for sample measurement. Now, the total uncertainty is somewhat the squares of all relative the standards uncertainty and that’s given here. So finally, the uncertainty for the overall process is by root of this, and is given here as 0.00202697. Now, we determined the concentration of the gold as 10,004 mg per liter. Now we multiply that by the uncertainty here so the overall uncertainty for the concentration that we have in our flask is 20.2771.

As I have mentioned earlier, the uncertainty measurement just doesn’t stop at combining them. You do have to multiply by a coverage factor and expand them. So the, here, we used a 95% concentration (?) and a coverage factor of 2. So when we do that multiplication here, you got a value of 40.554 and we round it up to 41 mg per liter. So, our ICP value for Gold is 10,004 mg per liter and the sum is 41 mg per liter. So the gold bars you saw earlier might have had some uncertainty component, if not 99.999 as you would think, it has this 41 mg per liter as uncertainty associated with it.

We have taken all this information from the reference given here. Our aim here is to determine uncertainty in the analysis of sample we prepared. From analytical parameters and conditions for analysis are not presented but then it varied from lab to lab. We have tried to give an overall approach and this could vary from process to process, lab to lab and by no means comprehensive. So you have to determine what you have and then continue the process that I have mentioned and pull it. So I hope I have… the old process and you are able to calculate uncertainty without any problem in the future. If you have any problem, of course you’re always welcome to contact SPEX CertiPrep. Thank you for listening to our talk. We are planning two more additional information. Here what we did we focused our attention only on the analysis by ICP and by no means the CRM is restricted only to analysis. You have uncertainty due to sample preparation. You have uncertainty due to homogeneity, stability, absolutely any method that you use to compute your certified value. I have not gone into details of all the other things because I am planning to save that for your additional seminars that you are planning here. Hope you enjoyed the seminar. Thank you, Peter.

Okay. Thank you, Vanaja. We’ve got a little bit of time for questions, so we don’t see any up at the moment but if you have any question at all, just type them into the question box then we’d be more than happy to answer them. Okay, looks like we have one here.

Okay the question here is, why do you use rules of standard uncertainty in some cases and not in other cases. *You know, I told you there are two process involved in the calculation here. One is the interim calculation and another is combining the interim...* *overall calculation. The number that is an interim calculation,* *we calculated the component from each of those as they are. Then when we combined it to the overall process then we took the relative uncertainty, the standard uncertainty, it is because usually in chemical measurements, contributions to overall uncertainty wherein proportion of the concentration of the level or the value measured. So it is sensible to use as our… standard deviation then we combined the uncertainty from various sources to one process.*

I got a couple questions here about whether or not we’re going to give out the slides. *And yes, we will be sending out an email a little later today with the links so you can download the files* *and also, another person is asking me about recordings to the webinar, and yes we’re gonna, this webinar is being recorded and sometime next week we’ll post it on YouTube. We will also send out an email letting you know when that’s available.*

Alright, we got another one here. Ah, what would be your approach on qualitative methodology such as the ones found in microbiology? *Okay, I’ll repeat the question here, what would you, your approach be on qualitative method as found in microbiology? As I said, you have to be familiar with the method, and I am not familiar with this method but you can use the same methodologies that I have described here so you outline the process, identify the sources, and then you will calculate it, you don’t(?) have to be familiar with the method in order to do this. *

Ah, another question, what is coverage factor? *The coverage factor is usually called K. And as I said in the, in some of the regulation... indicate that you have to have a central value flank by a plus or minus value that is… especially with health and safety regulations. There is no absolute true value, so they want here, an interval value. So when you calculate the method uncertainty, you expand it by 2 in order to flank your absolute value by that expanded uncertainty. *

Well, there’s another question here. What is the other sources other than what I have listed such as ceiling volume? *I think by ceiling volume, you mean, delivery of uhm solutions from the volumetric flask. If that is so, you have, what you have to do, we did some experiments here before proceeding to quantifying the uncertainty, we considered many factors and then we calculated uncertainty. And we used the Bohr graph to determine how significant that particular component is to the overall uncertainty. If you found them to be not very significant than what we did is because the complexity of the calculation gets too hard and you are not able to get a real practical input into that. So what I would do in anything you want to do, first go through the process, determine the uncertainty, and see if it’s really significant, significantly contributing to your process, then you would consider that. *

Okay, another question. I usually deal with rock powders, silicate rocks that are digested for ICP-OES analysis. From your experience, our errors are usually higher than dealing with single element standards for examples? *Yes, absolutely. The rock samples, as you know, they are not homogenous. You may get various sample compositions and so you don’t know what you are getting at. So even if you have, then you dissolve a certain amount of rocks, then you dissolve the rocks, and then you may want to do two-three variations, two-three replicates to see if you, if your value is good. But one more problem you may have here, rock samples have radius..., radius components and you, well, will not be able to recover it completely, so what we usually do in such cases, we do a...* *and compare it and then add it to this particular process. *

Another question here, when trying to calculate uncertainty or probe analysis, which is better to use? *By probe analysis, I think you mean any pH meter, conductivity meter, that type. If that is what you mean, again you have a sample, uncertainty due to sample preparation. Whatever sample you have prepared to analyze, you have to take into account as I have mentioned here by this process. Then you would take various readings. Consider the readings as the type of uncertainty and then calculate it. Unless we know the process completely, I won’t be able to say exactly what you do. But I think, from what I understand from your question, it is the overall approach I would take. *

Okay, uhm, what is the total, what is the percent total uncertainty of the gold measurement? *Yes, it’s about .4% *

Do you know which type the uncertainty to use for the flask, balance or pipet? *Okay, yes. Usually for the volumetric flask of the glassware, we used uhm triangular distribution with its normalizing factor which is root 6(?). For the balance, there are two approaches here. Some calibration agencies, they do not give the confidence, they do not state the confidence level. If there is no confidence interval stated in the calibration certificate, you would consider that as a rectangular distribution and use the normalizing factor square root of 3. But of late, many of the agencies that I have used have started giving the uhm confidence interval at 95% then I would use the coverage factor K=2 or 1.96. *

Do you consider stability into the uncertainty? *Yes, we usually do for our reference materials. In this particular case, I didn’t go to that because my focus here is to calculate the uncertainty only by the instrument method. Then as I said, we are planning two seminars on overall uncertainty from the sample preparation, the purity of the salt or the stability and homogeneity, in that we will cover all those other aspects as well. *

Okay, another one here, how do you estimate mu when they change concentration of the measurement? *How you estimate measurement of uncertainty when this change with the concentration of the instrument. Yes that is why we use the relative standard uncertainty. This is exactly why we use relative standard uncertainty when we calculate. *

So I believe, we got to all the questions. There’s to cover everything but there’s a. So if there’s a, we got a few more minutes here so if there’s anything else you want answered, please type it in and we’ll certainly.. we got a couple more. The question here is about the number of observations need, minimum needed to use the coverage factor 2. *Yes the coverage factor, the minimum needed is about number 7 because the K=2 is used when the degrees of freedom are safe. If it is less than 6, you have, if your observations is less than 6, then what do you do? You have to use the two tailed Student’s t at 95% confidence. Usually, with a decrease in number of measurements, you get a, greater uncertainty. That is why we tend to use about 7 or measurements of 7 or observations of 7 or more in order to reduce the uncertainty. *

Ah, this one is, a question that is related to that. What is the appropriate n numbers needed when using the standard distribution curve at say 95% confidence level. Is it a minimum of n=9 as you stated or the more the better like n=20? *Well, there is a statistical approach of how many minimum replicates are needed in order to arrive at a good value? What is the most benefit you get, do you get by increasing the number of measurements do you get a better value? No. I think and have decided, this is the calculation of...* *I can find it and send it to you. Minimum is greater than 6, replicate is greater than 6 does not increase any benefit in the accuracy. *

And how does one find K or the coverage factor? *Okay if your observation is less than 7, you go to the Student’s t, Student’s t it is available in any statistics book or I can email you that. *

Okay. Okay. It looks like we’re about it again. So uh, we got, we still got a couple of minutes, so feel free to type it in. I’ve got a few things that I can cover while waiting for you guys to type it in. So if you haven’t already received the copy of our new catalog, then you can request or download it right from our website and choose from the paper version, cd, or view the electronic flipbook. We also recently launched a consumer safety standards products where in manufacturers of pharmaceuticals and nutraceutical that must comply with USP 232. And Spex CertiPrep is social so come visit us on your favorite social networking sites. We have pages on Facebook, Twitter, YouTube, and LinkedIn. And anyone who is interested in any of our previous seminars inlcuding BPA and Phthalates in Consumer Products or The Art in Chemistry of Wine, you can watch them right now on our YouTube channel at YouTube.com/spexcertiprep. Okay, Vanaja, thank you for giving us yet another great presentation. As a reminder, we will be sending everyone a copy of the presentation slides and the recording of the webinar will be posted in about a week or so. We’ll also send an email out for that so that you’ll know when and where to view it. I’d like to thank everyone for attending today and we hope you are walking away with a better understanding of uncertainty and how it relates to instrumental analysis. We appreciate your time and we look forward to seeing you again in our future webinar. Thank you.