Calculating normalized error is common for laboratories participating in proficiency testing or interlaboratory comparisons. If you have participated in a proficiency test before, you may have noticed it in your final summary report either by name or abbreviated ‘E_{n}.’

When you participate in proficiency tests, the PT provider calculates normalized error for you. However, if you are participating in interlaboratory comparisons because there is not a proficiency test available, you may have wanted to know how to calculate normalized error to analyze your test results.

In this article, I am going to explain what normalized error is and how to calculate it. After reading this article, you should be able to calculate normalized error long-hand, using a calculator, and(or) using MS Excel.

### What is Normalized Error

Normalized error is a statistical evaluation used to compare proficiency testing results where the uncertainty in the measurement result is included. Typically, it is the first evaluation used to determine conformance or nonconformance (i.e. Pass/Fail) in proficiency testing.

Normalized error is also used to identify outliers in the proficiency test results. Sometimes, outliers are removed from the calculations of adjusted mean to prevent influence of excessive offsets.

### Why Normalized Error is Important

Proficiency testing is a requirement of ISO17025. If your laboratory needs to performed proficiency testing but is unable to find a PT provider, you may need to perform an interlaboratory comparison. Using the normalized error equation will allow you to evaluate the results in a manner that is acceptable to your accreditation body.

### How to Calculate Normalized Error

To calculate normalized error (i.e. E_{n}), use the formula below as a reference. If you need some help, keep reading; I am going to walk you through the calculation process.

##### 1. First, calculate the difference of the measurement results by subtracting the reference laboratory’s result from the participating laboratory’s result.

##### 2. Next, calculate the root sum of squares for both laboratories’ reported estimate of measurement uncertainty.

##### 3. Finally, use the value calculated in the first step (i.e. difference of measurement results) and divide it by the value calculated in the second step (i.e. RSS of Uncertainties).

If your results are satisfactory, the value of E_{n} should be between -1 and +1. If not, you may have a problem with your measurement process.

### Example Normalized Error Calculation

Need an example? I am going to calculate normalized error using data from one of my proficiency tests. In the proficiency test, I compared my Fluke 732A DC Reference Standard to a Fluke 732B sent to me by NAPT.

Looking at the test data, you can see that my Fluke 732A had a value of 9.9999361V with an uncertainty of 0.000028 V (i.e. 2.8ppm) and the Fluke 732B had a reference value of 9.99993365 with an uncertainty of 0.00000044V (i.e. 0.044 ppm). After calculating normalized error, or E_{n}, the result yielded a value of 0.09 which is between -1 and +1. So, my results were satisfactory.

### Calculate Normalized Error using Excel

Instead of calculating normalized error the long way, try using MS Excel. It is fast and easy to build a spreadsheet calculator where you can perform multiple calculations very quickly. To make life easy, use the image and equation below as a guide.

After you have completed the first line of data with the equation, simply copy and paste cell F3 down column F for as many calculations as you need. Using MS Excel is the fastest way to calculate normalized error for your interlaboratory comparisons.

### Evaluating Normalized Error

Evaluating the results of the normalized error equation is pretty easy. Calculated values between -1 and +1 are considered conforming or passing. Results outside of this range are considered nonconforming or failing.

### Conclusion

Calculating normalized error is not common unless you are a proficiency testing provider. However, if you do not have a proficiency testing provider and(or) participate in interlaboratory comparisons, you may need to know how to calculate normalized error. Even if you have a proficiency testing provider, sometimes it is best to double check their calculations. I have found some mistakes over the years which changed my test results from failing to passing.

Thanks for this article. Please, could you explain briefly, why a normalized error between -1 and 1 is considered conforming ?

Dear Mickaël,

it’s all about statistics: with ~95% probability, the En of labs with a bias consistent with the reported measurement uncertainty will be within limits. One can pick limits other than -1 and 1, but the cultural consensus amongst users of statistics is that 5% is a meaningful risk of wrong assignment (false negative / false positive).

Kind regards,

Christian

Good explanation about EN .

I want to know the correct process to calculate the K factor

Hi Francisco,

Thank you for your comments and your request. I will write an article in the next few months about how to calculate expansion factors (k-factors).

Great article.

Could you tell me if the En is used only for laboratories participating in proficiency testing or interlaboratory comparisons ? Can I use it for comparing data with a reference value?

Hi Rick,

Thanks for this article. Could you please tell me When we use En equation for interlaboratory comparison, the reference laboratory uncertainty should be always better for our uncertainty or not? Is it a requirement or rule for this comparison?

Very interesting article. Very good job you are doing for all of us.

Please, can you explain Uncertain and Normalized error how they relate?

Thanks. All the best for you in 2016

Hi Antero,

Uncertainty relates to the quality and/or confidence in a single measurement result or a series of measurement results. Normalized error is used to make an approximately normalized comparison of two measurement results and their respective uncertainties. I hope this helps.

Sorry, in my last comment about relationship betwen Normalized error and uncertainty I missed refering to “of a measurement”

In this example:

Reference value: 1.010 Uref: ± 0.006

Lab value: 0.996 Ulab: ±0.010

En = 1.2 therefore the lab is considered to have failed.

However if we take the lowest value of the reference: 1.010 – 0.006 = 1.004

and the highest value of the lab: 0.996 + 0.010 = 1.006

then we find that they overlap.

This indicates that the two results are mathematically equivalent and therefore we cannot determine if the ILC passed or failed.

I know that this argument is flawed, but I can’t explain why,

Please assist.

Hi Guy,

It is a good argument to claim that the result is indeterminate; neither Pass nor Fail. This would be similar to making statements of conformance for calibration results. If there is overlap, using your method, the result is indeterminate.

Try reading the ILAC G8 for more guidance. You can download it for free by clicking here.

I hope this helps.

Rick

Hi Richard

Thank you for your easy and clear explanation of Normalised Errors. Could you suggest any papers regarding En values. Also I know En values are not the only way to evaluate ones results in a PT/ILC scheme for example Z values. Could you also point me in a direction of some documents which shows when and why you will use a certain evaluation method?

Thanks

Corne

Hi Corne,

Your best bet is to read the ISO/IEC 17043:2010 for more information on En and z-scores. However, NAPT has a good guide for understanding their reports and it has a lot of great information in it that may answer your questions. Check it out here.

Also, I wrote a guide on proficiency testing that may helping you out as well. You can read it here.

If you need further assistance for your unique scenario, I would be happy to answer your questions and discuss some potential solutions for a fee. Just click here to reserve a consultation.

I hope this helps.

Best regards,

Rick