Best way to say something about the accuracy of a measurement

#1
Hi,
I'm an engineering student and encountered a problem to which I did not find a solution yet, and wondered if someone could give me the benefit of their expert opinion.

I'm going to perform an experiment in which I'll cook an egg using standing electromagnetic waves, only for a short period of time. This way there are only some regions that are cooked (regions A,B,C and D). The goal is to measure the distance between these regions as accurately as possible and tell something about the accuracy of the measurement. Assume we know the distance between regions A&B, B&C, C&D and D&A are the same.

Measurements.png

I came up with some methods to handle the data:
1) Visually make a guess of the centers of the cooked regions (red dots) and measure the distance between them. This creates an extra 'human' error, since it's just a visual guess. With those 4 values I could calculate a 95% confidence interval and thus am able to tell something about the accuracy.

But then I posed myself the question: Maybe we could use the fact that we can measure the green and orange distances to our advantage.

2) Measuring the longest and shortest distance between A&B, B&C, C&D and D&A. Then calculating the mean between the two and using these four values for calculating the confidence interval. The problem is, I won't be able to work in an ideal environment and the eggs aren't ideal either, so the shapes of the cooked regions will be irregular (like on the drawing).

So I had another thought: Maybe combining both methods?

3) Using the red dots to calculate the mean and using the other measures (green & orange lines) to say something about the confidence? Maybe use the confidence interval made with the red dot data and combine it in some way with the confidence interval data of the longest distances and the shortest distances respectively?

Can you come up with a suiting statistical method for interpreting the data of this experiment?
Happy to hear how you would tackle this problem!

Thanks in advance, John
 

noetsi

No cake for spunky
#3
If you can actually measure the distance between the areas why would they ever be inaccurate? Is there some reason you are unable to physically measure the distance, or your instrument is know to be inaccurate?
 

katxt

Active Member
#5
Or try this perhaps ...
Put green dots where the green lines meet on the diagram. Their exact locations are unknown but physics assures us (I think) that each is the same distance from the next, at an unknown distance L, say.
Near each green dot there is a red dot which approximates the position of the (unknown) green dot. An appropriate approximation might be the centroid of the irregular cooked region.
The problem now is to find the best place to locate the green dots such that the lengths are all L and the total error is a minimum . This will give a best estimates for L and the uncertainty of the red dots.
Three methods come to mind. Maximum likelihood, normal equations (you may have come across these if you have studied surveying), or minimization using Solver in Excel. If any of these appeal to you, post back and we can perhaps give you some ideas.
 
#6
So, your data is the four irregular areas? How are they given to you?
What would you do with just two area?
If you can actually measure the distance between the areas why would they ever be inaccurate? Is there some reason you are unable to physically measure the distance, or your instrument is know to be inaccurate?
Or try this perhaps ...
Put green dots where the green lines meet on the diagram. Their exact locations are unknown but physics assures us (I think) that each is the same distance from the next, at an unknown distance L, say.
Near each green dot there is a red dot which approximates the position of the (unknown) green dot. An appropriate approximation might be the centroid of the irregular cooked region.
The problem now is to find the best place to locate the green dots such that the lengths are all L and the total error is a minimum . This will give a best estimates for L and the uncertainty of the red dots.
Three methods come to mind. Maximum likelihood, normal equations (you may have come across these if you have studied surveying), or minimization using Solver in Excel. If any of these appeal to you, post back and we can perhaps give you some ideas.
The goal is to determine the distance between two local maxima of the oscillating electromagnetic waves. These are measured by placing a substance that melts or burns in the 'standing' electromagnetic field (the local maxima of the waves do not move). The four irregular areas are the result. To simplify, imagine A and B are the result, and the experiment is repeated four times, resulting in sets of two different irregular areas each time.

To determine the actual local maxima of the waves, we can measure the shortest and longest distance between the irregular areas (orange and green lines respectively) and take the average this way we tried to obtain the most accurate measurement. I changed my mind on the red dots since this new technique seems more accurate that guessing the centroid visually, like proposed in the original post. If we repeat the experiment we can determine a 95%-confidence interval for the actual value of the distance between two local maxima. My question is: Can we do something with the longest-distance and shortest-distance data between the irregular areas to maybe declare another type of boundary or confidence interval about the precision of the method. Could we use this data to say something meaningful about the experiment?

Hopefully this makes things clearer. Thanks a lot for your reactions :)
 
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katxt

Active Member
#8
To determine the actual local maxima of the waves, we can measure the shortest and longest distance between the irregular areas (orange and green lines respectively) and take the average this way we tried to obtain the most accurate measurement. I changed my mind on the red dots since this new technique seems more accurate that guessing the centroid visually, like proposed in the original post. If we repeat the experiment we can determine a 95%-confidence interval for the actual value of the distance between two local maxima.
Aha, I see what the green lines are now. I imagined that they were the true lines. I obviously didn't read the problem closely enough. Your solution is a typical engineering ad hoc idea which should give reasonable results of the true maximum points. However, if you want to estimate real 95% confidence intervals, then you will need to estimate the errors. This can be done using the method I outlined above except make some new lines and points representing the true situation.
 
#9
However, if you want to estimate real 95% confidence intervals, then you will need to estimate the errors. This can be done using the method I outlined above except make some new lines and points representing the true situation.
What extra lines/points would you use? I've performed the experiment and just got two regions, so I repeated the experiment to have a total of 4 measurements. (Then one of the components stopped working, so I had to pause the experiment). So imagine you have just the case with A and B as a result and repeat the experiment a few times. By measuring the shortest and longest distance between the two regions and taking the average we have a measurement for the experimental distance between the regions. Just using this data (unknown variance and some measurements for the mean), I think you could calculate the 95% confidence interval, using a t-distribution. But I was wondering, maybe you can use the measurement of largest and smallest distance between the regions to say something about the error.

PS: I've measured the largest and smallest distance on one line, that approximately goes trough the centroids, since the waves travel in a straight line trough space.
 

katxt

Active Member
#10
We have been going down different paths, I think. What you have outlined here is a simpler problem.
One question is whether the average of the the outside and inside measurements is a more accurate figure than the distance between the centroids. It is certainly easier to do, but it is vulnerable to odd lumps. The centroid averages out those lumps.
If you have several estimates of the distance between the two maxima then you can certainly use the t distribution to get a 95% CI. You have probably found plenty of calculators on line.
 
#11
It is certainly easier to do, but it is vulnerable to odd lumps. The centroid averages out those lumps.
I agree. However, since the results were almost circular, I went for the method described above.
If you have several estimates of the distance between the two maxima then you can certainly use the t distribution to get a 95% CI.
Yes, this is what I had in mind. But then I wondered, could you do something else with the max and min distances to define another kind of 'boundary', or is there no other use case for these values other than for calculating the distance itself in an experiment.
Taking the overall max and min distance (over all the experiments) will for example tell you "we've never measured anything above or below these distances". So I was wondering if we could use statistics to provide another use case to these min and max measurements.

After all, these min and max measurements are four measured values. By calculating the distance with them you reduce them to one value. To me this seems like loss of information, since infinitely many tuples of these four values lead to the same distance.
 

katxt

Active Member
#12
Taking the overall max and min distance (over all the experiments) will for example tell you "we've never measured anything above or below these distances". So I was wondering if we could use statistics to provide another use case to these min and max measurements.
This looks like a weak version of the confidence interval.
To me this seems like loss of information
No more so than finding the mean and SD of a sample. You do lose information about the actual numbers but you get more information about the population.