Help with this problem about error margins please

[Tsunnnami]

Homework Statement

A student performs a series of measurements, the obtained values are : 90, 110 and 180. The method that is used has a theoretical error of 5%.
What is the real value of the measurement and empirical error ? What are the conclusions ?

Homework Equations

We have no equations given

The Attempt at a Solution

No idea.
We haven't done any similar exercises. We only did experiments, where we measured different values and we got errors and we know that an error bigger than 5% means that something in the experiment went wrong, but that's it, honestly. We have never done something like this.

 

[Buffu]

Homework Statement

A student performs a series of measurements, the obtained values are : 90, 110 and 180. The method that is used has a theoretical error of 5%.
What is the real value of the measurement and empirical error ? What are the conclusions ?

Homework Equations

The Attempt at a Solution

Just as you filled the "1" part of the template, please fill the other two parts especially "3".

 

[fresh_42]

It might be a good start to define the terms you use. One could guess by what is meant by a theoretical error, but what is the empirical error?
Also your use of "the measurement" in comparison to the three measured values is a bit disturbing: Is it three or one measurement with three values simultaneously measured? And what is a real value if not the measured one?

 

[kuruman]

A student performs a series of measurements, the obtained values are : 90, 110 and 180. The method that is used has a theoretical error of 5%.
What is the real value of the measurement and empirical error ? What are the conclusions ?

I echo the comments of @fresh_42 and I add that I don't understand what is being asked. If a student performs an experiment are not th results of the measurements "real" values? Here we have three values, 90, 110 and 180. What makes one of these more "real" than the others? Furthermore, if the "real" value is something other than one of the three numbers, what makes it real if it is something that has not been measured? Maybe the purpose of the question is to raise these issues ...

 

[Buffu]

Homework Statement

A student performs a series of measurements, the obtained values are : 90, 110 and 180. The method that is used has a theoretical error of 5%.
What is the real value of the measurement and empirical error ? What are the conclusions ?

Homework Equations

We have no equations given

The Attempt at a Solution

No idea.
We haven't done any similar exercises. We only did experiments, where we measured different values and we got errors and we know that an error bigger than 5% means that something in the experiment went wrong, but that's it, honestly. We have never done something like this.

I guess you just need to find the mean error in the measurement and subtract that from each measurement to get the "real measurements" though I am not sure.
In other words, I am GUESSING that you need to find mean deviation of the the data given and subtract it from each data point to find the "real measurements".

$\bar{\delta a} = \displaystyle {1\over N}\sum^N_{i = 0} {|\bar{a} - a_i|}$

Where $\bar{a}$ is the mean measurment and $\bar{\delta a}$ is mean error.

 

[jbriggs444]

I guess you just need to find the mean error in the measurement and subtract that from each measurement to get the "real measurements" though I am not sure.
In other words, I am GUESSING that you need to find mean deviation of the the data given and subtract it from each data point to find the "real measurements".

$\bar{\delta a} = \displaystyle {1\over N}\sum^N_{i = 0} {|\bar{a} - a_i|}$

Where $\bar{a}$ is the mean measurment and $\bar{\delta a}$ is mean error.

You are estimating the true value of the measured quantity as $\bar{a}$. But with only three measured values to go on and with the variance in those being so high, such estimate will not be reliable. Garbage in, garbage out.

The sum of the absolute deviations from the sample mean is a biased estimator of the sum of the absolute deviations from the true value. It is biased low: On average, the deviation from the true value will be greater than the deviation from the sample mean.

You might try to remove bias from the estimator by dividing by n-1 rather than by n. But with numbers this poor, the result will still be garbage.

 

[Buffu]

You are estimating the true value of the measured quantity as $\bar{a}$. But with only three measured values to go on and with the variance in those being so high, such estimate will not be reliable. Garbage in, garbage out.

The sum of the absolute deviations from the sample mean is a biased estimator of the sum of the absolute deviations from the true value. It is biased low: On average, the deviation from the true value will be greater than the deviation from the sample mean.

You might try to remove bias from the estimator by dividing by n-1 rather than by n. But with numbers this poor, the result will still be garbage.

Or can I try standard deviation to improve results ?

But still it is only 3 data points with very large devations.

 

[haruspex]

Sounds to me like an exercise in rejection of outliers.

 

[kuruman]

The problem is asking

What is the real value of the measurement and empirical error ? What are the conclusions?

Suppose the highest value of 180 is rejected as an outlier. This leaves 90 and 110. They are not within the theoretical 5% of each other. One may ask and answer, "What additional empirical error is needed to reconcile the two?" However, I still don't know what is meant by "real" value.

 

[haruspex]

They are not within the theoretical 5% of each other.

You mean, they are not each within 5% of the same value, but I take your point.

 

[Buffu]

What are the conclusions ?

The percentage mean deviation error around mean is 9%. So maybe we can say that the experiment is not done properly.

 

[jbriggs444]

The percentage mean deviation error around mean is 9%. So maybe we can say that the experiment is not done properly.

If you include the outlier, the mean deviation percentage is much higher than 9%. So you must be excluding the outlier. Now you have two measurements that are still not compatible to within the theoretical expectation. How do you know that you removed the right outlier?

At least two out of the three measurements must be outliers. From the data at hand you have no way to guess which, if any, of the three values to accept.

 

[kuruman]

At least two out of the three measurements must be outliers.

I don't understand the "at least two" outlier number. I have no problem with it if the only source of error were the 5% theoretical error. For example, when a block is sliding down an incline, a theoretical error would be introduced if I assume that there is no friction when actually friction is about 5% of g sinθ. If I actually do an experiment to measure the acceleration, I am very likely to get values that differ from g sinθ by more than 5% because of random or systematic errors in the experiment. That extra bit is what I think is meant by "empirical" error in the statement of the question.

 

[jbriggs444]

I don't understand the "at least two" outlier number. I have no problem with it if the only source of error were the 5% theoretical error. For example, when a block is sliding down an incline, a theoretical error would be introduced if I assume that there is no friction when actually friction is about 5% of g sinθ. If I actually do an experiment to measure the acceleration, I am very likely to get values that differ from g sinθ by more than 5% because of random or systematic errors in the experiment. That extra bit is what I think is meant by "empirical" error in the statement of the question.

That's a fair comment. However, we are given three numeric values and a prescribed theoretical expected best case error bound. No two values are compatible to within that bound. We know that there is experimental error, but we have inadequate information from which to build a model for that error. Accordingly, all three measurements are suspect and should be discarded.

 

[kuruman]

Accordingly, all three measurements are suspect and should be discarded.

I think that's a bit extreme. What if I made a single measurement and got 90? Should I discard it? If the answer is "no" does this mean it is less suspect than when it is part of three measurements? If the answer is "yes", why even bother making the measurement in the first place? I think nothing should be discarded and that more measurements should be made until one can figure out what is going on.

 

[haruspex]

What if I made a single measurement and got 90? Should I discard it?

This gets into Bayesian territory. What if you made a second measurement and got 20000? You would now have far less faith in the 90 than before. What was your trust in the 90 based on? Experience.

 

[kuruman]

This gets into Bayesian territory.

Indeed it does. If I got 90, 20000 and then 180, I would examine what I am doing and how, fix the intermittent contact, throw out all three numbers as jbriggs444 suggested and start all over again. As you say, experience.

 

[haruspex]

Indeed it does. If I got 90, 20000 and then 180, I would examine what I am doing and how, fix the intermittent contact, throw out all three numbers as jbriggs444 suggested and start all over again. As you say, experience.

That wasn't my point. You were asking how much trust you should put in a single measurement, the 90. I am saying that the only reason for putting any trust in it is your experience, i.e. you have an a priori estimate of the error distribution. Without a Bayesian approach, you cannot justify that.