Assessing a measurement's precision

[physicsnerd26]

Homework Statement

I'm doing a quality of measurements coursework at the moment where I have to find the efficiency of the different volumes of water. To do this, I needed to use this formula:

After measuring the efficiency of different volumes of water, I'm asked to assess the measurement's precision, I assume this is asking me to assess my measurement of the initial and final temperature of the heated water, the quantified amount of voltage and current, and the time taken for the water to heat up.

Homework Equations

How do I exactly begin to assess this, how can I improve the precision? And where can I possibly see any systematic errors, I already looked up what systematic error means, and it seems way too complicated for me to understand.

The Attempt at a Solution

Before even reading the question, I already looked for any outliers using this formula:

I have two set of results and I found out there aren't any outliers. I kind of feel like this means that my results are reliable and I don't know exactly how else I can improve it if I've already gotten it right the first time around. Did I even get it right?

 

[mfb]

Please start with a detailed description of the experiment: what was its purpose, what was the experimental setup, what did you measure how, when, what equipment did you use, ...
I can make a guess based on your questions, but guessing is not a good way to start an answer.

 

[physicsnerd26]

Please start with a detailed description of the experiment: what was its purpose, what was the experimental setup, what did you measure how, when, what equipment did you use, ...
I can make a guess based on your questions, but guessing is not a good way to start an answer.

I added some more details, sorry for the vagueness.

 

[mfb]

There is still no description what you are doing, so let me do some guesses:

- you heat water with electricity and want to measure the efficiency of this heating process
- you have a DC current with measured voltage and current for a time t that gets measured in some way
- you have a thermometer in the water volume and record temperatures before and after the heating process
- you measured the amount of water at some point, or calculated it based on the geometry of the container
- you repeated this with different amounts of water
- did you also test different times or electrical currents?

If all those points are right, they form a minimal description of what you actually did.

To find experimental uncertainties, you have to dig deeper: for each value, how was it measured, what is the expected precision of this measurement and do you have some way to test this precision? Can this value be wrong in a systematic way because the measurement is not done at the right place (where your relevant quantity appears)? For example, could you measure the temperature at a place that is hotter or colder than others?
What exactly did you measure? Which times, currents, voltages, water volumes, ...?

To evaluate systematic errors you have to know the experiment as good as possible. Your description is way too vague to start with that.

 

[physicsnerd26]

There is still no description what you are doing, so let me do some guesses:

- you heat water with electricity and want to measure the efficiency of this heating process
- you have a DC current with measured voltage and current for a time t that gets measured in some way
- you have a thermometer in the water volume and record temperatures before and after the heating process
- you measured the amount of water at some point, or calculated it based on the geometry of the container
- you repeated this with different amounts of water
- did you also test different times or electrical currents?

If all those points are right, they form a minimal description of what you actually did.

Exactly right.

To find experimental uncertainties, you have to dig deeper: for each value, how was it measured, what is the expected precision of this measurement and do you have some way to test this precision? Can this value be wrong in a systematic way because the measurement is not done at the right place (where your relevant quantity appears)? For example, could you measure the temperature at a place that is hotter or colder than others?
What exactly did you measure? Which times, currents, voltages, water volumes, ...?

I assume the results, the data that I gained from the experiment should be used to get the precision, but I don't get what exactly I should do with them. I read a way to calculate the uncertainty last night, basically since I had two sets of results, I was able to do this.

so for example, the efficiency of 300ml in my first result is 44.6% and the efficiency of the second result is 46.3%, I found the mean, which is 45.45.
Then (46.3-45.45)+(45.45-44.6), which is 1.7, which means the uncertainty in 300ml of water is 1.7%, is this correct?

Yes, I did measure the time taken for the water to completely heat up, the current and voltage through a plug in power meter, and the change in temperature by measuring the initial and final temperature.

To evaluate systematic errors you have to know the experiment as good as possible. Your description is way too vague to start with that.

Am I right to assume that systematic errors are errors that I did myself using the instruments or the instruments not working properly? Would I identify these by comparing my sets of results to one another?

 

[mfb]

Then (46.3-45.45)+(45.45-44.6), which is 1.7

Which is just the difference between your two values.

which means the uncertainty in 300ml of water is 1.7%, is this correct?

No. But it is not completely unreasonable to expect an uncertainty roughly of this size - could be larger, but certainly not much smaller.

Am I right to assume that systematic errors are errors that I did myself using the instruments or the instruments not working properly? Would I identify these by comparing my sets of results to one another?

Systematic errors are errors that you repeat even if you do the same experiment many times. Instruments not working properly contribute, but also problems with the measurement setup.

 

[physicsnerd26]

Which is just the difference between your two values.

Sorry, what?

No. But it is not completely unreasonable to expect an uncertainty roughly of this size - could be larger, but certainly not much smaller.

So which part of it did I do wrong?

Systematic errors are errors that you repeat even if you do the same experiment many times. Instruments not working properly contribute, but also problems with the measurement setup.

Can you give an example? Even if it's totally out of context?

btw, is the assessment just evaluating the benefits and drawbacks of the instruments that I used?

 

[mfb]

Sorry, what?

Your two values are 44.6% and 46.3%, the difference between those two values is 1.7% (absolute difference). You don't need the mean value to get those 1.7%.

So which part of it did I do wrong?

You did not analyze where the deviation could come from.

Can you give an example? Even if it's totally out of context?

Your voltmeter could always show a value 2% too large, for example. You can repeat the experiment as often as you want, the result will always be off by those 2% even if everything else is exact.

btw, is the assessment just evaluating the benefits and drawbacks of the instruments that I used?

Estimating how precise your answer is is an important part of every experiment. A measurement result without uncertainty estimate is worthless.

 

[physicsnerd26]

Your two values are 44.6% and 46.3%, the difference between those two values is 1.7% (absolute difference). You don't need the mean value to get those 1.7%.

I just realized that, lol. Thanks for clearing that up for me.

You did not analyze where the deviation could come from.

So all I need is analysis but the equation that I used is legit?

Your voltmeter could always show a value 2% too large, for example. You can repeat the experiment as often as you want, the result will always be off by those 2% even if everything else is exact.

So in this case, the constantly changing amount of voltage in the plug in power metre is a systematic error since you won't really be able to get the exact voltage?

Estimating how precise your answer is is an important part of every experiment. A measurement result without uncertainty estimate is worthless.

My book says precision is a quality denoting the closeness of agreement between measured values obtained by repeated measurements. In the case of 300ml of water, 44.6% and 46.3% of efficiency, do I just analyse the consistency to evaluate the precision? Sorry for the lots of questions -__-

 

[mfb]

So all I need is analysis but the equation that I used is legit?

You got the (correct) difference between two measured values.
This is not the uncertainty for measurements, it can just serve as a very rough guess how large the statistical uncertainty might be. It does not tell you anything about systematic uncertainties because they will influence each value in the same way.

So in this case, the constantly changing amount of voltage in the plug in power metre is a systematic error since you won't really be able to get the exact voltage?

How is it constantly changing? If it is random, and you measure the average, the effect is not the same every time, then it would be considered as statistical uncertainty.

My book says precision is a quality denoting the closeness of agreement between measured values obtained by repeated measurements.

To draw conclusions based on measurement results alone, you should have at least 3-4 measurements with the same conditions, more are better.

 

[physicsnerd26]

You got the (correct) difference between two measured values.
This is not the uncertainty for measurements, it can just serve as a very rough guess how large the statistical uncertainty might be. It does not tell you anything about systematic uncertainties because they will influence each value in the same way.

That formula was actually for getting the precision, not uncertainty, sorry, just saw the site again.

And I found a way to get the uncertainty: Find the mean, subtract the mean to every piece of result you got on a certain value of the volume of water, then square root it. Is this right? For example, for 500ml, I got 50.8%, 52.1%, and 51.3%, the mean is 51.7333, then (52.1-mean)^2 + (50.8-mean)^2 + (51.3-mean)^2 = X, then X/3 -> then the square root of X/3 -> Mean ± (x/3)? Is that right?

How is it constantly changing? If it is random, and you measure the average, the effect is not the same every time, then it would be considered as statistical uncertainty.

How about how long you have the thermometer when you measure it's temperature?

To draw conclusions based on measurement results alone, you should have at least 3-4 measurements with the same conditions, more are better.

I've only got two measurements. I have some excess ones, but in most of the volumes, I only got two. Would that still work?

 

[mfb]

And I found a way to get the uncertainty: Find the mean, subtract the mean to every piece of result you got on a certain value of the volume of water, then square root it. Is this right? For example, for 500ml, I got 50.8%, 52.1%, and 51.3%, the mean is 51.7333, then (52.1-mean)^2 + (50.8-mean)^2 + (51.3-mean)^2 = X, then X/3 -> then the square root of X/3 -> Mean ± (x/3)? Is that right?

You get a better estimate if you divide it by "numberofmeasurements minus one" (here: 3-1=2).

How about how long you have the thermometer when you measure it's temperature?

If you don't share much more details of the experiment, I have no idea.

I've only got two measurements. I have some excess ones, but in most of the volumes, I only got two. Would that still work?

No way to tell without more information what exactly you have.

 

[physicsnerd26]

You get a better estimate if you divide it by "numberofmeasurements minus one" (here: 3-1=2).

But is this one good enough?

If you don't share much more details of the experiment, I have no idea.

Which part of it do you need? I'm not too sure on that part.

No way to tell without more information what exactly you have.

So, can I just make some of them up? I have to submit this on Wednesday, so..

 

[physicsnerd26]

I'm kind of confused now, the measurement that I'm assessing when checking the precision is the efficiency, or the factors that I needed to get it? (time, V, I, mass, etc)

 

[mfb]

But is this one good enough?

Why would you choose a value that gives a worse answer if you can simply plug in a different number to get a better value?

Which part of it do you need? I'm not too sure on that part

All the things I asked about in the previous posts.

So, can I just make some of them up?

?
You could share which measurements you have.

 

[physicsnerd26]

Why would you choose a value that gives a worse answer if you can simply plug in a different number to get a better value?
All the things I asked about in the previous posts.
?
You could share which measurements you have.

Here's the two that I have atm.

Just a side question, when talking about the greatest uncertainty, what points do you have to explore?

 

[mfb]

Okay, that looks quite odd. I would not expect the efficiency to increase like that. Do you have a sketch of the experiment? Where and how exactly where those values recorded, especially the temperature? What was the initial water temperature?

For each volume, you have two measurements - you can calculate the difference between those two and then compute the uncertainty based of this (using the approach from post #11). The average over 16 sets should give a reasonable estimate for the statistical uncertainty.

Just a side question, when talking about the greatest uncertainty, what points do you have to explore?

What is the context of the "greatest uncertainty"? On its own, it does not have a standard meaning.

 

[physicsnerd26]

Okay, that looks quite odd. I would not expect the efficiency to increase like that.

Was it increasing slower or faster than what you expect?

Do you have a sketch of the experiment?

What sort of sketch? Like a diagram of my laid out instruments?

Where and how exactly where those values recorded, especially the temperature? What was the initial water temperature?

I did them in different days of the week about 2 weeks ago, I just used the instruments I was given, namely, plug-in power meter, thermometer, large measuring cylinder, and kettle.

Do I need the initial temperature to draw out the raw results? Could the fluctuations in the temperature on different days affected the results I got in the change in temp?

For each volume, you have two measurements - you can calculate the difference between those two and then compute the uncertainty based of this (using the approach from post #11). The average over 16 sets should give a reasonable estimate for the statistical uncertainty.

Great, thanks.

 

[mfb]

Was it increasing slower or faster than what you expect?

I would not have expected differences that large, and the high efficiency values for the largest volumes surprise me.

What sort of sketch? Like a diagram of my laid out instruments?

Yes. Something where you can see what is where, connected in which way and so on.

Do I need the initial temperature to draw out the raw results?

No, but it would be useful to see how far away from the boiling point you were, for example.

Could the fluctuations in the temperature on different days affected the results I got in the change in temp?

In general, yes.I don't know if it is a language issue or something else, but I find it extremely tedious to get information about the experiment - things that you know for sure and could just write down, things that are relevant to answer your questions.