Need some advice on uncertainties

[lemonysage]

Hi, I have been given a random set of data. 10 numbers.
121
123
125
100
123
123
101
146
122
140

The question says.
1. Calculate the average of the data... No problem I can do that, my answer is 122.4
2. Determine the uncertainty for the average value. < This is the part that blows my mind. There are literally hundreds of methods for this and I am not sure what to use... My working out so far is as follows:

Using the information in this link.
http://virgo-physics.sas.upenn.edu/uglabs/lab_manual/Error_Analysis.pdf

It says.

The uncertainty in the each measurement is (Max-min)/2
The uncertainty in the the average = uncertainty in each measurement/(sqrt(N))
which is 23/(sqrt10) = 7.27 or 7.

answer to the whole question = 122 +/- 7

Am I thinking about this correctly? or missing it completely?

 

[jtbell]

There are literally hundreds of methods for this and I am not sure what to use...

If this is for a class assignment, can you ask your instructor which method he/she prefers?

I usually use the method on page 6 of the document that you linked (the one for "large data sets") even if the data set is small, just so I have only one set of formulas to remember. However, I think the method you used (from page 4) is probably "good enough" for small data sets, and it's certainly simpler to calculate.

 

[lemonysage]

If this is for a class assignment, can you ask your instructor which method he/she prefers?

I usually use the method on page 6 of the document that you linked (the one for "large data sets") even if the data set is small, just so I have only one set of formulas to remember. However, I think the method you used (from page 4) is probably "good enough" for small data sets, and it's certainly simpler to calculate.

Thank you for replying, my first question to the instructor was can you please provide a bit more information? and they told me that this was all that would be given.

Would you consider those numbers consistent? because one variation would be to select 0.5 as the uncertainty for the single measurements. what do you think?

 

[haruspex]

I would have expected some multiple, like 1, of the standard deviation. But 7 is only half of 1 sdev for these numbers, so 7 seems rather low to me.

 

[lemonysage]

I would have expected some multiple, like 1, of the standard deviation. But 7 is only half of 1 sdev for these numbers, so 7 seems rather low to me.

Do you think the number should be bigger?

 

[haruspex]

Do you think the number should be bigger?

I looked at the reference you posted, and you seem to have followed that correctly. Coming from the theoretical side, I find the idea of using the range of the numbers this way rather unsatisfactory. There's all that information of how the other numbers are scattered in between that's hardly being used (only for the mean). The values of the max and min observations have quite a broad distribution, so relying on them to compute the uncertainty really doesn't look good. But I deduce from your link that this is common practice (yuk).
Anyway, I compared the result with (what is to me) the more standard approach of calculating the standard deviation. [There are actually two different sdevs that can be computed from a set of numbers. One represents the scatter of the numbers, while the other is more geared to figuring out how the mean of the numbers is likely to differ from the 'real' answer. It's a relatively minor difference, only a factor of √0.9 in this case.]
From these numbers, using the second of the above, I get 14. For a Normal distribution, one sdev from the mean (+/-) encompasses 68% of the probability. I.e. there's a 68% chance that the real value is within +/-14 of the mean of this sample. The +/-7 range only gives a 40% chance of including the real value.
So it depends what you think is meant by 'uncertainty'.

 

[lemonysage]

I looked at the reference you posted, and you seem to have followed that correctly. Coming from the theoretical side, I find the idea of using the range of the numbers this way rather unsatisfactory. There's all that information of how the other numbers are scattered in between that's hardly being used (only for the mean). The values of the max and min observations have quite a broad distribution, so relying on them to compute the uncertainty really doesn't look good. But I deduce from your link that this is common practice (yuk).
Anyway, I compared the result with (what is to me) the more standard approach of calculating the standard deviation. [There are actually two different sdevs that can be computed from a set of numbers. One represents the scatter of the numbers, while the other is more geared to figuring out how the mean of the numbers is likely to differ from the 'real' answer. It's a relatively minor difference, only a factor of √0.9 in this case.]
From these numbers, using the second of the above, I get 14. For a Normal distribution, one sdev from the mean (+/-) encompasses 68% of the probability. I.e. there's a 68% chance that the real value is within +/-14 of the mean of this sample. The +/-7 range only gives a 40% chance of including the real value.
So it depends what you think is meant by 'uncertainty'.

Thanks for your detailed reply, I believe you have hit the nail on the head, The question is so ambiguous. I asked for more information; I was told that enough was given...