# Standard Deviation vs. Standard Deviation of the Mean (Standard Error)

#### tacom

##### Involved In Discussions
Suppose that we measure density d=m/v
m=mass
V= Volume
If we want to calculate the combined measurement uncertainty of the mass ?m?
There are three uncertainty sources, the repeatabilty, the readability and the contribution due to the uncertainty in the calibration function.
My question is related to repeatability ?s? of the balance.
İf I make ?n?(lets say n=10) measurement to determine repeatability of balance .
When combining uncertainty sources to calculate the combined measurement uncertainty of the mass ?m?
Should I use Standard deviation (S) or Standard deviation of the mean (Standard error)(S/ ^0,5 as repeatability component
Note: in routine analysis test sample is weighted only once

#### dgriffith

##### Quite Involved in Discussions
Suppose that we measure density d=m/v
m=mass
V= Volume
If we want to calculate the combined measurement uncertainty of the mass “m”
There are three uncertainty sources, the repeatabilty, the readability and the contribution due to the uncertainty in the calibration function.
My question is related to repeatability “s” of the balance.
İf I make “n”(lets say n=10) measurement to determine repeatability of balance .
When combining uncertainty sources to calculate the combined measurement uncertainty of the mass “m”
Should I use Standard deviation (S) or Standard deviation of the mean (Standard error)(S/ ^0,5 as repeatability component
Note: in routine analysis test sample is weighted only once

I was taught, and use in my uncertainty budgets, that if you are reporting the average(mean) of a sample size x, you also report the std dev of the mean.
There is a school of thought that says you may be underestimating the uncertainty, and should use the std dev of the sample size x instead.
I don't 'know', so I chose the former, and it seams to work for us.

But if you are weighing ten samples once, and not weighing 1 sample ten times, I might use the std dev, not sampling mean.

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
Suppose that we measure density d=m/v
m=mass
V= Volume
If we want to calculate the combined measurement uncertainty of the mass ?m?
There are three uncertainty sources, the repeatabilty, the readability and the contribution due to the uncertainty in the calibration function.
My question is related to repeatability ?s? of the balance.
İf I make ?n?(lets say n=10) measurement to determine repeatability of balance .
When combining uncertainty sources to calculate the combined measurement uncertainty of the mass ?m?
Should I use Standard deviation (S) or Standard deviation of the mean (Standard error)(S/ ^0,5 as repeatability component
Note: in routine analysis test sample is weighted only once

If you want to know the uncertainty of the weight of a SINGLE mass item, then you stick to individual standard deviation. You only divide by the square root of n if you are truly interested in how good you are estimating the AVERAGE (MEAN) measurement.

#### tacom

##### Involved In Discussions
Thanks steve and dgriffith
Although we weigh same object (ethalon for instance) 10 times in order to determine repeatability component of the measurement uncertainty of mass measurement for future use ;in laboratory practice samples or chemicals is weighted only once ; so I must use Standard deviation.
İnstead of Standard error. Am Iright?
TACOM

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
Yes, if you are estimating the error of weighing an object once.

#### dgriffith

##### Quite Involved in Discussions
Yes, if you are estimating the error of weighing an object once.
:mg:
I think I know what you meant, because I know you know that there is no standard deviation if you only have one reading.
What I think we have here, and correct me if I'm wrong, is a single object weighed 10 times and reporting the average (1 event), then use the std. dev.; if that same object will be weighed repeatedly (multiple events on same object), and each weighing is an average of 10 samples, then report the std. dev. of the mean.

#### tacom

##### Involved In Discussions
Suppose According to test method I must weigh approximately 10,5 g test sample to make a test. in laboratory practice sample (10,5g) is weighted only once[
Now, I want to calculate the combined uncertainty of 10,5 g. EURACHEM / CITAC Guide CG 4 says mass measurement has three uncertainty sources. The repeatabilty, the readability and the contribution due to the uncertainty in the calibration function.

So I want to find repeatability of 10,5 g. To fulfill this requirement I weigh 10 g (nearest to 10,5g) ethalon or any object( but not test sample ) ?n? times (usually n=10) . As a result I get Standard deviation as ?S? or Standard error ?S/ ^0,5?. This Standard deviation /error wil be used as a repeatability component of the 10,5 g (or 10.3g; 10.9g ) in the future use as a constant.
That is, when ever I conduct this test and weight approximately 10,5 g sample once and want to calculate combined uncertainty I will use this value as a repeatability component
I think we should use Standard deviation ?S? since the test sample taken is weighed only once

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
:mg:
I think I know what you meant, because I know you know that there is no standard deviation if you only have one reading.
What I think we have here, and correct me if I'm wrong, is a single object weighed 10 times and reporting the average (1 event), then use the std. dev.; if that same object will be weighed repeatedly (multiple events on same object), and each weighing is an average of 10 samples, then report the std. dev. of the mean.
Yes, I take multiple readings of a single test item to see what the estimate of the standard deviation is for the weighing device. I then want to predict what is the standard deviation of the estimate of the weight of the next item I weigh (which I will only weigh once), which won't necessarily have the same weight as the test object. I apply the standard deviation from the test as the estimate of the standard deviation for the weight of this item.

Note - I am making the assumption that the error for the weighing device is constant over the range of weights the weighing device is able to weigh. That may or may not be a good assumption.

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