# I Calculate measurement uncertainty? I made one measurement...

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1. May 3, 2018

### olgerm

How to calculate measurement uncertainty of m. I understand I should use these formulas to calculate it if I had data of many measurements, but when have only measurement then it becomes undefined, because of 0/0 in standard deviation formula.
$u(m)=\sqrt{u_a^2(m)+u_b^2(m)}$
$u_a(m)=\frac{s(m)\cdot Student(p;n_m)}{\sqrt(n_m)}$
$s(m)=\sqrt{\frac{\sum_{i=1}^{n_m}((m_i-\frac{\sum_{i=1}^{n_m}(m_i)}{n_m})^2)}{n_m-1}}$
$u_b(m)=\frac{_\Delta X_{scale}\cdot Student(p;\infty)}{\sqrt{3}}$
• p is confidence level
• $n_m$ is number of measurements.
Assuming that $u_a(m)$ is zero if I have only one measurement data is probably false because then uncertainty with one measurement would be smaller or equal to uncertainty with more measurements.

Last edited: May 3, 2018
2. May 3, 2018

### ZapperZ

Staff Emeritus
This is very vague. What TYPE of measurement did you make?

For example, take a ruler, and measure the length of your pen/pencil/whatever. Make only ONE measurement. Now, what do you think is the "uncertainty" in that ONE single measurement?

You are focused on the statistical spread in repeated measurement of the same quantity. However, in ONE single measurement, the uncertainty lies in the method and instrument that were used in the measurement. This doesn't change no matter how many measurement you made.

Zz.

3. May 3, 2018

### olgerm

In that cases result should be almost same however many measurements I made because all results are almost the same. Statistical uncertainty (a-type uncertainty) would be approximately 0. In that case it is not practical to make many measurements, but statistics formulas should still work.

4. May 3, 2018

### ZapperZ

Staff Emeritus
Are you sure?

Take both the pen and ruler, and give it to 10 people. Ask them to give you the measurement as accurate as they can using that instrument. I extremely doubt that the length of the object will fall EXACTLY on one of the tick marks on the ruler. If the smallest tick mark is the 1 mm scale, do you think an estimate of the length in sub mm by everyone will be identical?

We do this often in intro general physics class to introduce students to uncertainties and statistical spread. I have never found any instant where everyone got identical measurement.

Zz.

5. May 3, 2018

### olgerm

If they get different results then $u_a>0$ and $u_{many\ measurements}=\sqrt{u_a^2+u_b^2}$.
If I measured this pen only once with the same ruler then $u_a=0$ and $u_{single\ measurement}=u_b$.

Therefore $u_{many\ measurements}>u_{single\ measurement}$, which does not make sense.

6. May 3, 2018

### ZapperZ

Staff Emeritus
But these are NOT the same uncertainty! There is NO statistical uncertainty in ONE single measurement. The only uncertainty that is there is the measurement uncertainty.

I have no idea what is going on here.

Zz.

7. May 3, 2018

### olgerm

What you mean these are not the same uncertainties? these both are uncertainty of pencil length on same confidence level p. In both cases with probability p $l_{measured}-u(l)<l_{real}<l_{measured}+u(l)$ .

with probability p:$l_{measured}-u(single\ measurement)<l_{real}<l_{measured}+u(single\ measurement)$
with probability p:$l_{measured}-u(many\ measurements)<l_{real}<l_{measured}+u(many\ measurements)$

At least this what I note by u. Maybe my formulas are wrong. If so, then please note out the error in them.

8. May 3, 2018

### gmax137

The uncertainty in the measurement must account for the instrument being used.

Last edited: May 3, 2018
9. May 3, 2018

### olgerm

it is charactarised by $_\Delta X_{scale}$, that is in my my equations. It is taken into account.

10. May 3, 2018

### gmax137

oops sorry I missed that.

edit -- I see what you're saying now. There must be something wrong with the root-sum-square at the top of your post, since more measurements should get you lower uncertainty. I will let someone who understands this better than I do answer the question.

Last edited: May 3, 2018
11. May 3, 2018

### olgerm

I think it is untrue because then would making more measurements increase uncertainty.

12. May 3, 2018

### gmax137

What exactly does $u_a(m)$ represent?

13. May 4, 2018

### olgerm

a-type uncertainty.

14. May 6, 2018

### olgerm

I got an idea to count standard deviation of single measurement to be $s(m)=m\cdot \sqrt{2}$, beacause it could not be larger if you made more than one measurement and measured values could not be smaller than 0.
Do you think, it is correct?.