Find number of trials to be done to get specified uncertainty?

  • Thread starter sloane729
  • Start date
  • Tags
    Uncertainty
In summary: If the systematic error is different for each point, you would need to treat the systematic error separately in your calculation of the mean and standard deviation. In summary, to calculate the number of trials needed to achieve a desired uncertainty for a timed event, one must make N measurements of the quantity T and use the formula δT = (1/√N) times the error from a single measurement. If systematic errors are present, they must be treated separately in the calculation of mean and standard deviation.
  • #1
sloane729
7
0

Homework Statement


This has been bothering me for quite a while. I'm trying to work out how many measurements I will need to make to get my uncertainty under a predetermined value. If say I want the a fractional uncertainty [tex]\frac{\delta T}{T}[/tex] to be equal to or under some value for some timed event, how would I calculate the number of trials I will need to make to get that uncertainty.

Homework Equations


the standard deviation of some quantity to be measured T is
[tex] s = \left( \frac{1}{N-1}\sum_i (T_i - \overbar{T})^2 \right)^{1/2} [/tex]
then
[tex] \delta T = u = \frac{s}{\sqrt{N}}[/tex]

The Attempt at a Solution


Since to calculate the uncertainty [tex]\delta T[/tex] I would need first find the average of all values of [tex]T_i[/tex] then find the standard deviation divided by the square root of the number of trials which is equal to[tex]\delta T[/tex]. But the number of trials is what I need to find but I can't know it without first finding the average value which is not possible because I need the number of trials etc. It seems like a round about problem if I'm not mistaken
 
Physics news on Phys.org
  • #2
If you make N measurements of T then your error reduces to δT = (1/√N) times the error you get from making just 1 measurement.

So the answer to your question depends on how small you want the final δT to be. The error will continue to drop as 1/√N from your first measurement error:

(δT)N = (δT)1/√N.
 
  • #3
so s (from above) is the error from making just a single measurement not the standard deviation of all the T_i's?
 
  • #4
s is the estimated* standard deviation of a single measurement.
*as you use your own data to estimate this deviation.

But the number of trials is what I need to find but I can't know it without first finding the average value which is not possible because I need the number of trials etc. It seems like a round about problem if I'm not mistaken
If you know nothing about your measurements, you cannot determine the required number of measurements, right. If you have some way to estimate the mean and standard deviation (because you already took some data, or took similar data in the past, or have some theory prediction or whatever), you can use this formula with those estimates for mean and standard deviation.
 
  • #5
thanks for the help. I have one final question: (δT)1 is the random error of a single measurement but what if I wanted to include systematic error? Would I need to take the square of both errors and then take the square root of the sum since the above equations assumes only statistical fluctuations of T.
 
  • #6
Systematic errors which are the same for all data points? In that case, the method you described works both for the uncertainty of a single point and the uncertainty of the mean.
 

1. How do you calculate the number of trials needed to achieve a specific uncertainty?

The number of trials needed can be calculated using the formula: N = (Zα/2 * σ / E)2, where N is the required number of trials, Zα/2 is the critical value for a confidence level α, σ is the standard deviation of the sample, and E is the desired uncertainty.

2. What is the purpose of determining the number of trials in an experiment?

The number of trials is used to ensure that the results of an experiment are statistically significant and reliable. It helps to minimize the effects of random variations and increases the accuracy of the results.

3. How does the desired uncertainty affect the number of trials needed?

The desired uncertainty has a direct impact on the number of trials needed. A smaller desired uncertainty will require a larger number of trials, while a larger desired uncertainty will require a smaller number of trials to achieve a certain level of accuracy.

4. What are some factors that can affect the required number of trials?

The required number of trials can be affected by factors such as the variability of the data, the level of confidence desired, and the size of the effect being measured. It can also be affected by the complexity of the experiment and the accuracy of the measuring equipment.

5. Can the number of trials be reduced by increasing the sample size?

Yes, increasing the sample size can help reduce the number of trials needed to achieve a specific uncertainty. This is because a larger sample size provides more data points, which can lead to a more accurate estimate of the true value and reduce the effects of random variations.

Similar threads

  • Introductory Physics Homework Help
Replies
8
Views
853
  • Introductory Physics Homework Help
Replies
6
Views
891
  • Introductory Physics Homework Help
Replies
5
Views
824
  • Introductory Physics Homework Help
Replies
15
Views
919
  • Introductory Physics Homework Help
Replies
3
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
757
  • Introductory Physics Homework Help
Replies
2
Views
795
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
1K
Back
Top