Statistical Analysis Question: How many measurements to reduce uncertainty?

In summary, to reduce the uncertainty in the mean to +/- 0.003 m, we need to increase the number of measurements without changing the average. Mathematically, this can be achieved by making 6423 measurements in total.
  • #1
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Homework Statement


a. Mike makes six independent measurements of the diameter D of a leap year detector that made its way into the lab, obtaining D = 4.64, 4.78, 4.82, 4.68, 4.80, and 4.95m. What would result would he report in a lab writeup?
b. Alice does the same experiment as Mike, but makes only one measurement. Based on Mike's results, what would you expect for the uncertainty in Alice's single measurement of D?
c. How many measurements would have to be made altogether to reduce the uncertainty in the mean to +/- 0.003 m?

Homework Equations


D = [tex]\Sigma[/tex]x[tex]_{i}[/tex]/N
[tex]\sigma[/tex]D = [tex]\delta[/tex]D
[tex]\sigma[/tex]D=[tex]\sqrt{(1/(N-1))*\Sigma(xi-\overline{x})^{2}}[/tex]

The Attempt at a Solution



So, my real question doesn't come until part C, but I figure it definitely wouldn't hurt to make sure I have the first two parts right to base my work upon:

A. D = [tex]\Sigma[/tex]x[tex]_{i}[/tex]/N = (4.64+4.78+4.82+4.68+4.80+4.95)/6 = 4.78 m

B. [tex]\sigma[/tex]D=[tex]\sqrt{(1/N-1)*\Sigma(xi-\overline{x})^{2}}[/tex] = [tex]\sqrt{(1/6-1)*((4.64-4.78)^{2}+(4.78-4.78)^{2}+(4.82-4.78)^{2}+(4.68-4.78)^{2}+(4.80-4.78)^{2}+(4.95-4.78)^{2})}[/tex]
= 0.11m

C. So here's where my question is. I assume that we would use the same formula as in B, but how can we know what the new x[tex]_{i}[/tex] and [tex]\overline{x}[/tex] would be? Do you use a different formula to determine how to decrease the uncertainty? I know that each measurement induces a sqrt2 decrease in uncertainty, is this how you calculate it? Thanks so much!
 
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  • #2

Your solutions for parts A and B are correct. For part C, you are on the right track. To decrease the uncertainty to +/- 0.003 m, we need to decrease the standard deviation (sigma) to 0.003 m. This can be achieved by increasing the number of measurements (N) and keeping the same sum of the measurements (Sigma(xi)). In other words, we need to make more measurements without changing the average (x̄).

To calculate the new number of measurements needed, we can rearrange the formula for sigma (standard deviation) to solve for N:

N = (Sigma(xi) / sigma)^2 + 1

Substituting the values from part A and B, we get:

N = (4.78 / 0.003)^2 + 1 = 6422.67

Therefore, to reduce the uncertainty to +/- 0.003 m, we would need to make 6423 measurements in total.

I hope this helps and let me know if you have any further questions. Best of luck with your research!
 
  • #3


Based on the given data, Mike's result for the diameter D would be 4.78m, as calculated in part A. For Alice's single measurement, we can expect her uncertainty to be similar to Mike's, around 0.11m as calculated in part B.

To reduce the uncertainty in the mean to +/- 0.003m, we can use the formula for standard deviation, \sigmaD=\sqrt{(1/(N-1))*\Sigma(xi-\overline{x})^{2}}, where N is the total number of measurements. We can rearrange this formula to solve for N, which would give us N = (1/\sigmaD^2)+1. Plugging in the given uncertainty of 0.003m, we get N = (1/(0.003m)^2)+1 = 111,112 measurements. This means that approximately 111,112 measurements would need to be made altogether to reduce the uncertainty in the mean to +/- 0.003m. However, this number may vary depending on the precision and accuracy of the equipment and the experimental setup.
 

FAQ: Statistical Analysis Question: How many measurements to reduce uncertainty?

1. How many measurements should be taken in order to reduce uncertainty?

The number of measurements needed to reduce uncertainty depends on various factors such as the level of uncertainty, the type of data being analyzed, and the desired level of accuracy. In general, a larger number of measurements will result in a lower uncertainty, but the specific number needed will vary from situation to situation.

2. How does the sample size affect the uncertainty in statistical analysis?

The sample size, or the number of observations in a data set, has a direct impact on the uncertainty in statistical analysis. Generally, a larger sample size will result in a lower uncertainty as it allows for a more representative and accurate estimation of the true population parameters.

3. Is there a specific formula for determining the number of measurements needed to reduce uncertainty?

There is no one specific formula for determining the number of measurements needed to reduce uncertainty. However, there are various statistical methods and techniques that can be used to estimate the appropriate sample size for a given level of uncertainty and desired level of accuracy.

4. Can statistical analysis techniques be used to reduce uncertainty in all types of data?

Statistical analysis can be applied to a wide range of data types, including numerical, categorical, and continuous data. However, the specific techniques and methods used may vary depending on the type of data and the research question being addressed.

5. How can the uncertainty in statistical analysis be communicated effectively?

Communicating uncertainty in statistical analysis is a crucial aspect of scientific research. One way to effectively communicate uncertainty is by reporting confidence intervals, which provide a range of values that are likely to contain the true population parameter. Other methods include visual representations such as error bars or probability plots.

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