# Probability that two measurements have the same true value

1. Oct 14, 2017

### Daniel Sellers

1. The problem statement, all variables and given/known data
I am in a lab course studying Brownian motion. I have gathered data for for movement in two dimensions.
I currently have a fairly large data set for ∆x and ∆y and have taken the mean and standard deviation of each.

The lab asks what is the probability that these two sets of measurements correspond to the same 'actual value'?

2. Relevant equations

We have been introduced to gaussian probability functions, but I'm not sure how to apply them to find a probability in this way.

3. The attempt at a solution
I have tried to apply numerical solutions to the above mentioned equations but I keep getting confused. Do I need to find the difference between the mean values and compare them to the standard deviation somehow? Could use a solid starting place.

2. Oct 14, 2017

### Staff: Mentor

You cannot find such a probability with your dataset. You can only find a likelihood: "If the true values are the same, how likely am I to observe a deviation as large or larger than I saw?" That's something you can calculate with the means and standard deviations you have. If that probability is high, then it can indicate that the two true values could be identical. If it is small, they are probably not identical.

3. Oct 14, 2017

### Ray Vickson

Are the successive values of $\Delta x$ and $\Delta y$ independent? Are the $\Delta x$ and $\Delta y$ independent of each other? Are the values of $\Delta x$ and $\Delta y$ normally-distributed? If the answer to ALL of these questions is yes, then you can proceed using standard statistical techniques, and if you give some more details we can perhaps supply a few more hints or website links.

4. Oct 14, 2017

### Daniel Sellers

The values should be independent, the histograms I made seem to show a normal distribution.
As for details, I have pretty much given them. I have a set of about 150 of each, their mean and standard deviation.
That's encouraging. What process might I use to state that probability? Do I need to add the probabilities that the expected value is within one standard deviation of each mean? Or compare the total discrepancy between the mean values to their deviations?

Clearly, statistics is not my strong suit. Thanks for your replies.

5. Oct 14, 2017

### Staff: Mentor

Compare the observed difference to the distribution of the difference (for the same true mean value = 0 difference).

6. Oct 14, 2017

### Ray Vickson

See http://www.milefoot.com/math/stat/ht-means.htm
It has all the formulas you need, plus an example exactly like yours, worked out in detail.

If the data for $\Delta x$ and $\Delta y$ are "separate", you can perform an "approximate" t-test (as shown in the link), but if the $\Delta x$ and $\Delta y$ are linked as measured pairs $(\Delta x, \Delta y)$ you can perform a "paired-sample" t-test without making any approximations. This is also done in the link. The paired-sample test is more reliable.

Last edited: Oct 15, 2017