Variance of experimental data, compared to Theory

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Discussion Overview

The discussion revolves around methods for comparing experimental data to theoretical predictions, specifically in the context of confirming Benford's Law. Participants explore statistical approaches for assessing the fit between observed and expected distributions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes calculating the ratio of experimental readings to theoretical results and determining the standard deviation of these ratios to assess fit.
  • Another participant asks for clarification on whether the goal is to test the accuracy of a formula or to confirm a specific distribution of random values.
  • A later reply specifies the intention to confirm Benford's Law and mentions the expected distribution of leading digits.
  • One participant suggests using a Chi-Square test for goodness of fit to compare the expected distribution with the observed data.
  • Another participant reports issues with the Chi-Square test, noting discrepancies between observed and expected frequencies and questioning if low observed frequencies affect the test's validity.
  • A subsequent reply indicates that discrepancies in totals suggest a setup error, providing a formula for calculating expected frequencies based on Benford's Law.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate statistical methods to use, and there is no consensus on the resolution of issues related to the Chi-Square test and the setup of observed versus expected frequencies.

Contextual Notes

There are unresolved questions regarding the setup of the Chi-Square test, particularly concerning the total counts of observed and expected frequencies and the implications of low observed frequencies.

saad87
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I'm looking for a way to compare experimental and theoretical data and determining if it obeys a certain law.

In particular, I'm writing a computer program that does this and at the moment what I'm doing is, I'm calculating the ratio of each reading with the theoretical result and calculating the standard deviation of the various resultant ratios.

I compare each ratio with the mean to see if its within one standard deviation. Are there any better approached than this? I'm not really a mathematician, and am more of a programmer so any help would be much appreciated.
 
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saad87 said:
I'm looking for a way to compare experimental and theoretical data and determining if it obeys a certain law.

Welcome to the world of statistics!

Some more information about the problem is needed. In particular, are you looking to test a formula to see how accurate it is (e.g. whether the errors are "acceptable"), or are you looking to confirm is a sample of random values has a specific distribution?
 
bpet said:
Welcome to the world of statistics!

Some more information about the problem is needed. In particular, are you looking to test a formula to see how accurate it is (e.g. whether the errors are "acceptable"), or are you looking to confirm is a sample of random values has a specific distribution?

Basically I'm looking to confirm Benford's law in various files the user wishes to open.
 
If you are looking to confirm Benford's Law, then you have the expected distribution of numbers starting with 1,2,3,...,9, and you have the actual distribution given by your data.

I think a Chi-Square test for goodness of fit would be good here.
 
I just tried the Chi-square test, but the various online calculators basically tell me the total of my Observed and Expected frequences isn't the same. Is this because this test will fail if the observed freq. are too low than expected?

Thanks for all the help.
 
If your total expected does not equal your total observed, you have made some mistake in setting things up.

If you have [tex]N[/tex] (observed) numbers in total and the fraction predicted by Benford's Law to start with i is [tex]f_i[/tex], then the number expected to start with i is [tex]e_i = f_i \cdot N[/tex]. Since the [tex]f_i[/tex]'s sum to 1, the [tex]e_i[/tex]'s sum to [tex]N[/tex].
 

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