Probability that A is smaller than B?

  • Thread starter Thread starter wavingerwin
  • Start date Start date
  • Tags Tags
    Probability
Click For Summary
SUMMARY

This discussion focuses on determining the probability that the average of dataset A is smaller than that of dataset B without assuming a normal distribution. The participants confirm that if A and B are independent identically distributed (iid) random variables, then the probability P[avg(A) ≤ avg(B)] is at least 50%. They also discuss the applicability of non-parametric tests, specifically the Wilcoxon signed-rank test, for analyzing such datasets when normality cannot be assumed.

PREREQUISITES
  • Understanding of independent identically distributed (iid) random variables
  • Knowledge of ANOVA (Analysis of Variance)
  • Familiarity with non-parametric tests, particularly the Wilcoxon signed-rank test
  • Basic statistical concepts regarding probability distributions
NEXT STEPS
  • Research the application of the Wilcoxon signed-rank test for comparing two datasets
  • Explore the implications of iid assumptions in statistical analysis
  • Learn about non-parametric methods for hypothesis testing
  • Study the fundamentals of ANOVA and its limitations in non-normal distributions
USEFUL FOR

Statisticians, data analysts, researchers, and anyone interested in understanding probability comparisons between non-normally distributed datasets.

wavingerwin
Messages
93
Reaction score
0
Given two data sets A and B, we can, say, conduct ANOVA to see if the average is statistically different.

Is there a way to determine what is the probability that A is smaller than B?

Let's say that we can NOT assume anything about A and B e.g. if they follow a normal distribution.
 
Physics news on Phys.org
wavingerwin said:
Given two data sets A and B, we can, say, conduct ANOVA to see if the average is statistically different.

Is there a way to determine what is the probability that A is smaller than B?

Let's say that we can NOT assume anything about A and B e.g. if they follow a normal distribution.

? Do you mean the probability that the average of A is smaller than the average of B? If we know nothing at all, then we can conclude nothing at all.
 
Hornbein said:
? Do you mean the probability that the average of A is smaller than the average of B? If we know nothing at all, then we can conclude nothing at all.

Yes, something to that effect. Is it possible to say "A is smaller than B X% of the time" ?
 
wavingerwin said:
Yes, something to that effect. Is it possible to say "A is smaller than B X% of the time" ?
If we know that A and B are iid (independent identically distributed) random variables then we can say that avg of A is smaller or equal to avg of B at least 50% of the time.

Proof: Since they are iid, P[ avg(A)<= avg(B) ] = P[avg(B) <= avg(A)].

P[ avg(A)<= avg(B) ] + P[avg(B) <= avg(A)] >=1

P[ avg(A)<= avg(B) ] + P[ avg(A)<= avg(B) ] >=1

2P[ avg(A)<= avg(B) ] >=1

P[ avg(A)<= avg(B) ] >=1/2
 
wavingerwin said:
Let's say that we can NOT assume anything about A and B e.g. if they follow a normal distribution.
You can do non-parametric analysis, such as the Wilcox test.
 
I think Wilcoxon signed rank requires the samples be from a single population. Does that fit what we are looking at here?
 
jim mcnamara said:
I think Wilcoxon signed rank requires the samples be from a single population. Does that fit what we are looking at here?
I linked to wrong test. Here's the two sample version.
 
  • Like
Likes jim mcnamara
Much better - thank you. I thought maybe I had lost my last remaining brain cell.
 

Similar threads

High School Probability a randomly chosen natural number, which is not smaller than 𝑥, is greater than x?
  • · Replies 15 ·
Replies
15
Views
2K
High School Probability of hitting a target
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Adding random numbers: Tolerance analysis
  • · Replies 11 ·
Replies
11
Views
3K
Graduate Calculation of average damage using analytical methods
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Help me understand skewness in QQ-plots please
  • · Replies 1 ·
Replies
1
Views
2K
High School Relative And Absolute Probability (Probability Of Picking A Red Ball)
  • · Replies 8 ·
Replies
8
Views
3K
High School Why did it suddenly become subtractive? (Example of Bayes’ Theorem)
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Runners in a race, probability paradox
  • · Replies 3 ·
Replies
3
Views
2K
MHB Probability of misprints on each page
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Airplane crash probability fallacy
  • · Replies 20 ·
Replies
20
Views
4K