How to calculate Efficiency error?

Newbie
1

Hi, everyone

I use one detector like Scintillator, Aerogel Cherenkov counter.

When I get the total number of events N and the observed number of passed events k with cut condition, I can define efficiency.

efficiency=k/N

But I don’t know how to calculate Efficiency error.

I think TEfficiency method can calculate this, but I found method using two histogram.

If I have three data set (cut, N, k) instead of histgram.

How do I use TEfficiency and get efficincy error?

Please help me.

2

Hi,

You can call directly the TEfficiency static functions such as ClopperPearson for the frequentist method, where you pass the number of passed (k) and the total number (N) and it returns the upper or lower level of the interval.
See https://root.cern.ch/doc/master/classTEfficiency.html#ae80c3189bac22b7ad15f57a1476ef75b

For the description of the different method see the general TEfficiency documentation

Lorenzo

3

Thank you.

Your answer helped me.
I have two questions about link you sent.

First
I tried TEfficinency::ClopperPearson as fallows

cout << TEfficiency::ClopperPearson(100, 100,  95, true) << endl;

This return 1.
I think “true” option means upper bounary and efficiency=100/100=1.
So, This means the upper limit (error?) of efficiency is 1+1=2 ?.
I think upper boundary is zero if efficiency =0. I can’t understand this result.

Second
I can’t understand “P(x ≧ passed; total)=(1-level)/2” in document you sent.
Where does “(1-level)/2” term come from?

This is my lack of knowledge about statistics.
Please teach me if possible…

4

Hi,

The function it returns the upper value of the efficiency. In case the central value ( npassed/ntotal) is very close to 1, the upper interval value is then 1 as expected.
The upper/lower error value (computed at 68% confidence level ) are as following:

double efficiency = double(k)/N; 
double lower_error = efficiency - TEfficiency::ClopperPearson(N, k,  0.683,  false);
double upper_error = TEfficiency::ClopperPearson(N, k,  0.683,  true) - efficiency;

For the second question, P(x ≧ passed; total)=(1-level)/2 comes from the definition of the upper interval. The probability to get a value outside the confidence interval is 1.-condidence_level.
Then if you consider only the upper interval and by definition you want equal probability on the lower/upper side, then you divide by a factor 2.

Lorenzo

2 Likes
5

Thank you for your answer.
I forgot to change the third argument. I treated it as percentage notation.
And I misunderstood the meaning of upper/lower boundary.

And thank you for your second answer.
I study your answer as a reference.

Thank you very much.