Hi ,

These days I use “Pearson’s Chi-square” to calculate the “confidence levels”. And I find one thing which confuses me.

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There are three examples.

## x^2=\sum\frac{(O_i-E_i)^2}{E_i}

1>
E_i: 0.04 0.05 0.06 0.07 0.08
O_i: 0.08 0.09 0.01 0.07 0.02
x^2=0.0016/0.04+0.0016/0.05+0.0025/0.06+0+0.0036/0.08=[color=red]0.1587[/color]

2>
E_i: 4 5 6 7 8
O_i: 8 9 1 7 2
x^2=16/4+16/5+25/6+0+36/8=[color=red]15.87[/color]

# 3> E_i: 400 500 600 700 800 O_i: 800 900 100 700 200 x^2=160000/400+160000/500+250000/600+360000/800=[color=red]1587[/color]

Do they show that if all the bin Contents are small then the chi2 is small, and if all the bin contents are big then the chi2 is big too?

If yes, how should I handle the cases that all the bin contents are too small or too big?

If no, what is wrong?

Thanks a lot.

HI,
I don’t see anything wrong with your results.
If you expect 400 and you observe 800 you have a huge discrepancy compared to when you observe 8 and you expect 4.

If instead you have applied to the events a scaling factor you can’t use the formula like you wrote, but you must use:
x^2=\sum\frac{(O_i-E_i)^2}{\sigma^{2}_i}

where sigma is the standard deviation of your expected events. Therefore if you scale by a factor of 100, you must scale also sigma by 100, and you will obtained the same chi2 as in the case of unscaled events.

Best Regards

Lorenzo