# Astrophysics context : Introduction of a factor $\Delta\ell$ when summing equal distants $C_\ell$

I have the following formula which is the error on a $$C_\ell$$ :

$$\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)$$

where $$\Delta\ell$$ is the width of the multipoles bins used when computing the angular power spectra $$C_\ell$$.

I would like to understand why this factor appears. Normally, we can infer the expression of $$\sigma_(C_{\ell})$$ and get :

$$\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)}}\,C_{\ell}$$

That is to say, without having the $$\Delta\ell$$ factor under the root.

In my case, I have only a finite number of multipoles $$\ell$$ (60 exactly) in a range between 10 and 5000 and 60 corresponding $$C_\ell$$. From my tutor, to compute this standard-deviation, I have to take :

$$\Delta\ell=(4990/60)$$.

But I would like to understand why I have to use this factor when we have a finite number of equidistant multipoles with associated $$C_\ell$$ values.

For example, my tutor told me that, ideally, we would have a $$C_\ell$$ for each $$\ell=1..N$$, i.e with $$\ell=1,2,3,4,....N$$ with the formula :

$$\sigma(C_{\ell})^2=\frac{2}{(2 \ell+1)}\,C_{\ell}^{2}$$

But he mentionned that, like we have only a finite number of equidistant multipoles $$\ell$$, we are obliged to use this factor $$\Delta\ell$$ in our case (equation$$(1)$$).

My tutor told me it is like an average on the expression with multipole $$\ell$$, i.e on the factor $$\sqrt{\dfrac{2}{(2\ell+1)}}$$, but I didn't fully understand this meaning.

We could naively write :

for multipole interval $$[\ell, \ell+\Delta\ell]$$, we have :

$$C_b=\dfrac{1}{\Delta\ell)}\sum\limits_{\ell'=\ell}^{\ell'=\ell+\Delta\ell} C_{\ell}$$

and Variance follows (from equation $$(1)$$) :

$$\text{Var}(C_b)=\dfrac{1}{\Delta\ell^2}\sum\limits_{\ell'=\ell}^{\ell'=\ell+\Delta\ell}\,\text{Var}(C_{\ell'})$$

$$=\dfrac{1}{\Delta\ell^2}\sum\limits_{\ell'=\ell}^{\ell'=\ell+\Delta\ell}\,\dfrac{2}{2\ell'+1}(C_{\ell'})^2$$

$$\simeq \dfrac{1}{\Delta\ell^2}\,\dfrac{2}{2\ell_{mean}+1}\,(C_{\ell,mean})^2\,\Delta\ell$$

$$\simeq \dfrac{2}{(2\ell_{mean}+1)\,\Delta\ell}\,(C_{\ell,mean})^2$$

But I don't really undertand how $$\ell_{mean}$$ and $$C_{\ell,mean}$$ are obtained.

Classically, for a function $$f(x)$$, the mean is : $$ = \dfrac{1}{N}\sum\limits_{i=1}^{N}\,f(x_i)$$, isn't it ? How to make the link with above equations in order to introduce this factor $$\Delta\ell$$.

Any help would be really appreciated.