Battery pack provides 5V output using a dc/dc step up. battery itself can only provide 3.6 V. Thats why there is so much electronics in a battery pack.
That’s it!
These facts are forgotten very often.
Even the advertising do so, but this is normal! 
Individual batteries used in battery packs can be anywhere from 3250 - 3400 mAh (3.6-3.7 V).
13000 is an advertising number, the actual capacity can be higher than that.
Some cheaper battery packs also use 2600 mAh cells or 3000 mAh cells.
Check out this powercore 13000 and 26800 teardown
Essentially there are no repeated correct answers to this as different factors can come into the mix like heat, battery degradation etc…
…mine was based off the on the fly / rule of thumb option of dividing the phone battery size by the the PowerCores overall capacity…both of which are subject to change due to aging and external factors…
I think you are still not entirely correct. You are assuming that both the iphone’s battery and the 26800 operate at 3.8V, which is not true, since the battery pack delivers the output at 5V, that’s why the 5V is on my equation. The important part is not how much power does it hold but how it delivers it. I think you are close to the solution but not there yet.
No I’m correct. I do this professionally.
Irrespective of your device voltage, the actual battery is always operating at its nominal voltage. For iPhone that is 3.8V and for battery pack it is 3.6V.
By increasing the voltage to 5V, you cannot magically make extra energy.
Voltage x Ah = Wh (total energy) and this will be the bottom line. By increasing or decreasing Voltage and Current, you can extract this limited energy faster or slower but the amount of energy will only vary slightly due to higher or lower resistance.
5V is needed by many electronic components in circuit that’s why it is utilized.
Well I’m very proud of what I do and love my work. There is no shame in that
What exactly do you do? I assume you don’t sit around and calculate battery storages for a living .
Do you design batteries? Test them?
I’m a Ph.D. Battery scientist. I develop components for different energy storage devices including Li-ion, Lead-acid, Supercapacitors, Fuel-cells, primary lithium, alkaline batteries.
I also develop these devices for testing, test them and understand/publish the technology and patents for my company. I do both fundamental as well as applied research, meaning I make active materials (electrode coatings) for these devices as well as most of the components that goes in making these devices.
Testing batteries is a significant part of my work and I do have access to very sophisticated test equipments 
You are right.
There is always a lot of unqualyfied discussion about batteries. The output advertised is never what you really get.
We will let @kumar.sachin help us with batteries from now on .
Do you happen to know of any good videos that explain about these battery concepts?
Sure I’ll see what can be shared here.
Yes, I can get that information. My tester is at home through… I will be back middle of May
Long trip. Vacation??
Something like that… except this one is paid
There are plenty of videos on youtube that tries to explain li-ions but either they are done by amateurs who don’t know jack about it or by battery companies who don’t share enough.
There is this video by Prof Jay Whitacre (from Carnegie Mellon University) that is really good with a ton of technical information. I suggest going through first 30 min of it. It will provide information about the construction, working and testing of li-ions.
Before I watch it, I have a question. Will this explain about volts, Amps, Ah, Wh etc?
Or does it just explain how li-ion batteries work?
No it wont explain units. This video explains basics of li ions and then moves to advanced level. It is assumed that viewer know about these basics.
I can do it here
Ohm’s law: V = IR (V is voltage, I is current, R is resistance)
Power (Watt) = Voltage (V) x Current (I, or Amp)
Energy (Watt-hour) = Voltage (V) x Current (A) x hour (h)
With these basic equations you can roughly get an idea of energy stored in a device.
when li ion capacity is said in mAh unit, it don’t tell the voltage and is misleading since the voltage will tell you the energy that is stored in it. Also important is the discharge rate (A/sec) that is used. Higher discharge rate will result in lower energy (due to increased resistance).
If you have any specific question, I can answer it in simple terms.
