...This might be a bit long but please help.......
I had a NEW s5 mini off a shop on ebay,all new /sealed and unlocked,sold loads 99.8 feedback.

Had phone at 9.30 morning,put sim in and on charge,while on charge i set it up with my numbers and only 10apps as i don't use a lot,no games etc.
Bluetooth is on/wifi on/data on and power save on.
After doing this i left the phone charge,i was messing for a good 2 hours on charge.
At 3 phone was 100% so off charge and just linked up to Bluetooth car and put passwords in about 5 apps and a 20 mins call,this was timed close to test it,at 7 phone went to only 20% so i turned it off around 7.30 as i do.
This morning phone is dead flat,only way to turn on was charge up so lost the 20%
in 10 hours when off.

Is it an original battery or a cheap replacement?

It could be that it needs an initial 24 hour charge to establish full capacity.

Did you get a charger plug/cable with it?

Some cables or plugs don't give enough amps to charge phones properly.

Sainsbury's do a 2.1 amp plug/micro USB fast charge kit for £10 and my phone charges fully in 2 hours with 20-24 hours endurance even with heavy use.

Is it an original battery or a cheap replacement?

It could be that it needs an initial 24 hour charge to establish full capacity.

Did you get a charger plug/cable with it?

Some cables or plugs don't give enough amps to charge phones properly.

Sainsbury's do a 2.1 amp plug/micro USB fast charge kit for £10 and my phone charges fully in 2 hours with 20-24 hours endurance even with heavy use.

Its all genuine samsung stuff,lead,usb lead etc.

Just was seeing if messing on it as it was charging didn't give it the full charge when not messing.

Also whats this loosing 18% to 20% when off,i was going off apple my iphone 5 but i charged that the same time

and never turned it off and looked at it and still got 70% left,was trying to go away from apple,it's running more apps aswel.

On charge now and no messing and if this goes flat faster than my iphone 5 what's been on 24 hours it's bad.

Another thing to add,it takes 4 hours to charge from dead flat to 100%.

i know my S4 loses charge - its due to apps running in the background.

to close them down, keep the menu key pressed for a few seconds and let go, all previously used apps will apps and select the bin symbol to close them down. i do this throughout the day and it makes a big difference to my phone.

Use the left bottom button and close down any open Apps before you close down for the night.

To condition the battery always fully charge and then let it run out several times max out the battery. Also check for updates, i had one update that drained battery really quickly.

Doesn't sound right. Ive never had a problem like that with my previous and current Galaxy's.

I'd be a tad worried of a phone from eBay, they could install some background software to track your personal details,

worst case scenario there is a program constantly running in the background doing who knows what, draining power.

I would do a full system reset and make sure it is in factory condition.