There are many different formulae and calculations and interesting data in Tibia. For example how much experience you need for a level, how much mana you need for a magic level, how long time it takes to advance in different skills, how long time a day in Tibia is counted in real life time, etc etc.
Player Statistics
Experience
For the amount of experience needed to reach a certain level:
$ \frac{50lvl^{3}}{3}  {100lvl^{2}} + \frac{850lvl}{3}  200 $
$ = \frac{50\left(lvl1\right)^{3}  150\left(lvl1\right)^{2} + 400\left(lvl1\right)}{3} $
For the amount of experience needed to reach the next level:
$ 50lvl^{2}  150lvl + 200 $
 For details about the Level Formula, visit this page.
Magic Level
 Mana required to advance to next magic level:
$ 1600 \cdot b^{mlvl} $
 Total amount of mana spent at a certain magic level
$ 1600 \frac{b^{mlvl}1}{b1} $
Mana Training Time
This formula enable you to calculate how much time is needed to go a Magic Level up.
$ y = a \cdot b^x \cdot z $
$ y $ is the time needed to go a Magic Level up, in hours;
$ x $ is the current Magic Level;
$ z $ is the percent you have to go to the next Magic Level in decimal form (i.e. $ 47\% = 0.47 $).
$ a $ is the Vocation factor, depending on Mana regeneration:
$ a = \frac{1600}{\left(\text{MP per second}\right) \cdot 3600} = \frac{4}{9} \cdot \frac{1}{\left(\text{MP per second}\right)} $
 Mages: $ a = \frac{2}{3} $
 promoted Mages: $ a = \frac{4}{9} $
 Knights and Elite Knights: $ a = \frac{4}{3} $
 Paladins: $ a = \frac{8}{9} $
 Royal Paladins: $ a = \frac{2}{3} $
Melee Skill Levels
 Number of blood hits required to advance to the next skill level:
$ 50 y^{skill10} $
Magic Power
$ \max\left(1,\frac{lvl+4 \cdot mlvl}{100}\right) $
Usually displayed as percent.
Fishing
Number of tries required to advance to next fishing level:
$ 20 \cdot 1.1^{skill10} $
Number of tries from fishing level 10 to x:
$ 200 \cdot 1.1^{x10}  200 $
Hitpoints, Mana and Capacity
Vocation  Hitpoints  Mana  Capacity  

Per level  Total  Per level  Total  Per level  Total  
Paladins  $ 10 $  $ 5\left(2lvl + 21\right) $  $ 15 $  $ 5\left(3lvl  6\right) $  $ 20 $  $ 10\left(2lvl + 31\right) $ 
Knights  $ 15 $  $ 5\left(3lvl + 13\right) $  $ 5 $  $ 5\left(lvl + 10\right) $  $ 25 $  $ 5\left(5lvl + 54\right) $ 
Sorcerers and Druids  $ 5 $  $ 5\left(lvl + 29\right) $  $ 30 $  $ 5\left(6lvl  30\right) $  $ 10 $  $ 10\left(lvl + 39\right) $ 
Rookies  $ 5 $  $ 5\left(lvl + 10\right) $ 
What is worth noting, Rookies gain the lowest possible values per level among all vocations.
$ HP + MP + Cap = 5\left(9lvl + 77\right) $
Sum of HP, MP and Cap does not depend on vocation, providing balance between them (assuming, of course, they are each desired equally, which is often not the case).
Speed
 For details about the equipment increasing speed, visit this page.
Note that spells like Haste and Strong Haste increase your base speed without items and mount, you have to add the speed of the extra items and mount only after calculate your base speed with the spells.
Your speed helps determine how fast you walk ingame. The base speed for a level $ 1 $ player is $ 220 $.
The type of surface you walk on reduces your speed (i.e. you walk faster on pavement than you do on grass or mud). Below are the formulae for determining your speed in different states.
Unmodified
$ 220 + 2\left(lvl  1\right) $ This determines your Unmodified Speed, or "BaseSpeed".
Walking Time
The number of seconds it takes for a player to walk 1 square
$ \,=\frac{Tile Walking Time}{Player Walk Speed} $
Walking Time is entirely dependent on the type of Tiles you are walking on. Moving on a diagonal takes 2 times as long.
Haste
$ BaseSpeed \cdot 1.3  12 $
This determines the difference in your speed after casting Haste. It replaces any other haste spell.
Strong Haste
$ BaseSpeed \cdot 1.7  28 $
This determines the difference in your speed after casting Strong Haste. It replaces any other haste spell.
Swift Foot
$ BaseSpeed \cdot 1.8  32 $
It replaces any other haste spell.
Charge
$ BaseSpeed \cdot 1.9  36 $
It replaces any other haste spell.
Prey System
Timer Deductions
All bonuses have specific triggers upon which time is deducted from the remaining time of a prey bonus. Much like stamina, time is deducted based on the recent activity of the player. The amount of time deducted is equal to:
 $ D(currentTime,\ lastTriggerTime) = min(currentTime\ \ lastTriggerTime,\ 2\ minutes) $
Where:
currentTime
is the current system time. This is used to compute time relative to the last trigger time.lastTriggerTime
is the system time of the last relevant trigger event. The lists of relevant trigger events for each bonus type are listed below.
The triggers that are used to deduct time depend on the specific bonus. For example, when the user takes damage, the timer for damage reduction will be affected, but not the timer for experience gain. These triggers act on any event of the specified type, not just ones that would be affected. For example if an increased loot bonus is active for killing Rats, killing a Spider will affect the timer even though Spiders do not provide bonus loot. The triggers for each type of bonus are as follows:
Damage Reduction: time is deducted when:
 the user takes damage, whether PvP or PvE.
 the user gains experience from a slain creature.
 [NOT TESTED]: gaining experience from a quest or other noncombat event.
Damage Increase: time is deducted when:
 the user gains experience from a slain creature.
Loot Increase: time is deducted when:
 the user gains experience from a slain creature, irrespective of whether or not that creature has a corpse.
 [NOT TESTED]: gaining experience from a quest or other noncombat event.
Bonus Experience: time is deducted when:
 the user gains experience from a slain creature.
 [NOT TESTED]: gaining experience from a quest or other noncombat event.
Value and Effect
When bonuses are determined, two values are generated: the bonus type (damage increase, damage reduction, etc.) and bonus value. The effect of a bonus is a function of its value according to the following functions.
Damage Increase:
 $ E(value) = (5 + 2 \cdot value)% $
Damage Reduction:
 $ E(value) = (10 + 2 \cdot value)% $
Bonus Experience:
 $ E(value) = (10 + 3 \cdot value)% $
Improved Loot:
 $ E(value) = (10 + 3 \cdot value)% $
Bonus  Minimum  Step 

Damage Increase (%)  5  2 
Damage Reduction (%)  10  2 
Bonus Experience (%)  10  3 
Improved Loot (%)  10  3 
Combining these formulas provides a generalized formula:
 $ E(value) = (minimum + step \cdot value)% $
Where:

minimum
is the theoretical effect at value = 0 (equal to $ E(1)  step\, $). 
step
is the effect's step. For each value, the effect increasesstep
percent.
Tibian Environment
Time
 2.5 seconds = 1 Tibian minute
 30 minutes = 12 Tibian hours
 1 hour = 24 Tibian hours
House Rent
The formula for determining house rent is:
 $ rent = 100\left(\max\left(b, 1\right)  1\right) + r \cdot s + f $
Where:
 $ f $ is furnishing value. Most houses do not have one ($ f = 0 $), but some houses such as House of Recreation have one (in this case $ f = 5000 $).
 $ b $ is the number of beds.
 $ r $ is rate, the gp/sqm rate. Usually this is an increment of 5gp/sqm.
 $ s $ is size, the amount of tiles in the house.
Example 1: Darashia 5, Flat 01  25 sqm, 1 bed, 0 furnishing value.
 $ rent = 100\left(1  1\right) + 25r + 0 $
 $ rent = 25r $
To find the other value, use either the rent or the rate. Since the rent is readily available:
 $ 1000 = 25r $
 $ r = 40 $
Thus the rate is 40gp/sqm.
Example 2: Rathleton Plaza 2  56 sqm, 2 beds, 0 furnishing value.
 $ rent = 100\left(2  1\right) + 56r + 0 $
 $ rent = 100 + 56r $
To find the rent, you can also plug in the rate (if known), which is 45gp/sqm.
 $ rent = 100 + 56 \cdot 45 $
 $ rent = 2620 $
Spell/Rune Damage/Healing
NOTE: These formulae are based upon observed values after update 8.1.
All the spells have the same basic formula: $ \lfloor lvl \cdot 0.2 \rfloor + \left(mlvl \cdot x\right)+y $, where $ x $ is a decimal number and $ y $ is an integer.
For PvP damage, just divide the final result by $ 2 $.
Healing spells:
Light Healing
Max healing: $ x = 1.795, y = 11 $
Min healing: $ x = 1.4, y = 8 $
Intense Healing
Max healing: $ x = 5.59, y = 35 $
Min healing: $ x = 3.184, y = 20 $
Wound Cleansing
Max healing: $ x = 7.95, y = 51 $
Min healing: $ x = 4, y = 25 $
Mass Healing
Max healing: $ x = 10.43, y = 62 $
Min healing: $ x = 5.7, y = 26 $
Ultimate Healing
Max healing: $ x = 12.79, y = 79 $
Min healing: $ x = 7.22, y = 44 $
Instant attack spells
Strike spells
Max damage: $ x = 2.203, y = 13 $
Min damage: $ x = 1.403, y = 8 $
Divine Missile
Max damage: $ x = 3, y = 18 $
Min damage: $ x = 1.79, y = 11 $
Ice Wave
Max damage: $ x = 2, y = 12 $
Min damage: $ x = 0.81, y = 4 $
Fire Wave
Max damage: $ x = 2, y = 12 $
Min damage: $ x = 1.25, y = 4 $
Rune attack spells
Light Magic Missile
Max damage: $ x = 0.81, y = 4 $
Min damage: $ x = 0.4, y = 2 $
Heavy Magic Missile/Stalagmite
Max damage: $ x = 1.59, y = 10 $
Min damage: $ x = 0.81, y = 4 $
Icicle/Fireball
Max damage: $ x = 3, y = 18 $
Min damage: $ x = 1.81, y = 10 $
Holy Missile
Max damage: $ x = 3.75, y = 24 $
Min damage: $ x = 1.79, y = 11 $
Sudden Death
Max damage: $ x = 7.395, y = 46 $
Min damage: $ x = 4.605, y = 28 $
Thunderstorm/Stone Shower
Max damage: $ x = 2.6, y = 16 $
Min damage: $ x = 1, y = 6 $
Avalanche/Great Fireball
Max damage: $ x = 2.8, y = 17 $
Min damage: $ x = 1.2, y = 7 $
These spells have not been tested by me (yet) so I'll leave the old formulae here.
NOTICE: These formulae are based upon observed values with c calculated to one tenth with fair certainty
min: $ \frac{lvl}{5} + mlvl \cdot c $
max: $ \frac{lvl}{5} + mlvl \cdot d $
avg: $ \frac{max+min}{2} = \frac{lvl}{5} + \frac{mlvl(c+d)}{2} $
* c = Multiplier for min of the desired attack spell/rune
* d = Multiplier for max of the desired attack spell/rune
The c values are roughly:
0 for Explosion [adevo mas hur] (note: the min damage of this spell is 0).
2.5 for Energy Beam [exevo vis lux]
4 for Great Energy Beam [exevo gran vis lux]
4 for Divine Caldera [exevo mas san]
3.5 for Terra Wave [exevo tera hur]
4.5 for Energy Wave [exevo vis hur]
10 for Heal Friend [exura sio]
5 for Rage of the Skies [exevo gran mas vis]
7 for Hell's Core [exevo gran mas flam]
5 for Wrath of Nature [exevo gran mas tera]
6 for Eternal Winter [exevo gran mas frigo]
18.5 for Divine Healing [exura san]
The d values are roughly:
4.8 for Explosion [adevo mas hur]
4 for Energy Beam [exevo vis lux]
7 for Great Energy Beam [exevo gran vis lux]
6 for Divine Caldera [exevo mas san]
7 for Terra Wave [exevo tera hur]
9 for Energy Wave [exevo vis hur]
14 for Heal Friend [exura sio]
12 for Rage of the Skies [exevo gran mas vis]
14 for Hell's Core [exevo gran mas flam]
10 for Wrath of Nature [exevo gran mas tera]
12 for Eternal Winter [exevo gran mas frigo]
25 for Divine Healing [exura san]
Melee
NOTICE: This formula is based upon observed values.
 Minimum Damage:
$ Minimum Damage = \frac{lvl}{5} $
 Maximum Damage:
$ Maximum Damage = 0.085 \cdot d \cdot atk \cdot skill+\frac{lvl}{5} $ Where:
 $ atk $ = Weapon's attack
 $ d $ = Damage Factor:
 Full Attack: $ 1 $
 Balanced: $ 0.75 $
 Full Defence: $ 0.5 $
 Let's take for example a player level 80, holding a weapon which has an attack value of 50, skills of 85 and he is attacking on full attack.
 It would be like this: $ \left(0.085 \cdot 1 \cdot 50 \cdot 85\right) + \frac{80}{5} = 377.25 $ (377 rounded)
 Damage factor
So the damage you will do will be between 0 and the calculated number, please note that shielding and armor from the target will reduce your damage. The average damage you will deal is approximately half your maximum damage. Also note, this doesn't take into account monsters weakness or strength to physical damage and if you use blood rage, you'll have to use the appropriate skills in place of your normal skills.
Melee based spells
NOTICE: This formula is based upon observed values.
 atk = Weapon's attack
min: $ \frac{skill+atk}{3} + \frac{lvl}{5} $
max: $ skill + atk + \frac{lvl}{5} $
avg: $ \frac{max+min}{2} = \frac{2}{3}(skill + atk) + \frac{lvl}{5} $
min: $ 0.5\left(skill + atk\right) + \frac{lvl}{5} $
max: $ 1.1\left(skill + atk\right) + \frac{lvl}{5} $
avg: $ \frac{max+min}{2} = 0.8\left(skill + atk\right) + \frac{lvl}{5} $
min: $ 0.5\left(skill + atk\right) + \frac{lvl}{5} $
max: $ 1.5\left(skill + atk\right) + \frac{lvl}{5} $
avg: $ \frac{max+min}{2} = skill + atk + \frac{lvl}{5} $
min: $ 1.1\left(skill + 2 \cdot atk\right) + \frac{lvl}{5} $
max: $ 3\left(skill + 2 \cdot atk\right) + \frac{lvl}{5} $
avg: $ \frac{max+min}{2} = 2.05\left(skill + 2 \cdot atk\right) + \frac{lvl}{5} $
Note: As usual, damage is reduced by armor but none of these abilities can be blocked by shielding. Also note, this doesn't take into account monsters weakness or strength to physical damage and if you use blood rage, you'll have to use the appropriate skills in place of your normal skills.
Distance
NOTE: These formulae are based upon observed values.
 Minimum Damage:
$ Minimum Damage = \frac{lvl}{5} $
 Maximum Damage:
$ Maximum Damage = 0.09 \cdot d \cdot skill \cdot atk + Minimum Damage $
 $ atk $ = Weapon's attack
 $ d $ = Damage Factor:
 Full Attack: $ 1 $
 Balanced: $ 0.75 $
 Full Defence: $ 0.5 $
To calculate your damage in PvP, divide the result by $ 2 $.
Note that armor of the target will reduce your damage.
 Chance to hit (without weapon hit chance modifier)
Distance  1H weapon (throwing)  2H weapon (any bow or crossbow)  Any bow + Sniper Arrows 
1  $ \min\left(75, \left( DistSkill + 1\right)\right)% $  $ \min\left(90, \left(1.2 \cdot DistSkill + 1\right)\right)% $  $ \min\left(100, \left(1.35 \cdot DistSkill + 1\right)\right)% $ 
2  $ \min\left(75, \left(2.4 \cdot DistSkill + 8\right)\right)% $  $ \min\left(90, \left(3.2 \cdot DistSkill \right)\right)% $  $ \min\left(100, \left(3.2 \cdot DistSkill + 5\right)\right)% $ 
3  $ \min\left(75, \left(1.55 \cdot DistSkill + 6\right)\right)% $  $ \min\left(90, \left(2 \cdot DistSkill \right)\right)% $  $ \min\left(100, \left(2.25 \cdot DistSkill + 2\right)\right)% $ 
4  $ \min\left(75, \left(1.25 \cdot DistSkill + 3\right)\right)% $  $ \min\left(90, \left(1.55 \cdot DistSkill \right)\right)% $  $ \min\left(100, \left(1.5 \cdot DistSkill + 2\right)\right)% $ 
5  $ \min\left(75, \left( DistSkill + 1\right)\right)% $  $ \min\left(90, \left(1.2 \cdot DistSkill + 1\right)\right)% $  $ \min\left(100, \left(1.35 \cdot DistSkill + 1\right)\right)% $ 
6  $ \min\left(75, \left(0.8 \cdot DistSkill + 3\right)\right)% $  $ \min\left(90, \left( DistSkill \right)\right)% $  $ \min\left(100, \left(1.2 \cdot DistSkill  4\right)\right)% $ 
7  $ \min\left(75, \left(0.7 \cdot DistSkill + 2\right)\right)% $ 
Distance based spells
NOTICE: This formula is based upon observed values.
min: $ \frac{skill + 25}{3} + \frac{lvl}{5} $
max: $ skill + 25 + \frac{lvl}{5} $
avg: $ \frac{max+min}{2} = \frac{2}{3}\left(skill + 25\right) + \frac{lvl}{5} $
Armor
Note that:
 Armor reduction will only apply when damage crosses the shield.
 The damage armor needs to reduce may be lower than original damage because of the shield reduction.
 If you get a negative number as armor reduction, it means 0.
 Floor function is used here, basically it means that we remove decimals.
 Ceil function is used here, basically it means that we round up numbers with decimals to next integer.
Variables:
 r = minimum armor reduction
 R = maximum armor reduction
 t = total armor
 d = damage (after the shield)
 p = percentage reduction of item
Armor Reduction
$ r = \left \lfloor \frac{t}{2} \right \rfloor $
$ R = \left \lfloor \frac{t}{2} \right \rfloor \cdot 2  1 $
 where $ \left \lfloor x \right \rfloor $ denotes floor function (rounding down).
Percentage Reduction
This formula have to be applied for every single item that has percentage reduction.
$ d = \left \lfloor \frac{100  p}{100} \cdot d \right \rfloor $
Example
You get a hit with original value of 200 wearing only Zaoan Helmet and Protection Amulet
 t = total armor = 9
 $ r = \left \lfloor \frac{9}{2} \right \rfloor $
 r = 4
 $ R = r \cdot 2  1 $
 R = 7
Now we calculate the % reduction that Zaoan Helmet has
 p = 5
 $ d = \left \lfloor \frac{100  5}{100} \cdot 200 \right \rfloor $
 d = 190
Now we calculate the % reduction that Protection Amulet has
 p = 6
 $ d = \left \lfloor \frac{100  6}{100} \cdot 190 \right \rfloor $
 d = 178
For this example, damage becomes 178, minimum armor reduction will be 4 and maximum armor reduction will be 7, so the hit will hit you from 171 to 174.
Armor and Defense
 Damage reduction
Min:
$ totalArmor \cdot 0.475 $
Max:
$ totalArmor \cdot 0.95 1 $
 Attack of Creature = a
(Let's supose that a creature has a max damage of 500. The attack of creature will be a random number from 1 to 500)
 Defense = b (Defense value of your shield plus your weapon modifier or defense value of your two handed weapon)
 Total Armor = c (Make a sum of all your set arm value)
 Shielding = d
 Defense factor = e
Full Attack: 5
Balanced: 7
Defensive: 10
Now use this formula:
$ a  b \cdot d \cdot \frac{e}{100}  \frac{a}{100} \cdot c = averageDamage $
NOTICE: The max damage of a creature could be higher than the max damage in creatures page.