In *Pokémon Go*, each Pokémon has three values that determine how well it will battle: its moveset, its level, and its individual value (IV).

The moveset are the attacks that your Pokémon can use — things like *mudshot* and *fireblast* and *hydropump *— and can’t be changed. Every type of Pokémon has a moveset that is optimal (i.e. gives the highest possible damage per second of attacks available to you), such as *Water Gun/Hydropump* for the Vaporeon, and this will strongly affect how much damage you do.

The Pokémon’s level is marked by the arc in the background, which stretches from the very bottom left (Pokémon with 10 CP), to the bottom right (Pokémon that have the same level as their trainer does). When you click ‘power up’, you are increasing the Pokémon’s level, which will increase its HP and CP in accordance with its IV.

IV is the only piece of information about your Pokémon that is entirely hidden from you in the game, although it can be calculated using level, HP, CP and the information that your gym leader provides you, through external calculators. IV is broken down into three categories: Attack, Defence and Stamina, each of which take an integer value from 0 to 15, with the total IV simply being the sum of all these values (often presented as a percentage of the maximum value). So, what is a good IV for a Pokémon to have?

Thankfully, this is the same as rolling three sixteen-sided dice from 0 to 15, and we can use familiar maths to calculate both the average roll of *m **n*-sided dice, , and the standard deviation, . This gives an average IV of 22.5 (50%) and standard deviation of 8 (18%). This means that two-thirds of all Pokémon have an IV between 14.5 and 30.5 (32% and 68%).

To illustrate this graphically, let’s consider every single possible roll of these three dice, written in code here:

for x in 0:15 for y in 0:15 for z in 0:15 print(x+y+z) end end end

This results in the bell curve shown in Figure 1, because there is only one way to generate 0 (that is, all dice roll zero), but there are many ways to generate a roll of 22 (192 ways, in fact).

While it is useful to know how rare your Pokémon is, deciding on whether to keep it should be based on how likely it is to have *at least* its IV; otherwise, it’s easy to see that 0 and 45 are equally rare. The probability of a random Pokémon having as good an IV or better as your current Pokémon is given by the curve shown in Figure 2.

This suggests that Pokémon with 80% or 90% IV are incredibly rare, which matches up with what the Gym Leaders say: the top-tier evaluation of a Pokémon places it into this high (80%+ category). Let’s have a look at some particularly interesting IV:

- 23 (49%) — this is the median IV, or the 50th percentile: half of all Pokémon are equal to and better than it, while half are worse than it;
- 27 (60%) — this is the tier occupied by the top third of Pokémon;
- 30 (66%) — the top fifth of all Pokémon have an IV of thirty or higher, meaning 80% of all Pokémon have a
*lower*IV than 30; - 34 (76%) — your Pokémon is in the top 10% of all Pokémon;
- 37 (82%) — at this point, your Pokémon is considered to be top-tier by the Gym Leaders when you select ‘Appraise’, and they are in the 94th percentile;
- 41 (91%) — this is finally over the 90% IV mark, and represents the top 1% of all Pokémon. Anything with this IV almost has the best base stats of its kind;
- 45 (100%) — the ultimate Pokémon. Each Pokémon you find has a 0.02% (two in ten thousand) chance of having this IV, making them incredibly rare.

To illustrate how rare these values are, let’s look at Gyarados: you need to catch a hundred Magikarp to make a Gyarados, so what is the highest Magikarp you’re quite sure you can catch (>90% chance)?

For your first Gyarados, you would be expecting an IV of at least 38 (84%). Your second Gyarados should be even higher, with an IV of 40 (88%), but after this, your probability starts falling off sharply. Your third Gyarados should have at least 41 IV (91%), but you can’t be sure of getting a 42 IV Gyarados until your fifth, or a 43 IV one until your tenth! At that point, the chance of you having a 45 IV (100%) Gyarados is 20%, so if you’re lucky you may have a truly vicious sea monster!

For some anecdotal evidence, I have caught 4327 Pokémon, meaning that I should expect to have found about one 100% Pokémon by now. And I have!

A Zubat.