The answer is   
>!**126**  
>!  
>!Square each value, add together, divide by numbers of values  
>!  
>!(3<sup>2</sup> + 12<sup>2</sup> + 15<sup>2</sup>) / 3 = (9 + 144 + 225) / 3 = (378) / 3 = 126  

<strike>But the third clue seems to prohibit addition, so I'm not sure.</strike>  
The third clue was changed to only prohibit subtraction.  

Full list of equations:  
>!<pre>(1<sup>2</sup> + 3<sup>2</sup>) / 2 = 5  
>!(1<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup>) / 3 = 14  
>!(2<sup>2</sup> + 4<sup>2</sup> + 6<sup>2</sup> + 7<sup>2</sup> + 10<sup>2</sup>) / 5 = 41  
>!(7<sup>2</sup> + 13<sup>2</sup>) / 2 = 109  
>!(2<sup>2</sup> + 2<sup>2</sup> + 4<sup>2</sup>) / 3 = 8  
>!(1<sup>2</sup> + 4<sup>2</sup>) / 2 = 8.5  
>!(4<sup>2</sup> + 5<sup>2</sup>) / 2 = 20.5  
>!(2<sup>2</sup> + 4<sup>2</sup>) / 2 = 10  
>!(2<sup>2</sup> + 2<sup>2</sup>) / 2 = 4</pre>