The total influence of a real-valued function 
is the sum
of its coordinate influences under the discrete
uniform distribution, where 
is a positive integer.
If 
is a random vector uniformly distributed on 
and 
reverses coordinate 
, then
![I(f)=1/4sum_(i=1)^nE[(f(X)-f(X^({i})))^2],](/images/equations/TotalInfluence/NumberedEquation1.svg) |
where 
denotes the expectation value (O'Donnell 2021,
pp. 47-50).
For a Boolean function with values in 
, each coordinate contributes the probability
that reversing coordinate 
changes the output, so total influence equals average
sensitivity. Under this quadratic normalization, a function
with values in 
instead has total influence one quarter of its average
sensitivity. The pivotal-probability convention for a Boolean
function with values in 
uses average sensitivity
without this factor.
