 How to use it 
What percentage of men and women will have the same height in two normally distributed populations with known means and standard deviations?
# http://www.usablestats.com/lessons/normal
>>> men = NormalDist(70, 4)
>>> women = NormalDist(65, 3.5)
>>> men.overlap(women)
0.5028719270195425
The result can be confirmed empirically with a Monte Carlo simulation:
>>> from collections import Counter
>>> n = 100_000
>>> overlap = Counter(map(round, men.samples(n))) & Counter(map(round, women.samples(n)))
>>> sum(overlap.values()) / n
0.50349
The result can also be confirmed by numeric integration of the probability density function:
>>> dx = 0.10
>>> heights = [h * dx for h in range(500, 860)]
>>> sum(min(men.pdf(h), women.pdf(h)) for h in heights) * dx
0.5028920586287203
 Code 
def overlap(self, other):
'''Compute the overlap coefficient (OVL) between two normal distributions.
Measures the agreement between two normal probability distributions.
Returns a value between 0.0 and 1.0 giving the overlapping area in
the two underlying probability density functions.
'''
# See: "The overlapping coefficient as a measure of agreement between
# probability distributions and point estimation of the overlap of two
# normal densities"  Henry F. Inman and Edwin L. Bradley Jr
# http://dx.doi.org/10.1080/03610928908830127
# Also see:
# http://www.iceaaonline.com/ready/wpcontent/uploads/2014/06/MM9PresentationMeettheOverlappingCoefficientAMeasureforElevatorSpeeches.pdf
if not isinstance(other, NormalDist):
return NotImplemented
X, Y = self, other
X_var, Y_var = X.variance, Y.variance
if not X_var or not Y_var:
raise StatisticsError('overlap() not defined when sigma is zero')
dv = Y_var  X_var
if not dv:
return 2.0 * NormalDist(fabs(Y.mu  X.mu), 2.0 * X.sigma).cdf(0)
a = X.mu * Y_var  Y.mu * X_var
b = X.sigma * Y.sigma * sqrt((X.mu  Y.mu)**2 + dv * log(Y_var / X_var))
x1 = (a + b) / dv
x2 = (a  b) / dv
return 1.0  (fabs(Y.cdf(x1)  X.cdf(x1)) + fabs(Y.cdf(x2)  X.cdf(x2)))
 Future 
The concept of an overlap coefficient (OVL) is not specific to normal distributions, so it is possible to extend this idea to work with other distributions if needed.
