pylab.hist(data, normed=1). Normalization seems to work incorrect

I'm trying to create a histogram with argument normed=1

For instance:

import pylab

data = ([1,1,2,3,3,3,3,3,4,5.1])    
pylab.hist(data, normed=1)
pylab.show()

I expected that the sum of the bins would be 1. But instead, one of the bin is bigger then 1. What this normalization did? And how to create a histogram with such normalization that the integral of the histogram would be equal 1?

Solution 1:

See my other post for how to make the sum of all bins in a histogram equal to one: https://stackoverflow.com/a/16399202/1542814

Copy & Paste:

weights = np.ones_like(myarray)/float(len(myarray))
plt.hist(myarray, weights=weights)

where myarray contains your data

Solution 2:

According to documentation normed: If True, the result is the value of the probability density function at the bin, normalized such that the integral over the range is 1. Note that the sum of the histogram values will not be equal to 1 unless bins of unity width are chosen; it is not a probability mass function. This is from numpy doc, but should be the same for pylab.

In []: data= array([1,1,2,3,3,3,3,3,4,5.1])
In []: counts, bins= histogram(data, normed= True)
In []: counts
Out[]: array([ 0.488,  0.,  0.244,  0.,  1.22,  0.,  0.,  0.244,  0.,  0.244])
In []: sum(counts* diff(bins))
Out[]: 0.99999999999999989

So simply normalization is done according to the documentation like:

In []: counts, bins= histogram(data, normed= False)
In []: counts
Out[]: array([2, 0, 1, 0, 5, 0, 0, 1, 0, 1])
In []: counts_n= counts/ sum(counts* diff(bins))
In []: counts_n
Out[]: array([ 0.488,  0.,  0.244,  0.,  1.22 ,  0.,  0.,  0.244,  0.,  0.244])

Solution 3:

I think you are confusing bin heights with bin contents. You need to add the contents of each bin, i.e. height*width for all bins. That should = 1.

Solution 4:

What this normalization did?

In order to normalize a sequence, you have to take into account the bin size. According to the documentation , the default number of bin is 10. Consequently, the bin size is (data.max() - data.min() )/10, that is 0.41. If normed=1, then the heights of the bar is such that the sum, multiplied by 0.41, gives 1. This is what happens when you integrate.

And how to create a histogram with such normalization that the integral of the histogram would be equal 1?

I think that you want the sum of the histogram, not its integral, to be equal to 1. In this case the quickest way seems:

h = plt.hist(data)
norm = sum(data)
h2 = [i/norm for i in h[0]]
plt.bar(h[1],h2)