Betweenness centrality in NetworkX: logical error

I calculate betweenness centrality for the Florentine Families Graph by:

import networkx as nx

# build up a graph
G = nx.florentine_families_graph()
bw_centrality = nx.betweenness_centrality(G, normalized=False)

Excerpt from the description of betweenness_centrality(...) in networkx,

Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v:

Therefore, betweenness centrality should be less than 1. However, I got the result: (the betweenness centrality of the red node, 'Medici', is 47.5)

The way I calculate the betweenness centrality is as follows,

node_and_times = dict.fromkeys(G.nodes(), 0) # a dict of node : the number of shortest path passing through node
sum_paths = 0

for s, t in itertools.product(G.nodes(), repeat=2): # all pair of nodes <s, t>
    paths = nx.all_shortest_paths(G, s, t) # generator of lists
    for path in paths:
        sum_paths += 1

        # stats nodes passing through shortest path
        for node in path[1:-1]: # intermediate nodes
            node_and_times[node] += 1

bw_centrality = {k : v*1.0/sum_paths for k, v in node_and_times.items()}

and I got the following result,

Am I right?

As mentioned by the answerers, removing normalized=False got the following result which is not consistent with my calculation.