34 Distributions
Ladislaus J. Bortkiewicz was fascinated by mortality tables. Not for him a gloomy topic,
they were a field of enduring scientific enquiry. He famously counted the number of
cavalrymen in the Prussian army that had been killed by horse-kicks. Then there was
Frank Benford, an electrical engineer who counted the first digits of different types of
numerical data to see how many were ones, twos and so on. And George Kingsley Zipf,
who taught German at Harvard, had an interest in philology and analysed the
occurrences of words in pieces of text.
All these examples involve measuring the probabilities of events. What are the
probabilities of x cavalrymen in a year receiving a lethal kick from a horse?
Listing the probabilities for each value of x is called a distribution of probabilities,
or for short, a probability distribution. It is also a discrete distribution because
the values of x only take isolated values – there are gaps between the values of
interest. You can have three or four Prussian cavalrymen struck down by a lethal
horse-kick but not 3½. As we’ll see, in the case of the Benford distribution we
are only interested in the appearance of digits 1, 2, 3,... and, for the Zipf
distribution, you may have the word ‘it’ ranked eighth in the list of leading words,
but not at position, say 8.23.
Life and death in the Prussian army
Bortkiewicz collected records for ten corps over a 20 year period giving him
data for 200 corps-years. He looked at the number of deaths (this was what
mathematicians call the variable) and the number of corps-years when this
number of deaths occurred. For example, there were 109 corps-years when no
deaths occurred, while in one corps-year, there were four deaths. At the
barracks, Corp C (say) in one particular year experienced four deaths.
How is the number of deaths distributed? Collecting this information is one
side of the statistician’s job – being out in the field recording results. Bortkiewicz
obtained the following data: