Digression on The Sigma Operator
For those programmers new to statistics, this section provides
background on the Sigma operator, Σ.
Equation 13.3. Basic Summation
The Σ operator has three parts to it. Below it is a bound
variable,
i
and the starting value for the range,
written as
i
=0. Above it is the ending value for
the range, usually something like
n
. To the right
is some function to execute for each value of the bound variable. In
this case, a generic function,
f
(
i
). This is read as
“sum
f
(
i
) for
i
in the range 0 to
n
”.
One common definition of Σ uses a closed range, including the end
values of 0 and
n
. However, since this is not a
helpful definition for software, we will define Σ to use a halfopen
interval. It has exactly
n
elements, including 0
and
n
1; mathematically,
.
Consequently, we prefer the following notation, but it is not
often used. Most statistical and mathematical texts seem to use 1based
indexing, consequently some care is required when translating formulae
to programming languages that use 0based indexing.
Equation 13.4. Summation with HalfOpen Interval
Our two statistical algorithms have a form more like the
following. In this we are applying some function,
f
(), to each value,
x_{i}
of an array. When
computing the mean, there is no function. When computing standard
deviation, the function involves subtracting and multiplying.
Equation 13.5. Summing Elements of an Array,
x
We can transform this definition directly into a for loop that
sets the bound variable to all of the values in the range, and does some
processing on each value of the list
of integers.
This is the Python implemention of Sigma. This computes two values, the
sum, sum
, and the number of elements,
n
.
Example 13.2. Python Sigma Iteration
sum= 0
for i in range(len(theList)):
xi= theList[i]
# fxi = processing of xi
sum += fxi
n= len(theList)

Get the length of theList . Execute the
body of the loop for all values of i in the
range 0 to the number of elements1.


Fetch item i from
theList and assign it to
xi .


For simple mean calculation, this statement does nothing.
For standard deviation, however, this statement computes the
measure of deviation from the average.

More advanced Python programmers may be aware that there are
several ways to shorten this loop down to a single expression using the
reduce
function as well as list
comprehensions.
An Optimization. In the usual mathematical notation, an integer index,
i
is used. In Python it isn't necessary to use
the formal integer index. Instead, an iterator can be used to visit
each element of the list
, without actually
using an explicit numeric counter. The processing simplifies to the
following.
Example 13.3. Python Sample Values by Iterator
for xi in theList:
# fxi = processing of xi
sum += fxi
n= len(theList)

Execute the loop assigning each item of of
theList to xi .


For simple mean calculation, this statement does nothing.
For standard deviation, however, this statement computes the
measure of deviation from the average.
