For those programmers new to statistics, this section covers the Sigma operator, Σ.

Equation 15.1. 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 half-open interval. It has exactly n elements, including 0 and n-1; mathematically, .

Consequently, we prefer the following notation, but it is not often used. Since statistical and mathematical texts often used 1-based indexing, some care is required when translating formulae to programming languages that use 0-based indexing.

Equation 15.2. Summation with Half-Open Interval

Our two statistical algorithms have a form more like the following function. 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 15.3. 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 Java implementation of Sigma. This computes two values, the sum, sum and the number of elements, n.

Example 15.1. Java Sigma Iteration

int sum= 0;
for( int i= 0; i != theList.size(); ++i ) { 
    int xi= ((Integer)theList.get(i)).intValue(); 
    // int fxi = processing of xi 
    sum += fxi;
}
int n= theList.size();

	Get the size of theList. Execute the body of the loop for all values of i in the range 0 to the number of elements-1.
	Fetch item i from theList. This is an Object, which we cast to be an Integer. We then get the int value of this item 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.

This is the Python implemention of Sigma. This computes two values, the sum, sum and the number of elements, n.

Example 15.2. 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 elements-1.
	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 sum function as well as list displays.

In the usual mathematical notation, an integer index, i is used. In Java or 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. In Java, the use of an iterator is shown below.

Example 15.3. Java Sample Values by Iterator

int sum= 0;
for( Iterator i= theList.iterator(); i.hasNext();  ) { 
    int xi= ((Integer)i.next()).intValue(); 
    // int fxi = processing of xi 
    sum += fxi;
}
int n= theList.size();

	Get an Interator over all items of theList. Execute the body of the loop for all items in the iterator.
	Fetch the next item from the Interator, i. This is an Object, which we cast to be an Integer. We then get the int value of this item 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.

In Python, the iterator is implied, and the processing simplifies to the following.

Example 15.4. 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.

These iterator-based formulations can be slightly faster when using a LinkedList because this class is optimized for exactly this kind of sequential access.