MATH 2441 | Probability and Statistics for Biological Sciences |
Review of Subscripts and Summation Notation
Subscripts
As in other branches of mathematics, statistics uses literal symbols to stand for variable quantities. Thus, for example, we might define:
let x = the concentration of salt in a specimen of tomato soup, in units of mg/litre.
Now, we may measure the value of x for several specimens of tomato soup, say 5 of them, getting the set of values
{245, 423, 615, 912, 621}
These five values are all values of x, but correspond to different observations of the variable x. To distinguish symbolically between such alternative values of a single variable x, it is common to use a subscript notation, x_{k}, where k = 1 for the first value, k = 2 for the second, and so on.
Thus, for this example
x_{1} = 245
x_{2} = 423
x_{3} = 615
x_{4} = 912
and
x_{5} = 621.
Think of these subscripts as numerical labels used to distinguish one of a set of values from the others.
Summation Notation
A lot of statistical computations involve the evaluation of sums. To make it easy to indicate the computation of a sum, a mathematical notation using the Greek letter &Sigma ("sigma" or "s" for "sum") is used in the form:
What this means is: write down the value of "expression" for each value of "index" between "first" and "last" (including "first" and "last"). Then add up the results.
It's not as complicated as it sounds. Here's some examples.
On the right-hand side of the first line here, the bracketed expressions are what you get when you substitute the values k = 3, 4, 5, 6, 7, 8, and 9 into the expression '2k'.
Notice that in this second example, the 5 is repeated -- the rule is "write down the expression to the right of the &Sigma for each value of the index" before summing. That means we have to write down the value of 3n+5 for each value, n = 0, 1, 2, 3, and 4, before adding.
Summation notation is quite useful when combined with subscript notations for a variable. For example to indicate the sum of the five values of x in the first section of this document, we can write:
Suppose we have a second set of values of some variable y:
y_{1} = 3
y_{2} = 7
y_{3} = 4
y_{4} = 6
and
y_{5} = 2.
Then,
Make sure you understand how the simple rule of interpretation given above leads to this result.
Sometimes when people work with a set of values of a variable x, they will use the shorthand symbols &Sigma x_{k} or just &Sigma x to mean "add up all of the x values."
From the basic definition of this summation notation, we can derive or deduce a number of useful properties. We will give only three of the most useful here.
First, if the "expression" part is a constant that doesn't depend on the value of the index, we can calculate the sum easily. In general terms, if c stands for a constant (that does not depend on the value of k), then
For example:
Here
last - first + 1 = 7 - 3 + 1 = 5
and so what the general formula gives
is in agreement with what we got by expanding the sum in detail. You see that the general formula above just results from the fact that the number of values of k between "first" and "last" inclusive is "last - first + 1", and the constant c is just added into the sum this many times.
Secondly, the sum of a sum can be broken up into two separate sums. In symbols, we can write
It doesn't matter if you form the sums (x_{1}+y_{1}), (x_{2}+y_{2}), etc. first, and then add these up, or first add up all the x_{k} values, and all the y_{k} values, and then add those two values together. The result will be the same.
We can demonstrate this with our two short data sets.
Alternatively
Thirdly, a constant coefficient can be pulled out of a summation. In symbols, if c is a constant,
Again, a simple example will make it clear why this is so.
In this example, we didn't bother to go to the point of plugging in the numerical values of the x_{k}, because the algebraic pattern we wanted to demonstrate is obvious.
Although you will not be responsible for being able to manipulate and simplify complicated expressions involving summations in this course, it is possible to use these three simple rules to develop some very useful and fundamental
This material is available in Microsoft WORD format here.