Calculations

Learning Objectives

This module shows how various statistical calculations are done using a hand-held or java-based calculator.

The Calculations module targets the following cognitive tasks:

Task Skills	Concepts
Calc-1:	Evaluate algebraic expressions	Understand algebraic expressions
Calc-2:	Evaluate expressions involving summation notation	Understand summation notation
Calc-3:	Evaluate expressions involving grouping symbols	Understand grouping symbols
Calc-4:	Evaluate expressions involving combinatoric notation	Understand combinatoric notation
Calc-5:	Evaluate complex expressions	Understand more complex expressions

Statistical Calculations

Statistics is a mathematical science in which calculations on sets of numbers are often required. The numbers collectively are called data which represent observed measurements or attributes on individuals. Although statistical computations are nearly always done on a computer, it is important to understand how the calculations are done using a hand-held or electronic calculator.

You should have paper, a pencil, and a calculator. If you do not have a calculator, you can use the Java Calculator. It operates similarly to a TI hand-held calculator.

Calc-1: Algebraic Expressions

Simple algebraic expressions consist of numbers, operators such as +, -, /, and *, and free variables which are conbined in such a way that evaluation is possible. Free variables can take on values from outside the expression. For example, is a valid expression with an binary operator / and free variables and . If and , then the expression evaluates to 5. Note that the expression is undefined when .

Calc-2: Summation Notation

Algebraic expressions can also contain bound variables, i.e., variables that have values defined from within the expression. In statistics, the most important algebraic expressions involving bound variables are those containing the summation symbol, denoted by . For example, in the statistical expression

is a bound variable and

is a free variable representing the sample size. The

represent the data values for an observed variable. For example, if

, then the sum is 11. We often drop the notation for the bound variable if it is understood that the summation extends over the entire range of the data, i.e., we shorten the above expression to:

Calc-3: Grouping Symbols

Grouping symbols are used to clarify the order of evaluation. For example, consider the expression:

. The parentheses make it clear that

is subtracted from each

before the differences are squared and summed.

Calc-4: Combinatorics

We often need to count how many ways certain outcomes can occur. For example, the sixth century mathematician Varahamihira asked how many ways four substances can be selected from 16. We denote this by:

Examples

Example #1 shows how to perform the following calculations on a set of numbers :

Evaluations are also done for and .

Self-test