Associative and Commutative Properties

Grouping Versus Ordering of Elements of Equations in Statistics and Probability

The associative property concerns itself with the regrouping of elements and an operation. C.K.Taylor

There are several named properties in mathematics that are used in statistics and probability; two of these types of properties, the associative and commutative properties, are found in the basic arithmetic of the integers, rationals, and real numbers, but also show up in more advanced mathematics.

These properties are very similar and can be easily mixed up, so it is very important to know the difference between the associative and commutative properties of statistical analysis by first determining what each individually represents then comparing their differences.

Commutative property concerns itself with the ordering of certain operations wherein the operation * is commutative of a given set (S) if for every x and y value in the set x * y = y * x. Associative property, on the other hand, is only applied if the grouping of the operation is not important wherein the operation * is associative on the set (S) if and only if for every x, y, and z in S, the equation can read (x * y) * z = x * (y * z).

Defining Commutative Property

Simply put, the commutative property states that the factors in an equation can be rearranged freely without affecting the outcome of the equation. The commutative property, therefore, concerns itself with the ordering of operations including the addition and multiplication of real numbers, integers, and rational numbers and matrix addition.

On the other hand, subtraction, division, and matrix multiplication are not operations that can be commutative because the order of operations is important — for example, 2 - 3 is not the same as 3 - 2, therefore the operation does not a commutative property.

As a result, another way to express the commutative property is through the equation ab = ba wherein no matter the order of the values, the results will always be the same. 

Associative Property

The associative property of an operation exhibits associativity if the grouping of the operation is not important, which can be expressed as a + (b + c) = (a + b) + c because no matter which pair is added first because of the parenthesis, the result will be the same.

Like in commutative property, examples of operations that are associative include the addition and multiplication of real numbers, integers, and rational numbers as well as matrix addition. However, unlike the commutative property, the associative property can also apply to matrix multiplication and function composition.

Like commutative property equations, associative property equations cannot contain the subtraction of real numbers. Take for example the arithmetic problem (6 – 3) – 2 = 3 – 2 = 1; if we change the grouping of our parentheses, we have 6 – (3 – 2) = 6 – 1 = 5, so the result is different if we rearrange the equation.

What Is the Difference?

We can tell the difference between the associative or commutative property by asking, “Are we changing the order of elements, or are we changing the grouping of these elements?” However, the presence of parentheses alone does not necessarily mean that an associative property is being used. For instance:

(2 + 3) + 4 = 4 + (2 + 3)

The above is an example of the commutative property of addition of real numbers. If we pay careful attention to the equation, we see that we changed the order, but not the groupings of how we added our numbers together; in order for this to be considered an equation using the associative property, we would have to rearrange the grouping of these elements to state (2 + 3) + 4 = (4 + 2) + 3.