Groupings in Statistics
A grouping represents the total number of distinct selections formed by choosing one element from each of two or more sets.
More broadly, if the first set contains \( n \) elements, the second \( m \), the third \( k \), and so on, then the total number of possible selections is:
\[ n \cdot m \cdot k \]
This follows directly from the product rule in combinatorial analysis, since each element of the first set can be combined with every element of the second, then with every element of the third, and so forth.
A practical example
Consider two sets, A and B:
- set \( A = \{ a_1, a_2, a_3 \} \) with 3 elements
- set \( B = \{ b_1, b_2 \} \) with 2 elements
A grouping is obtained by selecting one element from \( A \) and one from \( B \).
The possible pairs are:
$$ (a_1, b_1) \\ (a_1, b_2) \\ (a_2, b_1) \\ (a_2, b_2) \\ (a_3, b_1) \\ (a_3, b_2) $$
There are six distinct groupings
$$ 3 \cdot 2 = 6 $$
This example clearly illustrates the product rule, since every choice from \( A \) can be matched with every choice from \( B \).
Example 2
Now consider three sets: A, B, and C.
- set \( A = \{ a_1, a_2 \} \) with 2 elements
- set \( B = \{ b_1, b_2, b_3 \} \) with 3 elements
- set \( C = \{ c_1, c_2 \} \) with 2 elements
A grouping is formed by selecting one element from each set.
The possible triples \( (a_i, b_j, c_k) \) are:
$$ (a_1, b_1, c_1) \\ (a_1, b_1, c_2) \\ (a_1, b_2, c_1) \\ (a_1, b_2, c_2) \\ (a_1, b_3, c_1) \\ (a_1, b_3, c_2) \\ (a_2, b_1, c_1) \\ (a_2, b_1, c_2) \\ (a_2, b_2, c_1) \\ (a_2, b_2, c_2) \\ (a_2, b_3, c_1) \\ (a_2, b_3, c_2) $$
There are twelve distinct groupings.
$$ 2 \cdot 3 \cdot 2 = 12 $$
A useful way to visualize all possible selections is with a tree diagram, a standard tool in introductory combinatorics and probability.
In a tree diagram, each set corresponds to a level of branching.
In this case, the diagram begins with two initial branches, representing the two elements of the first set
\( A = \{ a_1, a_2 \} \).
Each initial branch divides into three second level branches, corresponding to the elements of the second set \( B = \{ b_1, b_2, b_3 \} \).
Each of these then divides into two third level branches, one for each element of the third set \( C = \{ c_1, c_2 \} \).
Note. In general, each element of the first set generates as many branches as there are elements in the second set, and each of those branches splits again according to the elements of the third set, and the process continues in the same way for additional sets.
And the same reasoning extends naturally to any number of sets.
