Page 65 - Mathematics of Business and Finance

Chapter 1 | Review of Basic Arithmetic 45

Example 1.5(d) Calculating the Average of All Numbers Given the Average of Two Sets of Numbers
If the average of a set of three numbers is 45 and the average of a different set of four numbers is 55,
determine the average of all the seven numbers rounded to two decimal places.

Solution If the average of a set of three numbers is 45, then the sum of the three numbers would be 45 × 3 = 135.
Similarly, the sum from the set of four numbers would be 55 × 4 = 220.
Now, the average of the seven numbers would be the sum of the seven numbers divided by 7.
135 + 220
= 50.714285... = 50.71
7
Therefore, the average of the seven numbers is 50.71.

Weighted Average (Weighted Mean)

The weighted average is often called the weighted arithmetic mean. It is similar to an arithmetic
Weighted average
is computed by average, but instead of each of the data points contributing equally to the final average, some data
multiplying each value points contribute more than others.
(x) by its corresponding
weighting factor (w), When all the values of the terms are not of equal importance, each quantity to be averaged is
summing the weighted assigned a different weighting factor. These weighting factors determine the relative importance of
values and then each value.
dividing by the total
number of weighting If all the weighting factors are equal, then the weighted average is the same as the arithmetic average.
factors. In most cases,
the weighting factors For example, a student's final evaluation in a subject is often based on different components, such as
add up to 1, or 100%. quizzes, assignments, tests, and exams. Each component will be assigned a value by the teacher, which
will help determine the student's final grade earned for the subject.
Quizzes and assignments may be worth a smaller percent of the total grade, compared to major tests
and exams which may carry additional weight in the final grade earned. This means that the tests and
exams carry more importance in determining a student's grade in the subject, although the successful
completion of the other components will allow the student to earn the highest possible grade.
The weighted average is calculated as follows:
w , w , w , … w are the weighting factors assigned to the terms and x , x , x , ... x are the values of
n
3
n
2
1
3
1
2
the terms in that order. The weighted value is the value multiplied by its corresponding weight. To find
the weighted average we divide the sum of the weighted values by the sum of the weighting factors. The
weighted average formula can be expressed using sigma notation as shown below.
Formula 1.5(b) Weighted Average
Sum of all weighted values w x + w x + w x + ... + w x
3 3
1 1
n n
2 2
Weighted Average = = =
Sum of the weighting factors w + w + w + ... + w n
1
3
2