Average

In this article, you will learn everything you need to know about the mean or average. We assure you that you will master the topic with great ease and even be glad to find it on your exam.

You probably know what the average is, but we are here to remind you from time to time.
The average, or mean, is a number that represents a certain numerical set. We can say that it represents the whole set.
For example, if we ask what Diana's average grade is, we want to arrive at a single grade that represents all of them.

• The average does not necessarily have to be one of the numbers in the set.
• If we add a number identical to the average to the set, there will be no change in the average.
• If we add a number that is larger than the average to the set, the average will increase.
• If we add a number that is smaller than the average to the set, the average will decrease.
• If we add a fixed number to each number in the set, the average will increase exactly by the value of the fixed number.
• The average can be a fraction.

First step - we will add all the numbers
Second step - we will divide the total sum by the number of addends. (Let's remember that $0$ is also a number)

$Average= \frac{sum~of~the~numbers}{number~of~addends}$

What is Daniel's average grade considering the following scores?:
$87$ in English, $55$ in math, $80$ in language, $0$ in physical education, and $70$ in literature

Solution 1:
We will proceed step by step. First, we will add all the numbers and then divide the total by the number of addends.

We will obtain:
$\frac{87+55+80+0+70}{5}=$

${292\over5}=58.4$

Daniel's average grade is $58.4$.
Note that we also considered the number $0$ when adding all the addends.

If Daniel had had another history exam in which he scored $40$, would his average have been lower than $58.4$ or higher than $58.4$?

Solution 2:
$40$ is lower than $58.4$, therefore, if we add it to the set, his average will decrease.

A frequency table is, in fact, a table that organizes the data in a clearer way.
When there are many data points, this type of table helps us see them more clearly, as happens, for example, with the grades of the whole class and not just those of one student.

The table tells us that:
One student scored $100$, $3$ students scored $90$, $7$ students scored $80$, $5$ students scored $70$ and $2$ students scored $60$.
Instead of noting: $60,60,70,70,70,70,70,80,80,80,80$.......... and the list still goes on.

How do you calculate the average in a frequency table?
Exactly the same way we did before:

What is the class's average grade?
The result of the sum – will be the total of grades
The number of addends - will be the number of grades

Solution:
Let's calculate the total of the grades: $100+3 \times 90+7 \times 80+5 \times 70+2 \times 60=1400$
We multiply the number of students who obtained the same grade by the grade itself and then add to get the total of grades.
Let's calculate the number of grades: $1+3+7+5+2=18$
The number of grades is, in fact, the number of students since each one received a grade.

We will divide the total by the number and obtain the average:
${1400\over18}=77.777$
The class's average grade $77.777$.

To find the highest possible average, we will add the highest number to the set and vice versa.
In these types of questions, there will be some condition that we must meet.

The grades are within the range of $30-100$
Romi's four grades are $80$.
Romi has another grade.
What is the lowest possible average of Romi's five grades?

Solution:
We will add the lowest possible grade $30$ and calculate the average.
$\frac{80 \times 4+30}{5}=70$