Calculating Mean & Illustrated Examples

In mathematics and statistics, the arithmetic mean is the ratio of the sum of all numbers to the number of elements in that set. It is the relationship between the total value of the set and the total number of elements within it.
Aside from its applications in mathematics and statistics, the arithmetic mean finds widespread use in economics, anthropology, and history, among various other fields. For instance, the per capita income is the arithmetic mean of the income of an entire population of a country.

¯a=a1+a2+a3+⋯+ann=∑an$\overline{a}=\frac{a_1+a_2+a_3+\cdots +a_n}{n}=\frac{\sum a}{n}$

• ¯a is the arithmetic mean of the sequence
• a1,a2,..an$a_1,a_2,..a_n$ is an element in the sequence
• n: is the number of elements

Note: When calculating the arithmetic mean, the numbers in the sequence must be real numbers, not variables.

2. Significance of the Arithmetic Mean

The arithmetic mean is commonly used as a 'representative' for indicators, especially when comparing similar indicators. It is utilized not only in mathematics and statistics but also across various fields in both professional and daily life.

For example, the average income per person per month.

Calculating the average lifespan of the entire population at a specific point in time.

Calculate the per capita income of a nation.

3. What is the Arithmetic Mean of Two Numbers?

According to the concept of arithmetic mean mentioned earlier, the arithmetic mean of two numbers is the sum of those two numbers divided by 2.

Example: The arithmetic mean of a and b will be (a + b) : 2.

II. How to Calculate the Arithmetic Mean & Illustrated Examples

1. Method for Calculating the Arithmetic Mean

Rule for calculating the arithmetic mean: To find the arithmetic mean of several numbers, calculate the sum of those numbers and then divide the sum by the number of terms.

Utilize the formula for calculating the arithmetic mean:

¯a=a1+a2+a3+⋯+ann=∑an$\overline{a}=\frac{a_1+a_2+a_3+\cdots +a_n}{n}=\frac{\sum a}{n}$

Calculating the arithmetic mean for a sequence is as follows:

Step 1: Identify the terms in the problem for which you want to calculate the arithmetic mean and compute the sum of these identified terms.

Step 2: Count the total number of terms in the sum, ensuring that you count accurately and include identical terms when counting.

Step 3: Calculate the arithmetic mean by taking the sum of the calculated terms and dividing it by the number of terms.

Step 4: Draw a conclusion.

2. Illustrated Examples

Example 1: Calculate the arithmetic mean of the numbers: 5, 7, 5, 4, 9

Solution:

- Sum of the numbers in the sequence: 5 + 7 + 5 + 4 + 9 = 30

- The sequence has 5 terms in total.

- Arithmetic mean = Sum of terms : Number of terms = 30 : 5 = 6.

Therefore, the arithmetic mean of the numbers: 5, 7, 5, 4, 9 is 6.

Example 2: Grade 1 at LDP Elementary School consists of Class 1A1 with 35 students, 1A2 with 34 students, 1A3 with 33 students, and 1A4 with 34 students. What is the average number of students per class in Grade 1?

Solution:

- Total number of students in all classes: 35 + 34 + 33 + 34 = 136.

- There are 4 classes.

- Arithmetic mean = 136 : 4 = 34.

Therefore, the average number of students per class in Grade 1 is 34.

III. Various Exercises on Calculating the Arithmetic Mean

Type 1: The problem provides a sequence of numbers and requires calculating the arithmetic mean of that sequence.

Apply the rule for calculating the arithmetic mean: “To find the arithmetic mean of several numbers, calculate the sum of those numbers and then divide the sum by the number of terms.” You only need to perform the arithmetic mean calculation as outlined in the steps above.

Example 3: Calculate the arithmetic mean of the following numbers: 16, 17, 18, 19, 20

Solution:

- Sum of the numbers in the sequence: 16 + 17 + 18 + 19 + 20 = 90.

- The sequence has 5 numbers.

- Arithmetic mean = 90 : 5 = 18.

Therefore, the arithmetic mean of the numbers: 16, 17, 18, 19, 20 is 18.

Type 2: The problem states the arithmetic mean, knowing the number of terms, find the sum of those numbers.

For this type, you also apply the formula for calculating the arithmetic mean and deduce that the sum of the numbers is equal to the known arithmetic mean multiplied by the number of terms.

Example 4: The arithmetic mean of 5 numbers is 15, find the sum of the numbers.

Solution:

Sum of the numbers = Arithmetic mean x Number of terms = 5 x 15 = 75.

Therefore, the sum of the numbers is 75.

Example 5: The arithmetic mean of six numbers is 14. Adding the seventh number makes the arithmetic mean of seven numbers 15. Find the seventh number.

Solution:

- Sum of the initial six numbers is 14 x 6 = 84.

- Sum of the 7 numbers is 15 x 7 = 105.

- The seventh number is 105 – 84 = 21.

Therefore, the seventh number is 21.

Type 3: Calculate the simple arithmetic mean of evenly spaced sequences.

To solve problems of this type, you only need to calculate the arithmetic mean of the first and last numbers in the sequence; this will be the arithmetic mean of the entire sequence.

Note:

- If the sequence has an odd number of terms, the arithmetic mean is the middle number in the sequence.

- If the sequence has an even number of terms, the arithmetic mean is equal to half the sum of the first two and the last two numbers in the sequence (which is the arithmetic mean of the first and last numbers in the sequence).

Example 6: Given the sequence 2, 4, 6, 8, 10, …., 98. Calculate the arithmetic mean of the sequence.

The arithmetic mean of the sequence is: (2 + 98) : 2 = 100 : 2 = 50.

Example 7: Given 6 numbers 2, 4, 6, 8, 10, 12. Knowing that 4 of them have an arithmetic mean of 6, calculate the arithmetic mean of the remaining two numbers.

Solution:

- The sum of the 6 numbers is: 2 + 4 + 6 + 8 + 10 + 12 = 42.

- The sum of the 4 numbers with an arithmetic mean of 6 is: 4 x 6 = 24.

- The sum of the remaining two numbers is: 42 – 24 = 18.

The average of the remaining two numbers is 18 : 2 = 9.

Thus, Mytour has shared with you the method of calculating the average, the average of two numbers, and the average of a series of numbers. The article has also provided illustrative examples of some types of exercises on averaging for you to better understand. Hopefully, after referring to this specific knowledge about how to calculate the average & illustrative examples, you will grasp the knowledge of averaging to solve various averaging math problems. Thank you for your interest and following this article.