The word mean is used a lot in today’s life. Although these words are from specialized keywords mathematics and Arman Computer Magazine, we discuss the concept of average and its applications. The arithmetic mean, average, or average all refer to a type of statistical calculation that is included in the group of central statistical indicators (central tendency). By means of the average index, we can get the result or the center of gravity of the given numbers. In several ways we can describe the average.

Note: Remember that the average calculation formula is the sum of digits divided by their number. So if x_i indicates numbers and n indicates the same number, the average is obtained from the following formula.

$$\sum_{i = 1 }^n x_i /n $$

• A method for smoothing

In this way, the definition of mean is a value that can replace all values. Pay attention to the figure below. The height of each set of pencils can represent a number. The average is calculated in a way that can, based on smoothing, straighten the values or number of pencils in each batch and assign a single number to all of them. On the left side, you can see the equality of numbers, whose average is supposed to be calculated. We have shown the first number (6) as a set of six pencils. The second number (11) is also specified as a group of eleven. The third number (7) is a set of seven pencils. On the right side, you can see the tie of groups with the same number of pencils. There are eight pencils in each category. In this way, three groups with different number of pencils became three groups with equal number of pencils. The number 8 is the average of the values 6, 11 and 7.

( 6 + 11 + 7 ) / 3 = 24 / 3 = 8

• Point (center) of gravity or result of numbers

In this case, when we consider the center of gravity or result, we pay attention to the point or value from which the sum of the distance of all numbers is equal to zero. In physics, the point where the result (sum) of the forces is zero is called the center of gravity. In fact, the center of gravity or the center of mass is a certain point of the object as if all the mass of the object is concentrated at that point. Assume again that we have the set of numbers from the previous section. We are looking for a value whose total distance from all values is equal to zero. If we set the guess for the average equal to 8, we will reach the following relationship.

(6 – 8) + (11 – 8) + (7 – 8) = -2 + 3 -1 = 0

This rule is valid for all numbers.

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Always the sum of the distance of values from their mean will be equal to zero

As noted, the mean can be a point that is the sum of all values. Just like the consensus method where all opinions are involved in the final decision, the average is made of all values and both large and small values affect the average. On the other hand, this influence may also cause deviation. Consider that in a classroom, the scores of ten students are as follows.

1, 17, 16, 20, 19, 18, 17, 19, 17, 16

The sum of these values is equal to 160, which when divided by their number (10), the average will be equal to 16. The presence of a different value from the rest of the scores (i.e. score 1) has caused the average deviation. Because this value is also used in calculating the average, and it seems that 16 cannot be a good representative for these grades. It is clear that 16 is less than or equal to all scores (except one).

(1 + 17 + 16 + 20 + 19 + 18 + 17 + 19 + 17 + 16) / 10

According to the calculated average (score 16), we can guess that this class is an average class. But if we remove grade 1 from the class, the total grades will be 159 and the average will be about 18. As a result, the class will become a successful and desirable class of students.

(17 + 16 + 20 + 19 + 18 + 17 + 19 + 17 + 16) / 9 ≅ 18

This issue shows the effect of unusual points and far from the rest of the numbers. Such numbers are known as outliers in statistics.

Average advantages

• It has a simple calculation.
• It is obtained based on the same numbers.
• The value obtained for the average is considered in the range of numbers.

Disadvantages of Average

• It can only be calculated for numbers (quantitative values).
• It is affected by outliers and deviates.