Concept of Average

Average refers to the mean value of the values, which we can also understand as the sum divided by the total number of values in a particular set. It is one of the most commonly used measures of central tendency employed to make a statistical summary of enormous data. A simple example of the average is how the report card focuses on the aggregate marks to simplify the interpretation instead of stating the different marks one has scored in multiple subjects throughout the year. This central tendency is, thus, beneficial in interpreting large value sets to come to valuable conclusions. 

Average in Statistics

To define average in statistics, we can say that it is the average of the data set provided. We can find it by dividing the sum of all the values in the data set by the total number of values. 

For n values in a particular data set namely x1, x2, x3, … xn, the average can be found by:

x̄ = x1+ x2+ x3+ … +xn / n

For calculating the average of grouped data, we must calculate the class mark. 

Class mark= (Upper Limit + Lower Limit) / 2

After calculating the class mark, the average can be calculated similarly using the above formula by replacing xi with the class mark.

Let us take an example to understand it better. 

Example 1: 

Find the average value of the given set: 3, 4, 6, 7, 8

average = (3+4+6+7+8)/5 = 28/5=5.6

Example 2: 

Find the average of the first five prime numbers. 

As we know, the first five prime numbers are 2, 3, 5, 7, and 11. 

average = (2+3+5+7+11)/5 = 28/5= 5.6

Average Properties

Let us now look at some average properties to understand the concept better.

1. If all the numbers in a given set have the same value, k, then the average would also be k. For example: The average of the five numbers 12, 12, 12, 12, and 12 will be (12+12+12+12+12)/5 = 12. 

1. The algebraic sum of the deviations of a given set from their average is always zero. It can be stated as (x1−x̄)+(x2−x̄)+(x3−x̄)+…+(xn−x̄) = 0. For ungrouped data, it can be written as ∑(xi−x̄) = 0, and for grouped data, it can be written as ∑fi(xi−x̄) = 0.

1. If each number in a set decreases or increases by the same value, the average would decrease or increase by a similar value. Suppose the average of a set is x1, x2, x3 ……xn is X̄, then x1+k, x2+k, x3 +k ……xn+k will also be X̄+k. 

1. If each number in a set gets multiplied or divided by the same value, then the average would also be multiplied or divided by a similar value. If the average of a set  x1, x2, x3 ……xn is X̄, then x1/k, x2/k, x3/k ……xn/k will also be X̄/k. As for division, the fixed value must be a non-zero number since division by 0 does not give a defined number. 

Advantages of average 

The average is helpful in statistics, mathematics, economics, experimental science, sociology, and other similar disciplines. Here are some benefits of average: 

1. The formula for finding out the average is rigid and does not change based on the position of the value in any given set. The average is a more stable and rigid central tendency than the median. 

1. average is constituted by considering every value present in any given set. 

1. The formula for calculating the average is simple. Any person with basic addition and division skills can find out the average. 

1. Average provides valuable results irrespective of the size of the data set. It helps in the interpretation of a significant value set with ease. 

1. Average can be used for further mathematical operations, unlike other algebraic expressions like mode and median. 

1. Average also has applicability in geometry. For instance, the centroid coordinates of a triangle are also the average of the vertex coordinates. 

Disadvantages of average 

Along with advantages, there are also some disadvantages of average, such as:

1. One of the main disadvantages of average is that it gets affected by large values in the data set. For example, if the marks scored by different students in a particular exam are 10, 20, 30, 20, 30, and 90, the average is (10+20+30+20+30+90)/6 = 33.33 that is majorly affected by 90, an extreme value in the set. 

1. Average value can solely be used for quantitative data and not qualitative data such as honesty, hard work, etc. 

1. Average value cannot be calculated even if a single value is unknown since every value impacts the average. 

1. There are no averages to locate the average, either graphically or through inspection. 

1. Average cannot be found out in the case of open-ended classes without making a rough assumption of the class size. 

   Conclusion: 

   Average is an easy and valuable concept helpful in multiple disciplines such as mathematics, statistics, economics, and geometry. It is a useful mathematical operation used in everyday life to find the average and interpret data effectively. Ranging from weather statistics to the average marks secured by the students in particular subjects, all require using the average. 

   Along with practising average, you must also practice arithmetic progression and geometric progression to understand this mathematical concept thoroughly.