Average – its Formulas and Applications

Several formulae and numerical calculations are essential for data analysis. Average in mathematics is the calculation of the arithmetic mean of the central tendency of a group of numbers by adding all values and dividing the sum by the total number of values. The concept and the average formula are connected to mean, median, and mode. The statistical concept is used across several applications and hence helps with analysing the data. 

This article talks about the definition, use, and formulae for average and how it is used in overall calculations.

Finding the Average 

The average is the central point or the mid-point of a set of values, and hence it is expressed mathematically through a few calculations. The ratio of the summation of all the data or values to the number of units present is known as average. For example, 

There are numbers – 1 to 5, for which one needs to find the average. Hence it is calculated by 

1 + 2 + 3 + 4 + 5 / 5 = 15/5 = 3 

All the numbers from 1 to 5 are added together, and then the sum is divided by the total number of values. 

It is crucial to understand three necessary concepts for averages:

• Mean – It is the measure of central tendency, equal to the sum of all the values divided by the total number of values. For example, for finding the mean of 2, 4 and 3, it is (2+4+3)/3 = 3
• Median – Median is the middle value from a given list of numbers when arranged in an order. Whether it is an ascending or descending list, it is the middle number and helps calculate the average. For example, if the numbers are 2,4 and 3, and when the number is put in order (2,3,4), the number 3 is in the middle. However, when there are even numbers or values, the two middle numbers are added and the sum is divided by 2. The average of both the middle numbers is considered the median.
• Mode – It is the number present most of the time in a sequence of numbers. For example, if the numbers are 2, 3, 4, 3, 2, 3, and 3. As the number 3 occurs four times, hence it is the mode. 

Step by Step Calculation of the Average 

Average is calculated by using three simple steps, which include:

• The first step is adding all the numbers to find the sum of the numbers. 
• The second step is dividing the sum by the total observations.
• The third step is calculating the average by dividing the sum by the given dataset of observations. 

Calculation of the Average Velocity Formula

The average velocity formula for finding the mean value of a group of items or numbers:

Average = Sum of values/Number of values

If there are “n” number of values like x1, x2, x3, x4, x5, …… xn, the average of the given data would be equal to:

 (x1 + x2 + x3 + x4 + x5 …… xn ) / n 

The average velocity is the average value of the known velocities. It is represented as the total displacement of a body in time “t”. The following shows the average velocity formula: 

Average Velocity = Total displacement / Total time

For example, finding the average velocity for any distance Xi and Xf with corresponding time intervals t1 and t2 is represented by using the average

Vav=Xf – Xi / t2 – t1 

where 

Xi = Initial Distance

Xf =Final distance

t1 = Initial time

t2 = Final time

Types of Averages 

For data analysis, the averages are divided into two types: 

• Mathematical average consists of the mean, and further, it can be divided into arithmetic mean, geometric mean and harmonic mean. The arithmetic mean is calculated by adding all the numbers and dividing the sum by the total number of observations in the dataset.
• The geometric mean is the central tendency of a set of numbers by using the nth root of all the products. The harmonic mean is calculated by dividing the numbers in a series by the sum of reciprocals of each of the numbers. 
• Positional average consists of median and mode. The median is the middle number from a series. Mode is the number present maximum times in a sequence of numbers.

Application and Use of the Average

• Businesses and companies have to calculate many values. The average calculation helps find information like average sales, salary, wages paid, increase in manufacturing, etc. 
• It helps calculate information like the average time required for performing particular activities or the time needed for planning or completing a task. 
• It is significant in a research company, investor or stock market. It helps find the performance of a company’s stock compared to the previous year. Stock market brokers frequently use it. 

Conclusion 

The average calculation is crucial for statistical and data analysis on a day-to-day basis. The implementation and understanding of the average formula are significant in data analysis and making necessary calculations. The average value is majorly considered when analysing a large database. The combination of mean, median and mode is necessary for calculating the average formula, and it is the measure of central tendency and is significant for data analysis.