Average: Definition, Formulas, Questions and More

An Average is a single number that expresses a bunch of numbers in simple terms. The Arithmetic mean of a group of numbers is commonly referred to as average in mathematics. The arithmetic mean is a measure of a group’s central tendency that is calculated by adding all of the values together and then dividing the total by the number of values.

However, other concepts, such as median and mode, are also used to find the average of a set of values. It is a relatively simple statistical concept with a wide range of applications in many fields. The equation below is one of the most commonly used definitions of the average:

Average\({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{Terms}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{Terms}}}}\)

Read on to learn more about the proper definition of average, formulas, examples and more.

Definition of Average

The term average refers to the mid or the central point of a set of values. Mathematically, it is defined as the ratio of the summation of all the data to the number of units present in the list. In other words, to find the average of a set of data, add up all the values and then divide this total by the number of values. In addition, the mathematical symbol or notation for average is \( \overline x\).

Also, mathematicians have realized that there are three useful ways of looking at the average of a set of data. They are called the mean (the other terminology used instead of average at times), the median, and the mode of the data. For most mathematical applications, the term average indicates the mean.

How Average is Calculated?

What does it mean to calculate the average? If technically speaking, we are taking the sum of the values and dividing by how many numbers are there. But in terms of the real world, it is more like distributing the values of the entire set evenly among its numbers and then going one step back to see at what value the numbers all ended up.

Sounds confusing?

Let us understand it with a very simple and easy illustration.

Teacher: Children, come one by one and pick up some chocolates from the carton box.
Priya: I have got \({\rm{10}}\) chocolates.
Preethu: I have got \({\rm{4}}\) chocolates.
Rachna: I have got \({\rm{1}}\) chocolate.
Teacher: Don’t worry, my dear kids! Let me distribute these 30 chocolates equally among you all.

So, the teacher divided \({\rm{30}}\) by \({\rm{3}}{\rm{.}}\)
\({\rm{30}} \div {\rm{3 = 10}}\)

Thus, we observe that on average, each child gets \(10\) chocolates.

Now let’s look at another example

The following diagram shows strips of equal-sized squares.

Now, re-arrange the strips of squares so that there is the same number of squares in each strip.

Now, if we check out the first diagram, the total number of squares is \(15\) and the number of strips is \(5.\)
Thus, the number of squares in each strip of the second diagram is

\({\rm{ = }}\frac{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{squares}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{strips}}}}{\rm{ = }}\frac{{{\rm{15}}}}{{\rm{5}}}{\rm{ = 3,}}\)

which gives mean of squares in each strip.

Average\({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{all}}\,{\rm{the}}\,{\rm{values}}\,{\rm{or}}\,{\rm{observations}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{values}}\,{\rm{or}}\,{\rm{observations}}}}\)

Relevance and Uses of Average Formula

Average is useful because it summarises a large amount of data into a single value and, thus, indicates that there is some variability around this single value within the original data. Some of the major application of average in the real-life world is given below:

1. Many companies use average to find out their average sales and average manufacturing of the product, average salary, and wages paid to labour and employees.
2. WHO (World Health Organisation) also uses an average to know the yearly death and birth rate for a particular year.
3. Average can be used to plan daily schedules for children to ensure sufficient time is provided for studies as well as extracurricular activities.
4. Average is used in the research company to know the average traded price of a particular stock.
5. Many investors use an average of stock from the previous year for making new investments.
6. Average also have relevance in the stock market because it is used by analyst frequently.

What is the Abbreviation of Average?

In Mathematical terms, avg stands for average which is also termed as mean calculated by dividing the sum of all the values by the number of values.
Avg \({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{Terms}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{Terms}}}}\)

How to Find Average of Negative Numbers?

Negative integers are a part of the integers family. Negative integers are usually written with a minus sign in front of whole numbers. In case there are negative numbers present in the list, then also we apply the same formula to find the average. Let us understand it with the help of an example.

Find the average of \({\rm{7,}}\,{\rm{ – 10,}}\,{\rm{5,}}\,{\rm{20,}}\,{\rm{ – 2}}{\rm{.}}\)

As we have learned from the formula of average, the average is equal to the sum of all the values divided by the number of values.
So, the sum of these numbers
\( = 7 + \left( { – 10} \right) + 5 + 20 + \left( { – 2} \right)\)
\( = 7 – 10 + 5 + 20 – 2\)
\( = 20\)
Total units \( = 5\)
Hence, average \( = \frac{{20}}{5} = 4.\)

What are the Types of Averages?

Technically speaking, the average can be divided into \(2\) categories.
1. Mathematical average
2. Positional average
Check out the below figure to understand the further classification of mathematical average and positional average.

Solved Examples

Q.1. A batsman scored \(51,\,47,\,39,\,55,\,60\) and \(48\). Find the average runs scored by him.
Ans: Sum of runs scored by the batsman in all the \(6\) innings
\( = 51 + 47 + 39 + 55 + 60 + 48\)
\( = 300\)
Number of innings played by the batsman \( = 6\)
Average\({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{all}}\,{\rm{the}}\,{\rm{runs}}\,{\rm{scored}}\,{\rm{by}}\,{\rm{the}}\,{\rm{batsman}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{innings}}\,{\rm{played}}\,{\rm{by}}\,{\rm{the}}\,{\rm{batsman}}}}\)
Average \( = \frac{{51 + 47 + 39 + 55 + 60 + 48}}{6} = 50\)
Hence, the required average of the batsman is equal to \(50.\)

Q.2. Find the mean of first \(6\) multiples of \(5\).
Ans: The first \(6\) multiples of \(5\) are \(5,\,10,\,15,\,20,\,25\)and \(30.\)
Sum of these multiples \(= 5 + 10 + 15 + 20 + 25 + 30 = 105\)
Number of multiples \( = 6\)
Average\({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{6}}\,{\rm{multiples}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{multiples}}}}\)
Average \( = \frac{{5 + 10 + 15 + 20 + 25 + 30}}{6}\)
Average \( = \frac{{105}}{6} = 17.5\)
Hence, the average of first \(6\) multiples of \(5\) is equal to \(17.5.\)

Q.3. Find the value of \( x\) if the average of \(10,12,13,x\) and \(17\) is \(14\).
Ans: We know that average \({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{data}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{data}}}}\)
\(14 = \frac{{10 + 12 + 13 + x + 17}}{5}\)
\(14 = \frac{{52 + x}}{5}\)
\(40 \times 5 = 52 + x\)
\(70 = 52 + x\)
\(x = 70 – 52 = 18.\)

Q.4. If the mean of \(y + 2,y + 4,y + 6,y + 8\) and \(y + 10\) is \(13\), find the value of \(x\).
Ans: Given that the mean of \(y + 2,\,y + 4,\,y + 6,\,y + 8\) and \(y + 10\) is \(13.\)
Mean \({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{numbers}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{numbers}}}}\)
\({\rm{13 = }}\frac{{y + 2 + y + 4 + y + 6 + y + 8 + y + 10}}{5}\)
On further calculation, we get
\(13 \times 5 = 5\,y + 30\)
\(65 = 5\,y + 30\)
\(5\,y = 65 – 30\)
\(5\,y = 35\)
\(y = 7\)

Q.5. Weight in kg of \(7\) boys in a locality are \({\rm{47}}\,{\rm{kg,}}\,{\rm{55}}\,{\rm{kg,}}\,{\rm{51}}\,{\rm{kg,}}\,{\rm{50}}\,{\rm{kg,}}\,{\rm{44}}\,{\rm{kg,}}\,{\rm{49}}\,{\rm{kg}}\) and \(54\,{\rm{kg}}{\rm{.}}\) Find their mean weight.
Ans: Weight of \({\rm{7}}\) boys in a locality \({\rm{=47}}\,{\rm{kg,}}\,{\rm{55}}\,{\rm{kg,}}\,{\rm{51}}\,{\rm{kg,}}\,{\rm{50}}\,{\rm{kg,}}\,{\rm{44}}\,{\rm{kg,}}\,{\rm{49}}\,{\rm{kg}}\) and \(54\,{\rm{kg}}{\rm{.}}\)
Average \({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{weight}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{boys}}}}\)
Average \(= \frac{{47 + 55 + 51 + 50 + 44 + 49 + 54}}{7} = \frac{{350}}{7}\)
Average \({\rm{ = 50}}\,{\rm{kg}}\)
Hence, the average weight of the boys is equal to \({\rm{ = 50}}\,{\rm{kg}}{\rm{.}}\)

Summary

In this article, we learned about the definition of average. We also learned about the formula to find the average. We learned to find the average of a negative number and became aware of the uses of average in real life. We also learned the different types of average.

FAQs

Below are the frequently asked questions on Average:

Q.1. What type of average is best?
Ans: Median is the best average to be used when we want to divide the data set into two equal groups.

Q.2. What is average in simple words?
Ans: In simple maths average means to add up all the values and then divide this total by the number of values.

Q.3. Which is the number that appears the most often in a given data set?
Ans: The mode is the number that appears the most.

Q.4. How average is calculated?
Ans: In simple language, the average can be calculated with the help of the formula given below.
Average \({\rm{=}}\frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{all}}\,{\rm{the}}\,{\rm{values}}\,{\rm{or}}\,{\rm{observations}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{values}}\,{\rm{or}}\,{\rm{observations}}}}\)

Q.5. What are the 3 types of averages?
Ans: In mathematical average, we call average as mean, and thus the three types of mean are:
1. Arithmetic mean
2. Geometric mean
3. Harmonic mean

Q.6. What is average in maths?
Ans: It is defined as the ratio of summation of all the data to the number of units present in the list. It can also be defined as the mean value, which is equal to the ratio of the sum of numbers in the given set to the total number of values present in the set.

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