In this article we are going to discuss about Average,you would often hear this term from people around you. folks generally say average temperature, average marks, average height, average salary etc.
Before we get into the definition of Average let's start with an example that will make you familiar with the concept in easier way.
Before we get into the definition of Average let's start with an example that will make you familiar with the concept in easier way.
Example 1 : Alex brings 5 identical balls of red colour for a total of £50 ( Ball 1- £12 , Ball 2 - £8 , Ball 3- £10 , Ball 4 - £17 ,Ball 5 - £13 ) and each of them has different price. when he reaches his home he forgets which ball corresponds to which price. His friend joy comes to him and asks for a ball. Now at what price should Alex sell one ball so that he makes no profit no loss ?
The problem above can be very easily solved if you have even a basic idea of Average.
Alex payed £50 for 5 balls.
Average Price for one Ball = £50/5
= £10/Ball
Alex should sell one ball for £10 .
Example 2 : Doe was asked to record temperature from Monday to Thursday But on Tuesday he was out of town, so couldn't record the temperature for that particular day. The temperature for other days Monday- 36°C, Wednesday- 37°C, Thursday- 34°C.
He wanted to know the approximate temperature for Tuesday So he calculated the average temperature for the 3 days which was
= 36+37+34
= 107/3
Average Temperature for one day = 35.6 °C
So the approximate temperature on Tuesday could be 35.7°C
If the examples are clear to you let's dive into it .
What is average
The mathematical expression to calculate the average is :
Average = \( \frac{\text{Total sum of quantities}}{\text{Number of quantities}} \)
Or
Average = \( \frac{\text{Total sum}}{\text{Total numbers}} \)
Let's put the formula into examples to calculate the average.
Example 3 : The heights of 4 students are as follows 5 feet ,5 feet,6 feet and 8 feet.What is the average height?
Sum of all students' height = 5 feet + 5 feet + 6 feet + 8 feet.
= 20 feet
Number of students. = 4
Average height = 20 feet/ 4
= 5 feet
Example 4 : Temperature for a particular week was recorded in London.the data for temperature from Monday to Sunday is as follows 21°c, 34°c, 30°c, 24°c, 29°c, 28°c , 26°c
Total sum of temperature = 21°c +34°c + 30°c +24°c + 29°c + 28°c + 26°c
= 192°c
Total number of days for which the temperature was recorded = 7
Average temperature = 192/7
= 27.4 °c
How to calculate Weighted average
The Weighted average is generally given by the following formula
If \( N_1 ,N_2 \) and \( N_3 \) have average \( A_1 ,A_2 \) and \( A_3 \) respectively then the Weighted average is:
Weighted average = \( \frac{W_1 + W_2 + W_3}{N_1 + N_2 + N_3} \)
= \( \frac{N_1 \times A_1 + N_2 \times A_2 + N_3 \times A_3}{N_1 + N_2 + N_3} \)
Let's take an example to explain it.
Consider average marks of 6 students out from different classes. Average marks of Ena and joe from class 4 is 60 and Average marks of John and Ray is from class 6 is 44.Average marks of Carl and Lina from class 8 is 52. Find the Weighted average
Sum of the marks of Ena and Joe = 2×60
=120
Sum of the marks of John and Ray
= 2×44
=88
Sum of the marks of Carl and Lina
= 2×52
=104
Weighted average of the marks of 6 students
= \( \frac{120 + 88 + 104}{6} \)
= \( \frac{312}{6} \)
= 52
Example : In a tech company,There are two teams namely green team and red team.The average salary of all 5 workers working in the green team is 40 k and salary of red team workers is 50 k and the weighted average salary of all employees is 45 k find the number of employees in the Red team.
Ans : Let the number of workers in the Red team= x
Weighted average = \( \frac{5 \times 40 + 50 \times x}{5 + x} \)
45 = \( \frac{200 + 50x}{5 + x} \)
225 + 45 x = 200+50 x
5 x = 25
x = 5
Number of workers in red team = 5.
Tricks Related to Average
Trick 1: The average of first n Natural number is found by the following formula:
Average of first n natural Numbers = \( \frac{n+1}{2} \)
Example: Find the sum of first 6 natural Numbers .
Ans : let's solve it through both traditional method and formula one.
The first 6 natural Numbers are : 1,2,3,4,5,6
Average = Sum of all numbers/ total numbers
Average = (1+2+3+4+5+6)/6
= 21/6 = 3.5
Formula Method: The average of first 6 natural Numbers (Here n =6)
Average = (n+1)/2 = (6+1)/2 = 3.5
Trick 2: If the given natural numbers are not consecutive but they form an Arithmatic Progression then we can quickly find the average with the help of this formula :
Average = \( \frac{\text{First term + Last term}}{2} \)
Consider this A.P: 3,6,9, 12 , 15, ... 45
we can easily observe theat it is an A.P with common difference \( d = a_2 -a_1\)
= 6-3 =3
First Term \(a_1\) of this A.P = 3
Last Term (l) of this A.P = 45
Average = \( \frac{\text{a + l}}{2} \)
= \( \frac{\text{3 + 45}}{2} \)
= \( \frac{\text{48}}{2} \)
= \( 24 \)
You will get the same answer if you apply the traditional method to solve it .
Average FAQs
Q1 : How to find the average when one Number is removed from the data set and old /Orginal average is given?
Ans : Supppose there are n elements in the data set and Let the original average is A and number removed is k the the new average is :
\( \text{New average} = \frac{nA - k}{n - 1} \)
Q2: How to find the average when a new number is added?
Ans : If the average of n elements is A and a new element added is k then the new average is is :
\( \text{New average} = \frac{nA + k}{n + 1} \)
Q3: If the given numbers are in A.P and we have to find their average then their average and Medium will be the same ?
Ans: Yes if the given numbers are in A.P then their average and Median are always equal.
Q4: How to calculate the average when each number in the data set is multipy by a number n?
Ans: When each number of a data set is multiply by the number n then the average is also multiplied by n ,Therefore the new average is :
New average = \( \text{Original average} \times n \)