Hello everyone, welcome back to a new video
In today's video, we will learn about Averages
What is Averages?
Averages are
You must have heard a word in Maths in your childhood
classes
Means when you were studying in school
Mean, Median, Mod
By doing this, you must have read a chapter in Stats
Mean, Median and Mod to find out
So, in that you used to find out Mean
When did you find out Mean?
Suppose, you are given a group of data
Or a collection of data
Suppose, you are given a data
Means, you are given a survey of anything
Suppose, you are given marks
So, marks are also a kind of data
Marks of the students
So, if we had to find out the mean of that data
So, how did we find out its mean?
As we find out its mean
What do we call it?
In other words, we can call it Average
Like, if we are given a set of data
So, some marks are given
Suppose, we are given marks of 10 students
We have to find out
Suppose, 10 students study in one tuition
So, in this tuition
What are the average marks of all the students?
Means, what do they get?
In which range do they vary?
So, how can we find out that?
We can find out its mean
We can find out its average
So, if we are given marks of 10 students
So, what do we have to do?
We have to sum all the marks of those students
And how do we divide it?
We have to divide it by the total number of students
That is total number of observations
We have to divide it by the total number of entries
So, what is the mean?
Average So, what is the mean?
Whatever data we are given
We have to sum it all
Then, what do we do?
We have to count all the things that we have summed
How many things are there?
And what do we do with it?
We will divide it by that
So, the simple formula of Average
What are we doing now?
We will not do Average in detail in this video
I will just give you
I will give you a small overview
What is its formula?
And I will tell you one or two cases
When the rest of the units will start
Like work and time
Or speed
In speed, you will get the concept of average speed
In age, you will get the concept of average
So, we will do it in more detail
We will do its questions
How to do detailed questions
How to do tough questions
With the help of Average
So, first of all, you should know the basic definition of
Average
So, suppose we have a data
Or you can call it as an observation
What data do we have?
We have one child
A, B, C, D
Suppose we have four children
They will have some marks
So, we are making data of their marks
Suppose marks of A are a1, b marks are a2, c marks are a3
and d marks are a4
We will have some numbers
And what are they?
A, B, C, D are children
So, if we have to find their average marks
Average marks
What will we find the average of?
We will find the average of the data which we are dividing
Suppose we have data of marks
So, we will find the average of the data which we are
dividing
So, what will happen?
First of all, its formula is
Sum of all the observation
Divided by total number of observation
This is the basic formula of Average
Sum of all the observation
Divided by total number of observation
Suppose we have marks of these four children
Then, count the number of observation or elements 1, 2, 3,
4, A, B, C, D
So, we have four children
Divided by total number of observation
Sum of all observation
Means we have to sum of their marks
Sum of all observation
Divided by total number of observation
So, what will be the average marks?
Means the average marks of the class
Suppose, the result of a child
So, he got marks of five subjects
If we see the average marks of the child
It is not necessary that all the subjects have good marks
or bad marks
As soon as the marks come, we have to find the average
marks of the child
So, how will we do that?
Sum of all the subjects
Divided by all subjects
So, what will be the average marks of the child?
Now, you will say that if the average marks are like this
Then, will this be the percentage of the child?
No, it will not be the percentage
Percentage is a different concept
And, Average is a different concept
You have to correlate both the concepts
But, do not mix up
Both the concepts will be different
If you find the average marks of a child
It will be different
And, if you find the percentage of the child
Then, the percentage will be different
Then, you will find the formula of average
So, let's move on
Suppose, we have this data
We had the data
Let me tell you one more form of Sum of All Observations
If we are summing something
Then, it has a notation
What is that notation?
For example, if I sum up A1 plus A2 plus A3 up to A4
What did I write?
A1 plus A2 plus A3
Suppose, it is not going up to A4
It is going up to A100
We will not write it like this
Let's write it like this
Suppose, it goes further
Then, we can write it in another way
With the summation sign
What do we call this?
Summation sign
If you rotate M or W, it will look like this
So, this is the summation sign
This is our summation
Summation means we are summing
What are we summing?
A1, A2
I have denoted all these with A
Particularly, why?
So that, it is easy for me to tell you
So, we can write it as summation small a
And, what is happening?
If I write it as I
Like, in the subscript, it is written as 1, 2, 3 and 100
If I write I here
In the subscript
Then, we will tell the range of I in the summation
That, where is I from?
It is from 1 to 100
This lower limit will come down and our upper limit will
come up
So, what will this sign denote?
If this is written in full
Then, what will this sign denote?
This will denote that
If we put the value of A1, I
A1 Summation means we have to sum
So, between all these entries
A plus sign should be put
So, first A1
Then, increase one more
After A1, A2
Then, A3 and up to
How far will it go?
That sum will go up to A100
Because, what is the upper limit of I? 100 So, this is done
How do we write summation sign?
If we have to write sum
Then, summation sign is this
You should know this for the future
So, if we write average
Then, what will be the average?
Summation AI Divided by total number of observations
Total number of observations
Let us suppose, we denote it from n
That, we will have n number of observations
Like, what was the value of n in this?
It was 4
Because, there were 4 students
So, total number of observations was 4
So, what was the value of n?
It was 4
Next, we will move on to
If we have cases
Suppose, we have cases
We have an average of a particular thing
average is it a good way to do it.
So, I will do it by 10.
See, we got this as equal to this.
This was only one value, means the initial average which we
got, this was its value.
So, we got the value of initial average, previous average
or initial average.
And this 5, what was 5? 5 was our increment.
So, what was 5? 5 was our increment.
And increment was not only in one term, it was in all terms.
That's why we could take it as common.
So, if we have a case where we have a set of data, in which
every observation, what is happening to a finite number? It
is increasing.
It is increasing.
So, what will be its formula? So, we can write it as, then
new average would be previous average plus, we did 5, what
was 5? It was our increment.
Increment in each of the term.
So, what will be our new average? When increment is done?
In each one of the data, each one of the observation.
In every observation, if we increment, then the formula
will be this.
Similarly, if we are incrementing, then we can also
decrement with the same number.
Or with the same number, we can decrease all the
observations.
So, what will happen in that? Here we did plus increment.
What will happen in that? We will minus from each one.
So, what will happen in that? If this is your first case,
when increment is done.
So, suppose increment is done in each of the observation.
OBS means short form of observation.
When same, we are doing decrement, but what should that
decrement be? It should be same.
Not that you are minusing different numbers.
Same number should be subtracted.
So, when same decrement is done in each of the following
observation, then what will be our new average? Then what
will be our new average? Previous average minus decrement.
Whatever we are doing
in that, suppose if we are doing 15 minus in each
observation, then we will do minus 15.
So, our new average will come.
Now, when you are doing sum, subtracting, then what is the
harm in multiplying and dividing? We can do that also.
If we are doing any set of data with same number,
multiplying and dividing, then what will happen? Same will
happen.
If you are adding increment, then do plus.
If you are doing decrement, then do minus decrement.
If you are multiplying, then do into that thing in previous
average. And if you are dividing, then what should you do
in previous average? Divide.
So, third or fourth case also you will write in the same
way.
I will tell you in this only.
I will write in next only for you.
Third, when a same number is multiplied with each of the
observation.
With each of the observation, we will multiply that same
number with each of the observation.
Then our new average will be, previous average, we are
multiplying, so with that same thing, suppose we have that
same number, you take that as capital N.
So, then what should we do with that same number? Multiply.
What should we do with previous
average? Multiply.
What should we do with that same number? Multiply.
We will get new average.
Now, same fourth case, when a same number is divided with
each of the observation.
Means, we are dividing same number with each of the
observation.
So, new average would be previous average divided by,
suppose we are dividing each of the observation with
capital M.
So, divided by M.
What should we do with that number? We have old observation
or initial observation, sorry, we have initial average.
Divide that number with that number which you are dividing
everyone.
So, it was easy that if we are multiplying or dividing same
thing with everyone, then we don't have to do anything.
We have to do same thing in average also.
Next, we will do a question before we do the next rule.
This question.
Read all of them.
The average marks of 50 students in a class is 72.
So, keep on writing what you are getting.
Average marks.
How many are given? 72 How many students are given? 50
students.
Till now we are talking about 50 students.
So, what is n?
We have 50.
Total number of students will be 50.
The average marks of boys and girls in that subject are 70
and 75 respectively.
Find the number of boys in the class.
First of all, if we have a class.
What will be there in the class? The class will be made by
boys and girls.
If it is a co-ed school.
If it is a boys then boys will be made by boys and girls
will be made by girls.
But, we are talking about co-ed.
So, it will be made by boys and girls.
So, we have been given boys and girls.
So, what are the average marks of boys in that subject?
Average.
First of all, we have written the average marks of the
whole class.
Average marks of whole class.
That is 72.
The next thing which we have been given is the average
marks of boys.
The average of boys is 70.
Similarly, what is the average marks of girls?
That is 75.
Suppose, we are denoting this with a.
Write b in the subscript.
That will be of boys.
And what will be the result of a and g?
A and g will be the average marks of girls.
So, what we have to find out is the number of boys.
So, whatever we have to find out.
What do we take with us?
We take x with us.
So, what do you take as let number of boys? x.
Let number of boys be x.
How many boys do we have? x in numbers.
If a class has 50 students.
And we have x boys in that class.
So, how many girls will be there?
If we have 30 boys in a class.
So, how many boys will be there?
The total number of boys is 50.
So, 50 minus 30.
If we subtract boys from the total number of boys.
The total number of girls will be there.
So, what will we find out from this?
We will find out the total number of girls.
We have taken boys as let.
So, what will be the total number of girls?
The number of girls would be 50 total students minus number
of boys that is x.
This will be the total number of girls in this class.
50 minus x.
So, we have the number of boys and number of girls.
When we have the number of boys and number of girls.
We have their average.
So, put up the formula of average.
What will be the formula of average? The formula of average
will be.
If we are looking at the average marks of the whole class.
The sum of all the marks divided by 50.
That will be the average marks of the whole class.
If we are looking at the average of boys and boys.
What will be the sum of marks of boys students?
The sum of marks of boys students.
Let us suppose that we have the sum of marks of boys
students.
Mb Let us suppose that we have the sum of marks of boys
students. Mb So, what will we do?
If we are taking out the sum of boys students.
We will count the quantity of boys students.
How many boys students were there?
Boys were x.
So, it will be divided by x.
So, what will be the marks of boys students?
The sum of marks of all the boys students will be.
The average of boys students will be multiplied by x.
The average of boys students will be multiplied by 50.
So, 50 multiplied by x.
What will we get?
The sum of marks of all the boys students.
The sum of marks of all the boys students will be 70x.
What is x in this?
How many boys students are there in that class?
That is x.
Similarly, you can find out the marks of girls students.
Find out the sum of marks of girls students.
What will be the average?
The average will be 75 for girls students.
The number of girls students will be 50 minus x.
So, take out the sum of girls students.
The average of girls students will be the sum of all the
marks secured by girls.
So, what will be the sum of marks of girls students?
Denote it as mg.
Divide it by 50 minus x.
Don't get confused by looking at these marks.
This is just a notation.
Why am I writing this small b? So that you will know that
this is for boys.
Why am I writing this g?
So that you will know that this is for girls.
Otherwise, you can give any other name.
You can give b or g.
But, you should be clear about the letter you are denoting.
That's it.
Now, what is the average of girls? The average will be 75.
So, multiply it by 50 minus x.
This will be the sum of marks of all the girls in the class.
This is the sum of marks of all the girls in the class.
Now, use your brain and find out the value of x.
Because, we have to find out the number of boys.
So, how will we do this?
So, we will do this.
Average marks is equal to 72.
We had to use this.
So, if the average marks of the entire class is 72.
The average marks of 72 were of the whole class.
How many children were there in the whole class?
The number of observations were 50.
So, if we have a class.
The average of the class is divided into boys and girls.
So, the sum of marks will be the sum of all the marks.
The sum of marks will be the sum of all the marks.
The total number of students will be 50.
Now, if this is the sum of marks of all the students in the
class.
The sum of all the students in the class.
Please correct this in English.
What is this?
Sum of marks of all the students.
Now, if we segregate the sum of marks of all the students.
So, if we do the sum of boys and girls.
And, if we add them together.
Then, what will happen?
The sum of marks of the entire class will be the same.
Then, what will happen?
Nothing will happen.
The sum of all the marks of boys will be the same.
We had denoted it as MB.
Plus, the sum of all the marks of girls.
We had denoted it as MG.
So, if we do the sum of all the marks.
Then, we get the sum of all the marks.
Divided by 50.
The total number of students.
The sum of marks of girls and boys is 50.
So, we have written it in the divide.
Now, we don't have to do anything.
We just have to put up the value.
What should you do? Put up the value.
And whose formula have we written?
Average.
So, what is the average of the total class? 72.
So, whose equal will this be? 72.
Solve it.
So, 72 is equal to...
What was MB with us? See.
MB was 70X.
70X plus...
What was MG? 75 into 50 minus X.
So, 75 into 50 minus X divided by...
Whole divided by 50.
If you want to solve it fastly, then 72 into 50...
50 will go here.
Is equal to 70X.
Solve the like terms together.
Then separate out the X and find its value.
Plus 75 into 50.
5 x 5 is 25.
5 x 35 is 37.
3 x 7 is 50.
Minus 75X.
So, these like terms are 70X minus 75X.
Solve these two.
What will it be?
Minus 5X.
And take this here.
Solve this also.
What will it be? 72 into 5.
That is 10.
35 x 3 is 3600.
So, this side will be 3600.
3600.
Here was plus. Here was minus.
3750 is equal to minus 5X.
Solve this.
What will be your minus? 150.
This would be minus 150.
Is equal to minus 5X.
Minus cancels.
5 and 150 will be 30.
So, what will be the value of X?
The value of X is 30.
And whose value was X?
The number of boys in the class.
We had to find out that.
So, total number of boys in the class.
What will be the answer? 30 is the answer.
So, this was our basic definition of average related
question.
And what did we learn today?
Definition of average.
In the next class, we will learn more particular cases of
average.
We will see their formulas.
And based on them, we will do questions.
Thank you.
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