Hello everyone, welcome back to another video on averages
In last average video, we have learnt the basic definition
of average and related to that we have done some questions
and types like person joins a group or leaves a group or
replaces a group
So today we will do the concept of averages that is how to
calculate averages on numbers means if we have a series of
consecutive numbers in that form so in that we will learn
how to calculate averages
So first of all you should remember the definition of
average of last class that sum first, average is equal to
sum of all the observations upon total number of
observations
So if I write the formula of average in front of you then
average is equal to sum upon total number of observations
where S denotes sum and N denotes total number of
observations
Let us suppose we have been given an average in a question
Average is given and N is given
We need sum to calculate sum
So we can use the help of this formula
How will we calculate?
Sum will be our average into total number of observations
So with the help of this we can calculate sum
Now you will go ahead and read sequence and series
So in series you will find that we do sum of numbers
First we write them in sequence and then we do sum of
numbers
Now you will go ahead and see arithmetic progression
In that also we find the formula of sum
So today we will see the formula
How it is derived and how it is not
We will leave that for those classes
Today we will just remember the formula of sum and apply it
in the formula of average
That's it
So let us see related to this
Till now what we have done is
We had a set of data and we were given the number of
observations
Let us suppose we have been given some data and in that
some observations are being repeated
For example, one observation is repeated 5 times and
another 7 times and some are repeated 10 times
So will you add them by adding them?
No Now what we will do is
As we have this
We have to find out 4 plus 4 plus 4 plus up to
How many times we have to add 4? 20 times
So you will do it one by one 4 plus 4 is 8 and then plus 4
is 12
What will you do directly?
You will do 4 into 20 directly
Then it will give the sum of all these
What will it be? 80
Similarly, if you add 4 20 times
Still 80 comes
But what has happened?
It has become convenient
It has saved our time
So if I give you a topic related to this
Suppose we have a set of observations
Let A1, A2, A3 up to so on
Means add up to AN
Now if there is any observation in this and I will move it
forward
AN, A1, A1, A1
Similarly, how many we have? 1, 2, 3, 4, 5, 6 6 times A1
and take A3 3 times A3
This is also an observation and Someone else can repeat in
the future
For example, A6 has been repeated 3 times 1 is here 1, 2, 3
So it has been repeated 3 times
So how will we find the average of all these?
Their average will be
What will be their average?
How many times it is repeated?
Means How many times it is repeated?
What do we call it?
What is its frequency?
How frequently is it repeated?
So its name is frequency
For example, A1
How many times is A1 coming to us? 1, 2, 3, 4, 5, 6
It is coming 6 times
So what is its frequency?
Frequency of A1
What is it?
It is 6
Means frequency is nothing
It just tells how many times it is repeated
So A1's frequency is 6 times
Let us suppose
We have given it as F1
We have denoted it as F1
F1 means frequency of A1
Similarly What is the frequency of A2?
A2 is coming only once
So what is the frequency of A2? 1 How do we denote the
frequency of A2?
We denote it as F2
F2 is 1
Now how many times is A3 repeated?
It is repeated 4 times
So F3 means frequency of 3 is 4
Similarly, we will denote the frequency of all
Let us suppose we have this up to n
So what we have to do for n?
For each individual entry
Observation Means for n observations
We have to see different frequencies
That how many times it is repeated
All of them will be repeated
In case some of them will be repeated
So we have to denote them
What will be the frequency of others?
It will be 1
So what will be the average?
You have to multiply the frequency
Like we did 4 x 20
That how many times it is added
Similarly, we have to find the average of A1
We have to find the sum
It will be 6 times plus
So instead of writing it as 6 times add
Means A1 plus A1 plus A1
Instead of writing it as 6 times
What can you write?
You can directly write 6 x A1
So we will write this
So what is 6?
It is the frequency of A1
So we will write it as F1 x A1
Plus F2 x A2
Means add the frequency of A2
How long will this go on?
Till Fn x An
Fn x An
Whatever is our last term
Till there
Whatever is our last observation
And how will we divide?
Whatever is the total number of observations
Total number of observations
So how will you add the total number of observations?
Total number of observations
How will you add?
You count them
We will have n
Its denotation will be done
We will denote it with n
So when we have any observation
How many number of times it is repeating?
It is with a certain frequency
So we will put the average in it
This formula
This formula of average
Sum of frequency
Frequency x observation
And the whole sum divided by
Total number of observations
Second We will move ahead
The topic is
Related to natural number
Related to series of natural number
Let us suppose
You will go ahead and read
That we have
What is the sum of first and natural numbers?
First and natural number
What will we have? n natural number
If we have given any finite n
Then natural number up to that
Suppose we are told first 15 natural number
Then what will be the value of n? 15 So we have to take the
sum of
Natural number up to 1 to 15
So first n natural number
What is this new topic?
Average Of Of First n Natural First n natural number
First n consecutive It will be consecutive because we are
talking about first n
Or Consecutive n natural number
One and the same thing
So first
What will be the sum?
First we have to take the sum of first n natural number Sum
Divided by
Total numbers
How many numbers are there? 1 to n
Total number of observations
We have n
Sum by n
The formula of sum
First n natural number
What will be the sum? 1 plus 2 plus 3 plus 4 plus 5
Up to
This sum will go up to n
So the formula of this sum
We have n plus 1
Divided by 2
This is the formula of
Sum of first n natural number
This is the formula
If we put up in the formula of average
So see
Divide by n
So this will be n plus 1
Divided by 2
This will be
Average of first n natural number
This will be n plus 1 divided by 2
Similarly What can you get?
First n natural number
This is consecutive
You can get
Even consecutive numbers
Odd consecutive natural numbers
Now we are doing only natural numbers
What will be their sum?
Now they get second
Square Cube These are the general standard results
Which we have to remember in this series
you will be able to easily calculate the sum of first and
natural number, sum of odd terms, sum of even terms.
So, with the help of this you will be able to easily
calculate the average.
So, next see, next what will happen, this was your first
average of first and natural numbers.
Second see, second if we have, we have to find the average
of their square, sum of squares of first and natural
numbers.
So, what will happen, 1 square plus 2 square plus 3 square
plus 4 square plus up to n square.
Their sum we have, their sum we have n, n plus 1, n plus 1
divided by 2, divided by 6 sorry, it will come divided by 6.
This will be the sum of the series of squares.
So, if we find its average, what will be the average?
Average will be sum of all the numbers n, n plus 1, n plus
2, n plus 3, n plus 4, n plus 5, n plus 6, n plus 7, n plus
8, n plus 9, n plus 10, n plus 11, n plus 12, n plus 13, n
plus 14, n plus 15,
n plus 16, n plus 17, n plus 18, n plus 19, n plus 20, n
plus 22, n plus 23, n plus 24.
So, what will be the average? Average will be sum divided
by n, then n will cancel with n.
So, what will be the result we will have? n plus 1, 2n plus
1 divided by 6.
This is done, we have sum of squares of first and natural
numbers or consecutive numbers.
Now, we will see the cube.
Cube is similarly sum of natural numbers.
Like the sum of cube, what was their sum? n, n plus 1
divided by 2.
What will be the sum of cube series? Its whole square will
be the sum of cube series.
So, with the help of this we will find the average.
So, write the third point.
What is the third point? If we find their cube, that is 1
cube, 2 cube plus 3 cube plus up to n cube.
So, their sum will be n, n plus 1 divided by 2's whole
square.
That is n square, n plus 1 square upon 4.
So, this is their sum.
What will be their average? Average would be sum n square,
n plus 1's whole square divided by 4.
And what will be the average? Sum upon n, so divide by n, n
will cancel with n.
This will be the answer of your average.
Average would be n, n plus 1's whole square divided by 4.
It is not necessary that you just remember the formula of
average.
You just remember the formula of sum.
With the help of the formula of sum, you can derive the
formula of average. Because, the sum is very important for
us to do in the series, in arithmetic progression, in APGP.
So, remember the sum.
Rest will be done.
Next, moving ahead, what will we do?
Which number is this? 3, 4th number.
We have done 1 plus 2 plus 3, n consecutive natural numbers.
Suppose, we have to find out even numbers, that is 2 plus 4
plus 6 plus 8 plus up to n.
Will we write up to n or what? We will not write n in this
case.
Because, n can be anything.
It can be even or odd.
But, specifically what is told in this? That we are finding
out the series of even numbers.
So, what is the specialty of even number?
Even number is always multiple of 2.
So, to show even number of 2, what we will write here?
2n.
So, their sum will be n, n plus 1.
Similarly, if we do 5th case, I am writing both together so
that it will be easy to remember.
So, what is the sum of even numbers? Similarly, we can also
get odd numbers.
What will be odd numbers? Odd numbers will be 1 plus 3 plus
5 up to, how far will we go? Like we denoted even number
with 2n.
So, if one of them is more, then what will we get? We will
get odd number.
So, we will denote its last with 2n minus 1.
This is the series of odd numbers.
What is their sum? Their sum is n square. So, when sum of
both is there, then what will be the average of even
number? It will be n plus 1 by 2.
n will get cancelled.
Sorry, 2 will not come.
2 comes in it.
Because n will get cancelled from n, 2 is not there.
So, its average will be n plus 2.
And, the average will be n square divided by n, that is n.
So, this is the average of odd number and this is the
average of even number.
So, their square and cube also exists.
You will see once what happens.
You will learn how to get their sum in series.
Because there is no point
in memorizing the formula.
You will not learn how to get it.
I will make you write it.
But, what will you do after some time? You will forget it.
How much formula will you memorize?
This is basic.
You should remember this much.
Formula related to sum.
That is why I did this.
Now, next moving ahead.
What will we do? Which number was this? This was 5th number.
So, your 6th number would be.
Let us suppose we have n observations.
Whose average is
A. And, m number of observations whose average is B.
We have n observations whose average is A.
And, m observations whose average is B.
So, what we have to find? We have to find average of total
number of observations.
Average of total.
Total n or m.
Total n plus m observations.
Means, average of n numbers of observations.
We have given some particular average.
And, we have given other average of other.
So, if we have to find total average
of n numbers of observations.
So, what will we get? We do not have to do anything.
First, multiply this by this.
Multiply n by A.
Plus, multiply m by B.
Plus, m by B.
Divided by total number of observations.
Earlier, what was there? There was n for n and m for m.
What will be their total? Their sum.
So, divided by total number of observations.
That is n plus m.
So, this will be related to this case.
When we have averages of different sets.
Means, some observations have different averages.
Some have different averages.
So, what will be their combined average? Its formula will
be this.
Related to this formula, we will see some questions.
What are your questions? First, let us see the questions.
This is your question.
The average of eleven numbers is 63.
The average of first six numbers is 60.
The average of eleven numbers is n.
n is equal to 11.
The average of first six numbers is
60. Let us suppose you have eleven numbers.
So, you can denote them like this.
What will be the result? A1, A2, A3, A4, A5, A6, A7, A8,
A9, A10 and A11.
These are your eleven observations.
The average of all these is
The next line is the average of first six numbers.
First six numbers means a1, a2, a3, a4, a5, a6.
The starting six numbers, what is their average? Let us
denote their average from a1.
Their average is given as 60.
Their average is given as 60 for n is equal to 6 means
starting six observations.
Then the last six numbers is 65.
Means last six, the last six numbers are the observations.
What is their average? We have their average as 65.
Their average is 1, 2, 3, 4, 5, 6.
So, the observations from a6 to a11, their average is
denoted as a2 and that is given to be 65.
And what is n for this? n is equal to 6 because this is the
average of six observations.
So, what we have to find out? We have to find the sixth
number.
So, first of all, see which is the sixth number? We are
getting the sixth number as a6.
a6 is the required number which we have to find out.
What will be its value?
So, pause the video and think how you will do this question.
Try to write it in a copy and then match that step with me
that whether you are doing it with long method or short
method.
How is it comparatively? Is it lengthy or short? Then you
will see that the answer is correct.
So, what will be the solution? The solution would be when
we are taking their average that is a1 plus a2 plus a3 up
to a6.
So, we are getting the average as 60 and the previous one
is 65.
So, if we are given the average of total.
In the average of total, if we are given the average of
these two, what will happen? One is of these six and other
is of these six.
So, we add them.
If we are taking the average of these two, then what is the
common part in it? The common part a6 is repeated in both
of them.
If we do this, it is like total average.
It is made up of the formulations of these two.
But what is happening in these two? One number is repeated
in both of them.
So, it cannot be repeated twice.
Then it will affect our average.
It should be repeated only once in total.
So, what we will do? What will happen to the a6 that we
have to find? First, we will take the sum of these two
averages.
So, how many observations will we get in total? 11
observations plus a6.
It is repeated twice.
So, it is the sum of 12 observations.
If we minus the total average, then the extra a6 that is
coming, we will get the remaining
left. So, what we will do? First of all, we will take the
sum.
Why are we taking the sum?
Because we have given the average a1.
So, what will we take out of the first six numbers?
We will take out the sum.
So, the sum of the first six numbers, denote it with s1.
What will that become? What formula did we read? Average is
equal to sum upon n.
So, what will that become? Average into n.
So, a1 is 60 and n is 6.
First six numbers.
So, it becomes 60 into 6.
That is 360.
Similarly, what we will do? We will take out s2.
Sum of last six numbers.
What will that become? a2 into n.
That is a2 into 6.
Because we have 6.
So, 65 into 6.
What will that become? 390.
So, we have the sum of first six observations and last six
observations.
Now, if we do the sum, then see what I just told you that
a6 is coming in the first one and a6 is coming in the
second one.
It is coming in both.
So, we will have to subtract it from 1.
So, if we see this, then what is the sum? If we do s1 plus
s2, then what will be the sum? It will come from a1 to a11.
Plus, what will a6 become? a6 will become an extra.
So, what will this become? a1 plus a2 plus a11 plus a6.
Because a6 is coming twice.
So, what is s1 plus s2? We have this.
Now, we have to
subtract the value of a6.
If we put their value, then the value of a6 will come.
We can subtract the value of s1 plus s2.
What will this become? Solve it.
How much will we get? 750.
s1 plus s2 will come.
We will have 0, 5, 1.
This is 750.
s1 and s2 will come.
We will have 750.
a1 plus a2 up to a11, where will the sum come from? It will
come with the help of this.
With the help of the average that we were given earlier.
So, what will the sum be? 63 into 11.
That is 693.
So, we have the total sum of 693.
But, what we have is 750.
So, we will have a6.
Put the value of a6 here.
We will have 693.
Is it 693 only?
Yes, 693.
So, what do we have in front of a6? We have 750 minus 693.
That is 57.
So, what should be the value of our last 6 number a6? It
should be 57.
You just understand that according to the question, what
formula average we will get and what we will do.
That should be in your mind.
Was this question easy? It was easy.
It can be a little difficult for
you. But, slowly you will find it very easy and you will be
able to do this calculation very quickly.
What is the next question? Read it.
It is saying the average of the marks obtained in an exam
by 8 students was 51.
That means the average of 8 students is 51 marks and by
other 9 students was 68.
And, the average of other 9 students is 68.
Once you have read this much, think about the formula which
should be used in this.
Which formula should be used? See, the average of some
number is given.
The average of first 8 students is given and the average of
other 9 students is given separately.
So, which formula should be used in this? It should be NA
plus MB divided by N plus M formula.
I have done that one set of observation has a different
average and the other set of observation has a different
average.
So, we will use that formula.
Suppose, this 8 is our N.
Its average was A that is 51.
We have 9 as M.
What was its average? It was B that is 68.
So, what we have to find out is the average mark of all the
17 students.
How will 17 students come? 9 plus 8 will be 17.
So, the average mark of all the 17 students would be A1.
what will happen to A1? We will have NA plus MB divided by
N plus M. N plus 1 is 17 only.
You solve this. 51 into 8 plus 9 into 68. It would be 408
plus 72.
54, 612 divided by 17, Inka sum karke divide by 17 kar lena
aapke paas average marks aajayega.
Now moving on to new rule humne ek question kar liye next
jo rule tha kyaunse number pe rule tha humara?
Kitne rule hogaye? 6, 6 tak hogaye the humare formulae then
hum dekhte hain 7 number, 7th abhi tak jo humare paas
number the natural number ke jo bhi the wo the starting
from 1, 1 se start ho rahe the ya to even wali series 2 se
shuru ho rahe the but
let us suppose ab hume consecutive numbers ka sum national
numbers ka sum nikalna hai but wo kya hai consecutive
matlab ek ke baad ek aane chahiye but wo kya hai 1 se start
nahi hai wo kisi aur digit se start hai let us suppose
humare ye hai 5 se start
hai 5, 6, 7, 8, 9 and up to lagalo 25 tak hai humare paas
ye saare numbers hai hume inka sum nikalna hai ab hume 1 to
25 tak ka toh pata hai uska hum nikal lenge us wale
formulae se sum kyunki average ke liye hume sum ki zarur
hota hai aur phir hum usme
agar 1 se lekar 4 tak ka sum nikal ke subtract kar de toh
humare paas aa jayega but wo karne saaj bhajaye agar hum
direct iska koi formulae lik de toh wo humare le sahi
rahega na toh iska jo direct formulae ki in sab ke
consecutive sum kya aayega jo inke
sum ka formulae hai wo aapke paas kya ho jayega aapko bhis
kya karna hai sum nahi average nikal rahe hain hum side
average iska kya ho jayega jo bhi sequence ki first term
hai first term kya hai humare 5 wo first term le lo uska
naam de diya a last term
hogi jo uska naam le liya l.
first and last sum ka hume sum karna hai first and last
term ka sum karna hai and divided by 2 matlab inka average
le rahe hain matlab inki mean value le rahe hain indirectly
words toh wahi kya aayega inki average nikal jayega this is
our 7th point.
Let us suppose humare paas ek 8th case hai ye 8th case lik
lijiye humare paas abhi kya hai ye set of numbers hai ye
hai na aur ab jo 8th point hai usme humare paas kya ho raha
hai humare paas kisi particular number ke kya hai multiples
multiples of a
particular number x kitne multiples hai koi finite terms
hai matlab suppose kar lo n multiples hai
matlab sabse pehle 10 multiples hai 2 k uska matlab kya ho
gaye 2 1 2 se lekar 2 10s tak saare jo uski table mein aaye
2 4 6 8 and so on 20 tak wo saare uske 10 multiples ho gaye
2 k toh waise hi hume x ke kitne multiples le ne n
multiples toh x ke n
multiples hunge
sabse pehle toh x aayega phir 2 x aayega phir 3 x aayega
and so on ye kab tak aayenge n x tak n into x tak ab hume
inka average nikalna hai toh same yahi rule apply karke
inka average kya ho jayega first term plus last term
divided by 2 toh inki first
term hai x last term hai n x divided by 2 toh iska average
humare paas kya ho gaya x plus n x divided by 2 or in other
words x ko common le liya 1 plus n divide by 2 ab ya toh
aap isko independently independently alag formulae ki tarah
treat kar lo ya
toh aap ye yaad rak lo aur iski help se ye solve kar dena
okay aap jab humne ye kar liya toh is jaise aapne hi wala
question kya 5 se lekar 25 tak ke hume natural number ke
kya le na hai average le na hai toh ek baar ye solve kariye
ki ye average aayega kya iska agar mai ye solve kar riyon
toh ye aaraye 5 plus
25 divided by 2 that is 30 by 2 which is 15 toh iska
average aayega 15 agar mai isko achhe se likhu achhe se ye
series kya ban rahi hai 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 and 25.
yaha tak humari ye kya hai ye hume terms of the
observations de rekhi hai aap dekhiye humne iska kya
average nikala iska average according to our formula nikla
15 ye nikla tha iska average suppose hume ye sara de rakha
hai aur hume kya dikha na hai iska
average agar hume nikal na hai toh hum kya dekhte hai aadi
aap kya karte rao aap sort karte jao jaise ek number yaha
se hataya
toh ek number yaha peeche se hata diya phir ek number
doosar number hata hai phir doosar number peeche se hata
diya toh aise equal terms aap kya karte rao hataate jao
phir jo aapka bachega aise equal terms yaha tak hataenge 1
2 3 4 5 6 7 8 9 10 10 terms
humne aage se hatai 10 ye terms peeche se hataenge 1 2 3 4
5 6 7 8 9 10 toh 16 tak ke terms humne peeche se hata diya
16 ke baad wale toh 10 terms aage se hatai 10 terms peeche
se hatai toh jo mid value bach gayi wahi kya ho gayi humare
average ho gayi
mid average hota hi kya mean hi nikal liya na indirectly
toh aap is tariki se bhi kar sakte hai formula laga ke bhi
kar sakte hai waise itne basic question toh aate nahi hai
but aapko seekhne ke liye ye sab pata hona chahiye kaise
aaye
ab let us suppose humare paas koi questions hai humare paas
koi data abhi toh ye line wise consecutive terms ka data
tha toh humne ye aise karke kar diya but let us suppose
humare paas consecutive terms ka data na ho matlab saath
saath ke nahi hai ye
hum alag alag hai
that is humare paas suppose 110 120 124 130 135 ye humare
paas kaha se kaha tak hai 150
148 and so on kitte ho gaye 5 6 10 kar leti ho 155 160 and
126 and 130 132 humare paas ye set of observations hai
humein kya karna hai humein ka find out karna hai average
ye bhi ho sakte hai abhi toh chhote number of observation
le rekhe hai that is
maine sirf 10 numbers of observation le rekhe hai but
humare paas iske andar aur bhi finite terms ho sakte hai
jise 100 ho gaye ya kuch bhi ho gaya toh hum kya karenge
unka average find out karne ke liye toh ya toh hum inka
pura ke sum calculate kare
aur phir divided by total number of terms 10 se kar lo toh
usse bhi aapka average nikal jayega aur ek doosra method
hota hai aapka deviation wale method ka matlab assumed mean
wale ka sabse pehle aap kya karu aap isme ekdum minimum aur
sabse bada number
ek bar find out kar liye toh agar iska hum sabse chhota
number dekhe toh wo hai humare paas 110 aur sabse bada
number kya hai 155 ab humein kya karna hai hum iske beech
ki koi bhi value kya kar lete hain beech wali value ko hum
assume kar lete hain ki
wo iska average hai a usko naam de dete hain a1 se take
assumed average toh wo kya ho jayega inka beech ka koi
number le lo jaise 110 hai aur 155 hai sorry 155 nahi hai
highest last jo humare hai wo hai
toh 150 nahi hua iska largest iska hua largest humare paas
160
theke 160 hua toh ek ho gaya 110 aur ek ho gaya 160 inka
beech ka number kya le le
130 le le kya difference kya ho gaya inka 50 ka 50 toh
divided by 25 ka ho gaya na toh 135 ho jayega inka beech ka
average beech wali value toh 130 bhi le lo usme bhi koi
dikkar nahi hai calculation easy ho jayegi toh aap aisa
kare aap kya le le 130 le
lete hain taake calculation humein toh easy pade 135 bhi le
sakte 140 bhi le sakte jo bhi aapka mann kare aap wo le
sakte toh 130 humne inka beech ka number le liya ab aap kya
kijiye 130 ka in subse kya nikalte jayiye difference aur
difference kaise agar
dekho 130 me se 110 minus karenge malak kitna humein peeche
jana padega 130 se 110 tak toh humein 20 steps peeche jana
padegena toh jitna aapko peeche jana hai toh aap negative
ka sign rakhi
So, you will see it's not less.