## Median from Group data with mid value and frequency

I wonder Kumar sharing with you a

question from my student Joseph Joseph

thanks a lot for sharing this question

it is indeed unique since here we are

given the mid values or the class marks

and we need to find median of a group

data so let's look into this question

the question here is find the median of

the following data mid values sometimes

also called as class marks are given to

us and their corresponding frequency is

given to us so from here we need to find

median now whenever the mid values are

given we can find the difference between

the consecutive values twelve point five

minus seven point five gives us five

right so that means we know the class

interval is of five correct so from here

we get class interval

which is you can calculate from 12.5

minus 7.5 and that is 5 so 5 is the

class interval now to get the classes

what we do is let's call this class

interval as H we add H by 2 value and

subtract H by 2 value to get the class

so what is half of H it is half of 5

which is 2 point 5 correct so what we

will do here is we'll add and subtract

half of this value to get the class

right so we'll include a column here so

say class group so if I add 2 point 5 to

7 point 5 we get 10 and if I subtract 2

point 5 we get 5 so the class group will

be from 5 to 10 you didn't idea right so

so that is how we can build this class

group here so now it will be from 10 to

15 and now likewise we can actually

create this class group right so making

these dotted lines just to align them 15

to 20 now this is 20 to 25 25 to 30 30

to 35 35 to 40 and from 40 to 44 right

so this is the first step to get the

class groups we know frequency so now

from frequency we can get cumulative

frequency right so let me make these

columns so we'll add here a column and

just saving on space so we'll call

cumulative frequency as the next column

so so 5 is 5 5 plus 6 is 11 and 11 plus

15 26 plus ten thirty six plus five

forty one plus four forty five plus two

forty seven plus two forty nine so we

know the Sigma or the sum of the

frequencies is indeed 49 right to find

the median we are always looking midway

between this frequency is correct

so we can say n is equal to 49 so what

is n by 2 so n by 2 will be 49 divided

by 2 so that should be twenty four point

five right so n by two is twenty four

point five now twenty four point five

helps us to figure out the modal class

so we are looking to the frequency

column variable the frequency is just

more than twenty four point five that

becomes our modal class gray so in this

particular case the modal class is is

this this the one right

since n is 24.5 the frequency between

after 11 is 26 cumulative frequencies so

somewhere here between 15 and 20 lies

our median right now within this class

group there are 15 elements and they are

assumed to be uniformly distributed

that's the whole concept right and that

gives us the formula so the median

formula is L plus n by 2 Joseph also

wants to understand how do we get this

formula right what is the base for this

formula so let me explain you the base

of the formula also as we move along so

definitely we have understood one thing

that somewhere here is our answer so we

know it is definitely more than 15 right

so L is that lower limit so this is L

for us lower limit L so so we know L is

15 for us right now second is that we

are assuming that all these 15 elements

are uniformly distributed so 15 elements

all these 15 elements are uniformly

distributed in this group so our

assumption is uniform distribution right

so that is why now we know we are

looking at a position which is twenty

four point five now twenty four point

five we know 11 is before this group

right so after eleven somebody

so this eleven is actually equal to the

capital F so somewhere after eleven 24.5

is n by two right so n by two is twenty

four point five so this group uniform

distribution means n by 2 minus F so

that is to say a twenty four point five

minus eleven divided by number of

elements in this group which is fifteen

so that gives us uniform distribution

and since the class interval is of in

this case five units we'll multiply this

by H which is equals to five so I hope

the concept is clear to you right so so

here what we have here is the value of F

is is eleven the frequency or number of

elements within our model group are

fifteen and the class interval is equals

to five now we know all these values by

substituting these values we can

calculate our answer and therefore we

can say median is equal to lower

interval ela 15 this is of the world

class lower limit n by two is twenty

four point five so it is plus twenty

four point five minus eleven divided by

total number of elements within this

group which is 15 times five which is a

class interval so now we can calculate

this answer and get the value right so

we have here twenty four point five

minus eleven divided by 15 times five

and then we'll add 15 to it which gives

us 39 over two in decimals is

nineteen point five so nineteen point

five becomes the median for us so it is

important to note that the total

cumulative frequency of this group was

twenty-six we are looking for twenty

four point 0.5 right which is actually

if you see it is closer to twenty does

make sense to you so nineteen point five

seems to be a reasonable approximation

so every time when we calculate medium

it is always an approximate value great

so our answer will be approximately

let's say approximately nineteen point

five right so I hope the steps are

absolutely clear the idea is from the

given mid values find the interval half

of that interval should be added and

subtracted to each Center value to get

the class growth right from frequency

table as such we have to find cumulative

frequency half of the number of elements

gives us the position for median and

also helps us to figure out the model

class lower limit of model class is L n

by two is midway F is the frequency of

cumulative frequency of just before the

model class right F is the frequency of

the mantras and H is class interval with

that formula we can exactly approximate

the median so I hope the steps are

absolutely clear so I hope you find it

interesting and useful feel free to post