01 - Cumulative Frequency Distribution - Less than type Ogive - Statistics - Class 10 - Maths Video Lecture

hello everybody welcome to this session
of statistics and today we will learn
about the cumulative frequency and its
representation in a graph the graphical
representation of a cumulative frequency
gives us an ogive ogive are of two types
the less than type and the more than
type today we will specifically learn
about the less than type
now before I tell you why the name less
than type let's start with an example so
I have an example of daily income of
some of the workers in terms of Indian
rupees now this particular data is a
group data because it is divided into
small groups every group is having a
class size of 20 so there is a
difference of 20 rupees so 100 to 120
120 to 140 140 to 160 160 to 180 and 180
to 200 and this particular income group
also show us the number of workers that
fall in the corresponding category so
the column number of workers is nothing
but the frequency column denoted as f
you
frequency is nothing but the rate at
which something occurs that means 12
number of workers have a daily income of
100 to 120 rupees
Bodeen number of workers have a daily
income of 120 to 140 rupees and so on
now let us start by calculating the
total of the frequency so the total
frequency is denoted as Sigma F so 12
plus 14 plus 8 plus 6 plus 10 will give
us 50 so this tells us that this is the
frequency distribution now a frequency
distribution can be easily converted
into a cumulative frequency distribution
cumulative frequency is nothing but the
running frequency that means we have to
add frequencies at every single stage so
let's form a column for cumulative
frequency
cumulative frequency is also denoted as
CF
now before we calculate the cumulative
frequencies let's come back to the point
why the name less than type because in
this case we have to actually take the
upper limits whenever we say less than
something that means that something has
to be of the higher value so we are
going to say lesser than 120 lesser than
140 lesser than 160 180 and 200 so now
let's form a column for all these upper
limits
one way is upper limit for the first
group 140 is the upper limit for the
second one 64 third one eighty four four
and 200 for fifth now let's start with
the cumulative frequency we know that
the frequency till 120 is 12 so the
cumulative frequency for the first group
is 12 now if we want to calculate the
cumulative frequency till 140 so we will
add the frequencies from 100 to 140
that means frequency of the first group
and frequency of the second group so 12
will be added with 14
which will give us 26 now similarly the
cumulative frequency till 160 would be
everything from 100 to 160 so 12 plus 14
plus 8 will be added so 26 plus 8 will
give us the cumulative frequency till
160 which is 34 similarly we will do for
all the groups
now if you notice the last running
frequency actually matches the total
frequency after forming the columns for
the upper limit and the cumulative
frequency we will represent it in a
graph now upper limits will be actually
represented in the x-axis because it is
an independent variable whereas
cumulative frequencies is a dependent
variable so it will be located on the
y-axis now let's plot all these points
in the graph so we have this graph ready
with us and we need to form the x-axis
and the y-axis now x-axis we can see
that there is daily income category
and on the y-axis which is the vertical
axis is the cumulative frequency
before we start drawing a graph we shall
set the scales on the x axis and the y
axis now for me I have set a scale on
the x axis as one centimeter equal to pi
rupees
so we shall write scales for
x-axis is one centimeter
equal to PI rupees and y-axes
one centimeter is equal to 2.5 units
now in our x-axis if we notice there is
a small discontinuity because we have
started the graph right from the hundred
so because there is a difference of ten
rupees
between every daily income group but
there is a difference of 100 rupees from
zero to hundred so we put a
discontinuity here now the first point
in our graph was 120 and 12 so we will
plot 120 from the x-axis and 12 is
something here on the y-axis which is
this point
so this is the x-coordinate and this is
the y-coordinate the second point was
140 and 26 140 is zero and 26 is
somewhere here
Third Point is 160 and 34 which falls
here
after we plot the points we have to form
a smooth freehand graph
now this is an increasing curve which is
characteristic of a less than type oh
joy
now this is the graphical representation
of a cumulative frequency distribution
from the same graph we can find the
median of this data to find the median
we have to find n by 2 now n in this
case is the total frequency which is 50
so n by 2 is nothing but 50 divided by 2
that is equal to 25 so 25 will be the
half of total frequency now we will draw
a line from this point 25 and we will
touch the curve and from that point we
will touch the x-axis and that will be
the median so let's start by forming a
line
you
at this point on the x-axis is nothing
but the median which is 137 point five
so the median of this data is rupees one
thirty seven point five now let's
quickly summarize the less than type
method first identify the upper limits
from your data second calculate the
cumulative frequencies now mark the
upper limits on the x-axis and
cumulative frequency on the y-axis of
your graph then set the skills for your
graph and then plot these points onto
the graph now form a freehand smooth
curve now this resultant curve would be
the less than type oh joy we can also
find the median of group data by the
formula n by to locate n by two and
touch the curve and then touch the
x-axis that would be the median of your
data that was all about the less than
type method to learn more please visit
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