Construction of statistical tables, Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com

in this video I'm going to demonstrate
how to build a frequency distribution
table that can be used to make a
histogram a couple of terms I'm going to
use are the number of data points and
this is called the sample size and it's
typically denoted n another thing I'll
refer to is the precision of the data
and that refers to how many decimal
places your data contains so in the
numbers you see for this example these
data values all go out 0 decimal places
and so a value like 36.1 let's say
that's one decimal place another value
like 74 point zero five two would be
three decimal places and so on now the
example I'm going to use are these are
the ages of 32 participants attending a
retirement seminar the starting age with
the lowest data value in this set let me
just point out is 48 and the highest
value is 74 now as I said there were 32
participants so my end value in this
case is going to be 32 there are two
things you need to consider when you
decide on how to construct your
frequency distribution table the first
thing is going to be the starting age or
starting value of your first class and
the second will be the width of the
classes so to choose the starting age of
your first class or generally that would
be referred to as the lower class limit
of the first class the simplest thing to
do would be to simply choose the lowest
value in your data set and in this case
that would be 48 depending on what your
data is about you could choose a value
that makes it convenient to compare
several data sets at the same time so
for example if there were multiple
seminars given and you're collecting
data on the ages for each of those
seminars you might want a wider range so
maybe you would start your frequency
distribution table at let's say 40 and
then if you constructed all your tables
the same way
you could compare them or you might have
some practical considerations and
physical constraints on what the data
could be and so it you know it might be
more convenient depending on what the
data is to choose value like zero or 50
or 100 or something like that but for
this data set I'm going to choose 48 as
our lower class limit that's a simple
and convenient choice to make note the
value doesn't always have to be an
integer in this case it makes sense to
choose an integer value but you win at
whatever your data is you want to
consider how many decimal places your
data has and what the range of values
are going to be now the next thing we
have to figure out is our class width
and in order to decide the class width
we start by deciding on the number of
classes our table is going to have so
the number of class is a good rule of
thumb for choosing that is to take the
square root of n now in this example
square root of 32 falls between 5 & 6
and I'm going to choose one of those
values that's it's convenient to choose
an integer value when you're
constructing a distribution table for
this example I'm just going to go ahead
and choose 6 once you've chosen the
number of classes your class width can
be determined by taking the range and
dividing it by the number of classes let
me fix my handwriting here okay so now
the range in this example our range of
Ages is from 48 to 74 so I'm going to
take the upper the highest value and
subtract the lowest value when I get is
26 and so my class width would be
determined by taking 26 and dividing by
the number of classes I'd chosen which
is 6 and if I do that what I get is
about 4 and a third 4.33 so what I want
to do once I've got that value is again
rounded either up or down
I could choose 4 from my class width or
it could choose 5 I'm going to choose
for this example if I choose for I run
the risk that I'll actually end up with
more classes than I wanted I choose for
I might it I might end up with seven
classes so I'm going to choose five in
this case so our class width is five the
next thing we want to do is to list the
lower class limits of all of our classes
and you'll recall my first class was
going to start with 48 and what I'm
going to do is simply add that class
width value to get each of my lower
class limits so 48 plus 5 is 53 plus 5
again is 58 and 63 68 and my 6th class
would start at 73 ok the next thing I do
is list the upper class limits and how
I'm going to do that is by considering
where the next class starts and
subtracting point 1 so for my first
class I look at my second class that
starts at 53 and I'm going to subtract
point 1 from that and I'm going to get
52.9 likewise for my second class while
my third class starts at 58 so I
subtract point 1 and I get 57.9 and if I
continue that pattern I'm going to get
these values and you'll note that all of
these values are spaced by 5 as well and
so that helps me figure out what my last
ones going to be by add five to seventy
two point nine I get seventy seven point
nine so now I have six classes with the
upper and lower class limits determined
let me point out a couple of variations
on the last class it is acceptable to
give this last class with the last value
rounded up to 78 it's also acceptable to
say 73 and older and that's an
open-ended class okay in case you have
some outlier values that you want to
account for but that are it's not worth
setting up a separate class for those
values but to keep it simple I'm just
going to go with the class limits that
I've already determined up to seventy
seven point nine
now by doing it this way you're going to
ensure three things first that your
classes all have the same with second
that the classes don't overlap and third
that there aren't any gaps between the
classes and so that every data point in
your data set has one and only one class
where it's going to fall let me point
out something else when you're looking
at your limits your class limits for
your classes I want you to recognize
where you see the class with note that
the difference between the lower class
limits is five because we kept adding
five to get those values right it's also
the case here the just the difference
between these values upper class limits
is always fiving what the class width is
not is not the difference between the
upper class limit and lower class limit
of any one class that value is always
going to be 4.9 but that's not what the
class width is it's this value you're
comparing either the lower class limits
or you're comparing the upper class
limits okay all right now how did I
choose to subtract point one to get my
upper class limits remember I took 53
and subtracted point one to get fifty
two point nine likewise 58 minus point
one gave me my upper class limit of
fifty seven point nine and so on well
what I did was I considered that my
original data went out zero decimal
places and I want one decimal place more
and subtracted point one I would have
been acceptable to do something like
this suppose I
set up my class limits to look like this
this would be acceptable too because
again there's no gaps and there's no
overlap and every data value will fall
into one and only one class okay so the
rule here is you don't consider how many
decimal places your original data is and
consider the distance between in between
the classes and just make sure that gap
between the upper-class limit and the
next lower class limit isn't going to be
bigger than the precision of your data
okay now that's a little confusing so
let me give you another example of just
that one concept so let's suppose I have
data that looks like this let's say it
has goes out to one decimal place etc
okay I might start my lower class limit
at five I might choose a class width of
0.5 now since my data goes out one
decimal place I might subtract point
zero one going out two decimal places to
get my upper-class limit so five point
five minus point zero one is five point
four nine
likewise six minus point zero one is
five point nine nine and so on so I took
my
data and I want one decimal place more
when determining my upper-class limits
okay let me go back to my original
example I'm going to go back to the
first set of class limits that I
determined and I'm going to go ahead and
bin or organize the data okay so when I
do that I get these frequency values 5 4
10 7 5 and 1 and then what you want to
do once you've you've tallied your
frequencies is to add all those values
up 5 plus 4 plus 10 plus 7 plus 5 plus 1
adds up to 32 so that means all my data
points are accounted for they all fall
into one class or another now in the
second step you might do at this point
is to determine your relative
frequencies and how you do that is to
take your frequency values and divide
each one by n and in this case n was 32
so your relative frequencies are
determined by this what I would do is to
report the relative frequencies is to
give a decimal value so 5 over 32 is
about 0.16 when I round to two decimal
places and so if I do all these
calculations here are the values I'm
going to get and these are my
frequency's okay
and these should add up to one if we've
done it correctly although because I'm
rounding these values to two decimal
places it's possible it will round off
to 0.99 or one point zero one again due
to rounding error but that's okay so
these should add up to one or pretty
close to one okay so you can see that
there are a few choices you have to make
when building a frequency distribution
table and so that means there's no one
exactly correct way to do this but here
are the things you want to avoid first
make sure that your classes all have the
same width the second thing you want to
make sure is that the classes don't
overlap and you want to make sure that
there are no gaps between your classes
again this is to make sure that every
data value falls into one and only one
class now the next thing to avoid is
having too few classes now in this
example I chose six classes if I had
chosen three or four classes instead of
six what would happen is in my histogram
I'd have fewer bars and the bars would
all be taller and it would be harder to
discern where the peak of the
distribution distribution would be it
would just make it kind of lumpy the
other thing you want to avoid is having
too many classes so again instead of six
if I had chosen let's say eight classes
or more what that's going to do is give
me a lot of bars in my histogram that's
going to flatten out the distribution
and it's going to get hard to discern
any kind of shape to that distribution
so again I recommend the strategy of
using a value that's close to the range
divided by the number of classes and
you'll choose that number the number of
classes by considering the square root
of the number of data points that those
are good rules of thumb to follow when
making those decisions
the last thing I'll mention is that when
deciding these parameters your
calculator the ti-83 and 84 is are
particularly useful because you can
change these parameters on the fly and
see what your distribution looks like
once you've entered your data into a
list it's helpful to sort the data
there's a command for sorting the data
in a list that makes it really helpful
for counting up the number of values in
each class once you construct your
histogram in the window menu x-men is
that lower class limit of your first
class so that's something you can change
and test likewise X scale is actually
your class width so again a value you
can adjust to see how it changes your
distribution and help you decide which
are the best parameters to choose I hope
you found this helpful
thanks for watching