Harmonic Mean - Measures of Central Tendency, Business Mathematics & Statistics Video Lecture | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

in this presentation we're going to look
at the harmonic mean the harmonic mean
will go to the note here as H there's
actually a couple other ways of denoting
it for example an x-bar soup subscript H
and we just will just take with H at the
time and it's not the only way of
denoting the harmonic mean but it's just
the easiest for the time being and the
formula is simple but not exactly the
most simple and formula we've ever seen
so suppose we have a sample here x1 x2
x3 x4 all the way up to xn what we do
here is we will get the reciprocal of
each and add them up for example suppose
we have x1 is 2 the reciprocal of that
would be 1/2 suppose the next item is 3
then we'd add 1/3 1 over 3 so this is 5
w plus 1 over 5 and so on so that is the
denominator of these expressions the
numerator is actually quite simple the
sample size how many items there are so
if there's 5 items of the data set in
your sample then the numerator of your
expression is simply 5 it's
straightforward enough actually and ok
let's do a quick example so suppose we
have data set here
7 + 9 / - easy to say M 2 example a 2
items there in our sample at 7 and 9 and
so 1 over 7 plus 1 over 9 and let's just
start this again numerator is the number
of items - ok so at the first item is 7
so 1 over 7
the next item is 9 so 1 over 9 ok and
let's work that out and so - bit of
cross multiplication helps here with the
numerator so cross multiplying that out
we would get and 16 over 63
okay and we can go a little bit further
there we can multiply a button below the
numerator denominator by 63 so we get 1
to 6 over 16 you can work that out in
your calculator what to do there I'm
going to just sort of put a little
counter example suppose I add in m3 okay
what what happens then
okay so we have to start again here
what's the harmonic mean now so we have
1 over 7 plus 1 over 9 we'll also add in
1 over 3 and the sample size is now 3 so
3 goes up here okay working that out we
will get at 3 over and let's put
everything over 63 that worked the last
time 7
so 9 plus 7 plus 21 over 63 so it would
work out to be 3 over 37 divided by 63
or in other words that would be 189
divided by 37 again little bit the
calculator work the answer should be
around 5.5 between 5 and 6 something
like that ok let's look at a real world
example this is a where we would use the
harmonic mean so pump machine so this is
two machines the first machine takes 6
hours to fill them with fill a swimming
pool with water okay so 6 hours the
second machine will take 4 hours to fill
the same swimming pool with water ok so
4 & 6 what's the harmonic mean of that
what we're interested in here is how
long does it take it to fill the
swimming pool
if using both machines together
you
okay so using machine 1 and Machine 2
together necessarily it should speed up
the job ok but how do you do that well
the answer is what we do is we compute
the harmonic mean okay and divided by
the number of machines this is where you
would use the harmonic mean actually
these sort of the engineering jobs so
there's a sort of rationale for that
that I sort of talked a little bit about
at the end so aren't mean divided by the
number of machines or the number of
servers or a number of operators okay so
it's a straightforward enough job so
here where I've just done that H equals
2 the first machine was going to do it
in six hours
ok the acetic machine could do the four
hours so working that out we get a
harmonic mean of 40 is over 1048 at 4.8
ok but we're using both machines
together okay so it's the on average
it's essentially if it's n chily 2.4
hours
ok what it's how long it would take to
swim you through a filter swimming pool
if we're using both machine 1 and
machine 2 together okay I bet sort of a
little layer sort of more put a bit more
rationale onto that in it shortly
suppose we have a Thursday pub machine
and that can fill the swimming pool in
three hours now using all three machines
how long will it take to fill up the
swimming pool so this time we have three
machines okay and one takes three hours
will takes four hours won't take six
hours so the harmonic mean of that a
little bit of algebra 3/8 at 1/3 is 8
over 24 a quarter is 6 over 24 and 1/6
is 4 over 24 that is 18 over 24 if you
have them all up
72 is 72 over 24 is 3 sorry to put that
as that's 72 over 24 so the harmonic
mean here is for hour at 4 hours but
we're using 3 machines so
along to take the suit yes f upswing
bill H over N four hours divided by
three so the answer is 1.33 hours let's
just put a sort of as a sort of set of
promised a a bit of it it's sort of a so
for our machine can fill one quarter of
the pool in one hour the six hour
machine can fill one six the pool in one
hour and the three hour machine can fail
one third of the pool in one hour
okay so a quarter plus 1/6 plus a third
that's actually three quarters that's
actually something we worked out already
that's 18 over 24 which is 3/4 so how
many so if the evil three machines
working together and in the first hour
1/4 of the pool will be filled up so our
site requires Ahlul will be filled up in
the first hour so for the remaining
quarter you just need another third of
an hour that's it really is how it sort
of makes sense it's actually once you
think about it like that it's actually
quite easy all right that's the harmonic
mean and this is just a sort of one last
remark actually the harmonic mean is
used in computer science as this thing
called the F score or F measure I have
that done separately
it's the harmonic mean again it's very
simple application of the harmonic mean
but in this instance I'm just
concentrating on the harmonic mean how
to calculate