in the last film we looked at variations
in people's weight and discovered that
we could represent a whole population of
people by a normal distribution we can
explore more the statistics of many
phenomena using this concept if we've
measured the diameter of all the trees
in the wood and then plotted that
information as a frequency distribution
what would it look like it would conform
to something like a normal distribution
you have a few very small trees a few
very large ones and lots and lots of
them
somewhere in between we can use these
principles to consider the probability
of shares making a given return the way
in which wealth and income is
distributed order likely intelligence
quotient of any person or group of
people the list is endless the normal
distribution is one of the most
important concepts and statistics
shoplifting is a problem for retailers
everywhere but the problem is not taking
one once one pieces a one piece of
merchandise apparently taking the whole
stack and running that's all you know
they comment every retailer suffers from
it what's often happened with I mean
thieves nowadays is you steal to order
and you will find that you lose a lot of
books say you would spot that and
because we know what should be on our
shelves news agents also suffer from
shoplifting people come in steal
magazines newspapers etc how do they
control it at one thing they can do is
install security devices like this one
but it costs and so whatever they do
it's going to reduce their profits
preventing a discouraging theft is
costly businesses respond to shoplifting
in many ways with security systems with
item tags and with trained staff and
surveillance
even University Libraries lose books to
thieves
in response this library has installed
electronic item tanks and electronic
gates most businesses simply accept that
some theft is going to take place and
include such losses in their business
calculations these self-service things
saves the shop quite a lot of money it
saves them on library costs the trouble
is it also makes it much easier for
people to shoplift so why do they have
these things here because the savings on
the labor costs are greater than the
losses on the shoplifting shoplifting
adds to the cost of running a retail
business this cost is estimated at about
1/3 of 1% of sales
it doesn't sound much but it has around
a hundred of thirty pounds to the
average family's annual shopping bill
how big a problem is shoplifting for the
individual store it varies from store to
store it varies from day to day but the
pattern conforms to a normal
distribution and that means that we
could say quite a lot because we can use
our normal distribution to examine a
whole range of problems about the size
of the loss but if we're going to do
that we first have to transform the
normal distribution into the standard
normal distribution let's see how we do
that the standard normal distribution
has a mean of 0 and a standard deviation
equal to 1 we can look at the area under
the curve by converting into what are
called z-scores an any normal
distribution can be converted using the
transformation that Z equals X minus mu
divided by Sigma
where X is the score of the original
normal distribution mu is the mean of
the original normal distribution Sigma
the standard deviation of the original
normal distribution and Z is the
standardized variable to be found
be aware whilst any normal distribution
can be transformed using this formula
the formula will not convert a skewed
distribution into one that is not skewed
so we're looking at our theft problem
because the theft data conforms to a
normal distribution
as the storm losses on average about 200
pounds a day and we know the standard
deviation of the daily loss is 30 pounds
now let's see if we can solve a number
of problems for the store owner first
what are the chances that the store on
any given day will lose more than 250
pounds second what are the chances that
the storm will lose between 250 and 350
pounds
third what's the minimum amount that the
store can expect to lose the majority of
the time let's say 95% of the days we
want to know the probability P that X is
greater than 250 pounds and to find this
we'll have to transform the normal
distribution into a standardized normal
distribution and find the z-score thus
we look for the probability P of Z being
greater than X minus mu over Sigma
which is the probability that Sid is
greater than 250 - 200 the average store
loss over 30 the standard deviation
which means we want the probability that
Z is greater than one point 6 7 now if
we look up one point six seven in our
standardized normal table we get a value
of naught point naught 475 that is the
answer is four point seven five percent
of the time on any given day there's are
just less than five percent chance that
the store will lose more than two
hundred and fifty pounds
now let's look at the possibility that
the loss in the day will be somewhere
between 250 and 350 pounds we'll do this
in three steps step one we know that the
probability of more than 250 pounds
being lost is naught point naught 475 we
just did that in the first problem so
the probability of a loss of less than
250 is 1 minus the probability that X is
greater than 250 1 minus naught point
zero four seventy five equals naught
point nine five two five
step two now find the probability of the
loss being more than 350 which is the
probability that Z is greater than X
minus mu over Sigma in our case 350
minus 200 over 30 which is the
probability that said is greater than 5
which is someone likely that
approximates to 0 so step 3 the
probability that X will lie between 250
and 350
is one minus the probability of X being
less than 250 plus the probability of X
being more than 350 which is 1 minus
naught point 5 2 5 plus 0 which equals
naught point naught 475
in this last problem were interested in
the left hand tail of the normal
distribution the lowest 5% now given
that a lower tail value of minus one
point six five gives a tail area of five
percent we have minus one point six five
equals x minus 200 over thirty minus
forty nine point five is therefore X
minus 200
x equals a hundred and fifty point five
oh so the store can expect to lose at
least a hundred and fifty pounds and
fifty pence on 95 percent of the days if
we know the data conforms to a normal
distribution and we know the mean and
the standard deviation we can derive
considerable volumes of information
about losses in libraries or shops or a
book store and so on but it's not just
information about losses we can also see
many other things the distribution of
wealth and income the probability of
returns for some investment in the stock
market the amount of intelligence that
people have the list is endless the
normal distribution then is one of the
most important concepts its statistics
you