getting an aircraft off the ground is
something that millions of passengers
take for granted every day but actually
it's quite a complex problem for the
designers of air and for air travel
regulators aircraft have to operate
within a wide margin of error to ensure
safely and the ability of an airplane to
be on airport is affected by many
variables the airplane is required to be
able to accelerate up to what we call
decision speed v1 and then stop on the
available runway the heavier you are
obviously the longer it's going to take
to accelerate the faster you have to get
going initially anyway since you're
carrying more weight off the ground and
the longer it's going to take to stop so
if you only have a fixed amount of
runway you're going to have a certain
weight limit the other requirement that
we have is we have to be able to
maintain a certain clime gradient after
takeoff if we lost one engine which
means the airplane can only weigh X
amount for a given day it's you know
certain temperature in elevation the
airplane will only climb X altitude you
know XP per minute based on its weight
the airport in Aspen sits at nearly
2,400 meters above sea level surrounded
by high mountain peaks this high
altitude and terrain makes operating
passenger aircraft quite challenging at
this altitude aircraft performance
declines and is affected even more when
the temperature rises with less air to
generate lift and mountains all around
the performance of the aeroplane becomes
critical
the FAA requires certain performance
minimums out of all the airplanes and in
to meet that performance requirement
when it gets warm and at high elevations
airplanes have to leave weight behind in
order to make that so the warmer it gets
the less weight that an airplane can
takeoff at and at the same altitude
essentially and being up at 8,000 feet
that means a lot of times especially
during the summers the commercial
airlines are limited in the maximum
weight they can carry obviously you
can't shave off much fuel so the only
thing leave behind is people in baggage
and that's what they have to do as a
consequence commercial aircraft must
leave passengers behind what's a safe
number of people to work this out we
have to appreciate the idea of a normal
distribution people on average are
getting bigger indeed according to one
recent study one in ten of the world's
adults is obese that is having a body
mass index or BMI at more than 30
researchers have studied body mass index
cholesterol levels blood pressure and
tended to find it in many countries
blood pressure and cholesterol levels
have been falling but the body mass
index has tended to rise in the UK for
example the proportion of the population
that's obese has more than doubled
it's easy to test whether medics
consider your obese overweight healthy
or underweight and says hey 50p to
measure weight high and BMI or I'll be
mild eighteen point five to 25 is
considered healthy 25 to 30 is
overweight over 30 is obese let's see
what happens
you can find your BMI by taking your
weight in kilograms and dividing by the
square of your height in meters and BMI
equals W over H squared where BMI is
body mass index W equals weight in kilos
and the height H is the height in meters
so I weigh 79 kilograms and I'm 1.8
meters high so my BMI is W over H
squared which equals 79 over
1.8 squared which equals 24.4 so I need
to be very careful now if I have to stay
healthy
although obesity rates in many countries
across the world are increasing and
although obesity is linked to high blood
pressure and cholesterol mean average
blood pressure and cholesterol levels
have fallen in some parts of the world
not everyone is overweight indeed summer
unhealthily underweight I already look
like if we took the BMI for the whole
population and plotted that distribution
many would cluster around the mean
average weight some below average and
some above a few way below and a few way
above so this is what the picture would
be like with a mean average BMI for many
societies of above 25 as the BMI the
value of the variable increases from
very low levels so the number of people
the frequency of the observations
increases the BMI that occurs with the
greatest frequency as where the curve
reaches its maximum so the mode is the
same as the mean beyond that the number
of observations
declines
indeed weight would conform to the same
general pattern with a mean weight of
around 80 kilograms cholesterol levels
and as we'll see in the next film a
great many other things come close to
doing so as well this normal
distribution or Gaussian distribution as
it's sometimes called is a bell curve
because the bell describes its shape if
the great majority of observations are
close to the mean then we'll have a
narrower higher curve with a relatively
low standard deviation if many
observations are some distance from the
mean the curve will be lower and broader
and the standard deviation greater let's
use this concept to see how many
passengers it can safely carry when
taking off from aspirin airports it's
relatively easy to determine the actual
gross weight the maximum weight that
airplane can take off on a given day at
a certain temperature with the wind at
the elevation and then they just take
the gross weight subtract out the empty
weight of the airplane the fuel and what
they have left is payload and that's
they take that number and figure out how
many passengers they can carry based on
average weights let's look at the
capacity of the typical aircraft used at
aspirin the BAE 146 300 has a maximum
takeoff weight of forty four thousand
two hundred and twenty five kilos at
Aspen's altitude of 2,400 meters and a
temperature of 27 degrees Celsius the
maximum gross weight drops to thirty
three thousand four hundred and
seventy-five kilograms
so this means that the airline must shed
weight carrying less fuel passengers and
baggage to calculate how many passengers
and bags to leave behind the airlines
use averages provided by the FAA until
1995 they used an average of 72 point 6
kilos per person then 77 kilos until
2004 and then it was increased to 86
point 2 kilos in the summer and 88.5
kilos in the winter to include the extra
clothing now these changes of course
reflect the increasing obesity of the
population so if an airplane must shed
fifteen hundred and fifty one kilos it
must leave eighteen passengers behind
during the summer they automatically
block out 15 seats they don't sell them
on certain flights if they have a warmer
day than they anticipated or the winds
Iran they will bump somebody
or bump their baggage occasionally they
just bump the baggage then instead of a
few unhappy individuals you have an ol
airplane full of them the FAA I used a
national study on health and obesity to
determine the new averages for passenger
weights now let's see why the
authorities are so strict about the
loading requirements suppose given the
flying conditions the weight of the crew
the baggage etc a smallish aircraft can
take 6,000 kilograms of passenger weight
to be conservative and using some
average of the weights used by the FAA
let's suppose that the average passenger
weighs ninety one point six six
kilograms
if the number of people allowed to fly
is restricted to say 60 60 such average
people would weigh 60 times ninety one
point six six kilograms which equals
five thousand five hundred a but in
practice they won't all be the same
weight and the standard deviation of the
passenger weights is let's say fifteen
hundred kilograms
now by setting a maximum of 60
passengers is there any chance that a
random selection of passengers might
weigh more than the 6,000 kilograms of
weight that's regarded as safe it
doesn't seem very likely but let's check
it out by using our understanding of the
normal distribution
diagrammatically the problem looks like
this any combined weight of over six
thousand kilograms is a problem the mean
is five thousand five hundred kilograms
so typically they'll be fine but we're
interested in the area that's greater
than six thousand kilograms that's the
area in which there will be a safety
problem now the total area under the
curve gives all the possibilities and is
therefore equal to one how can we find
the probability for the occurrence of
anything represented by just the shaded
area if weight can be represented by a
bell curve all we need to know is the
mean average and the standard deviation
and we can calculate that area
that clearly will have a different area
for every value of mean and standard
deviation
but there's one table of probabilities
for a standard normal distribution which
has a mean of 0 and a standard deviation
equal to 1 so first we can convert our
values into standard units usually
called z-scores and our formula for
doing this is Z equals X minus mu over
Sigma where X is the score from the
original normal distribution mu is the
mean of the original normal distribution
and Sigma is the standard deviation of
the original normal distribution and Zed
is the standardized variable it
represents the number of standard
deviations above or below the mean a
particular score is let's see how that
works out in our example here X is 6,000
kilograms mu the mean is five thousand
five hundred kilograms Sigma the
standard deviation is fifteen hundred
kilograms so we have Z equals six
thousand minus five thousand five
hundred divided by fifteen hundred but
is to say Z equals five hundred over
fifteen hundred Z equals not 0.33 now if
we look in the normal tables we can look
down the left-hand column for Z equals
nine point three and then across under
naught point O three and the number in
the table is the tail area for Z equals
naught point three three which is not
point three seven zero seven this is the
probability that the weight will exceed
six
and kilograms so our answer is that the
probability that the total weight
exceeds 6,000 kilograms is not 0.37
correct to two decimal places or 37% so
given our assumptions we would find that
such a restriction on passenger numbers
would result in over one-third of these
planes flying with a weight regarded as
too great to be safe so the concept of a
normal distribution is highly valuable
in making sure the passengers are able
to fly safely one way of taking more
passengers safely is to extend the
runway recently the airport began
construction to add a thousand feet or
305 metres to the length of the runway
well the longer runway will will help
for the regional jets that is an
airplane that is runway limited out of
this Airport and consequently a longer
runway means they'll be able to take off
if more weight on a given day it'll
allow me to take off slightly heavier or
most people to take off slightly heavier
this additional runway length makes it
possible for some aircraft to take off
with additional load one aircraft can
carry an additional one five eight eight
kilograms or an additional 18 passengers
however the additional Run Road length
is only partly effective because it only
affects one limitation the ability of
the IRA to obtain decision speed doesn't
change the angle of climb and the
ability to clear the surrounding
mountains there are many instances where
weight is crucial not just aircraft for
example here in the business school we
don't the lift to the staff office is
continually breaking down how can we
reduce the chances that it'll be
overcrowded we can use our understanding
of normal distribution we can work out
from the knowledge of the way the school
operates that the mean number of people
using the lift at any time is four with
a standard deviation of two people
here's the problem if the manufacturer
and the architects of the building have
calculated that the lift can hold up to
eight people or six hundred and thirty
kilograms one of the chances that eight
or more people will use the lift at the
same time
notice that there's an assumption that
each person here weighs on average
seventy eight point seven five kilograms
given a normal distribution we can first
find the z-score with our formula Z
equals X minus mu over Sigma so Z in our
case is 8 minus 4 over 2 which gives 2
looking at the Zed table we find out
that the z-score of 2 corresponds to
naught point nine seven seven two
that is to say ninety seven point seven
two percent of the time less than eight
people are wanting to use the lift at
the same time but we were asked what's
the proportion of the time that more
than eight people wish to use the lift
this is easy to calculate since the area
under the curve is one or a hundred
percent then the proportion of eight or
more people wishing to use the lift is
exactly one - naught point nine seven
seven two which equals null point zero
two three or two point three percent so
it's only very occasionally that more
than eight people will turn up at the
same time to use the lift the chances
are only around one in 50 notice there
are normal distribution tables giving
p-values greater than X as we used in
our first example and tables giving
p-values less than X that we've used to
solve our lift problem
we began by saying that aircraft have to
operate within a wide margin of error to
ensure safety an understanding of the
normal distribution makes that very
clear
but as we shall see in our next film
there are many other examples of data
that will conform to a normal
distribution not just weight so the
techniques based on such a distribution
or widespread use across the social
sciences
Oh
you