Introduction - Measures of Central Tendency, Business Mathematics & Statistics Video Lecture - Business Mathematics and Statistics - B Com

let's take a look at an introduction to
measures of central tendency here's a
histogram of survival times in days of
60 guinea pigs that were infected with
tuberculosis this was from an experiment
investigating the effect of tuberculosis
on guinea pig survival in addition to
plots we often report a numerical
measure of central tendency for instance
we might report the average survival
time or the median survival time so
let's take a look at some of those
measures I'm going to assume in this
video that the data is from a sample and
does not represent the entire population
this is almost always the case in
practice if the data represented the
entire population
the notation would be a little bit
different the sample mean x-bar is just
the regular old ordinary average we add
up all of the observations and divide by
the number of observations this is the
arithmetic mean as opposed to other
kinds of means like the harmonic mean or
geometric mean the median is the value
that falls in the middle when the
observations are ordered from smallest
to largest if n is odd there will be one
value in the middle and the median is
that middle value if n is even there
will be two values that fall in the
middle and the median is the average of
those two middle values there isn't
Universal notation for the median you
might see the median labeled as X with a
tilde or as a big M or as MD or possibly
something else the mode is the most
frequently occurring value in the sample
in some situations in statistics the
mode is an important quantity but in
most situations when we are summarizing
sample data the mean and the median are
much more meaningful let's work through
a simple example suppose we have this
sample of six observations since no
number occurs more than once they all
occur the same number of times we'd say
there is no mode to find the mean we add
all the values and divide by 6 the
number of values here
that works out to 6.5 so our sample mean
is 6.5 to find the median we first need
to order the values from smallest to
largest
which I've done here and since there are
six values here an even number of values
there are two middle values 5.0 and 6.2
the median is the average of those two
values or 5.6 let's look at those values
on a number line here's a dot plot of
the observations the median is right
here at five point six halfway between
the two middle observations the mean is
a little bit larger at six point five
will see that extreme values have much
more of an effect on the mean than the
median and to illustrate that let's see
what happens when we change the largest
value here fourteen point one to 26
point one or in other words we move this
green dot
way out here to twenty six point one
increasing the largest value to
twenty-six point one from fourteen point
one has greatly increased the mean which
is now eight point five the mean has
increased two units but the median
hasn't changed at all extreme values
have much more of an effect on the mean
than the median to illustrate that
another way let's suppose these dots had
physical weight an equal weight for each
dot and they were resting on a stiff
board of negligible weight a fulcrum
place that the mean would balance the
dots the mean represents the balance
point but can sometimes help us
visualize where the mean might be on a
plot if we shifted the fulcrum over and
placed it at the median then the weight
wouldn't balance and the board would tip
a fulcrum place that the mean would
balance the weights and a fulcrum placed
anywhere else in the board would tip
let's return to the guinea-pig survival
time data and here's the histogram again
the class in the histogram that has the
greatest number of observations is
sometimes called the modal class so this
class here
the 150 to 200 class is the modal class
for this data the median is the value of
the variable that has half the
observations to the left and half the
observations to the right you can't
figure out the median precisely just
from the histogram but I went to the raw
data on which the histogram is based and
found out that the median is 214 point
five right about there at two fourteen
point five fifty percent of the
observations lie to the left and fifty
percent of the observations lied to the
right I also went to the raw data and
found out that the sample mean is two
hundred and forty one point to note that
the mean is a little bigger than the
median here these larger values in the
right tail will have more of an effect
on the mean and the median they'll pull
the mean out toward them a little bit
this distribution is a little right
skewed and for right skewed
distributions the mean is greater than
the median here's a left skewed
distribution here again the median is
the value of the variable that has half
the observations to the left and half
the observations to the right and if we
went to the raw data and calculated the
mean we'd see that it's a little less
than the median these values out here in
the left tail will have more of an
effect on the mean than the median for
left skewed distributions the mean is
less than the median here's a perfectly
symmetric distribution we'd see here
that the median falls right in the
center and that the mean is exactly
equal to the median for a perfectly
symmetric distribution the mean and
median will be equal for an
approximately symmetric distribution the
mean and median will be close in value
to sum up compared to the median the
mean uses more information in the sense
that it uses the actual value of every
observation this has good and bad points
it uses more information but if there
are extreme observations present then
the mean will be influenced by those
values much more than the median will be
and sometimes it can be a misleading
measure
so when should we report the mean and
when should we report the median in many
situations it's reasonable to report
both measures the reader can then make
up their own mind about which is the
more appropriate measure of center in a
given situation here are a couple of
overall thoughts about when we'll be
using the median and when we'll be using
the mean the median is a very useful
descriptive measure of center it's often
reported in situations where there are
extreme values or skewness present like
in housing prices or salaries the mean
is more influenced by those extreme
values but it does have some very nice
mathematical properties and is often
used in statistical inference procedures
there are many other measures of central
tendency that are sometimes used and
here are a few there's the trimmed mean
in which a certain percentage of the
largest and smallest observations are
removed before calculating the mean for
example we might throw out the largest
10% and smallest 10% of the observations
then calculate the average of the
remaining values this would of course
result in a mean that isn't as sensitive
to extreme values but it comes with a
loss of information there's the weighted
mean where different values are given
more weight in the calculations and as I
mentioned earlier our usual mean the
sample mean x-bar is the arithmetic mean
but there are different types of mean
like the harmonic mean and the geometric
mean depending on the scenario one of
these might be a more appropriate
measure to use but for most situations
we'll encounter will find that the
sample mean x-bar and the median are the
most appropriate measures of center and
that's a brief introduction to measures
of central tendency