the measures of dispersion follow the measures of central tendency so the
common measures of dispersion are standard deviation and variance and if
you are ethically works at your college by now you know about the bell curve
which looks like this and you know often hear that this is one standard deviation
of the mean or within two standard deviation of the mean so let's see what
it means what is standard deviation and how we compute it let's look at the age
of an instructor so let's say if the average age is 52 what does it mean it
means that the individual ages may differ some may be 48 or maybe 55 or 75
so the average age is an estimate but what we also need is an estimate for the
dispersion in the data set the other thing to note is we range in in our data
set for example the difference of the range is from a minimum of 29 years of
age to a maximum of 73 years and this to you it's a distance or the difference
between the minimum and the maximum now unlike the mean the the main difference
between a variance for the population and sample is that when you compute the
population variance you divided by the total number of observations these are
the deviations between observation and the mean squared and then add them and
then divided by the total number of observations and for sample you divided
by n minus 1 we usually talk about squares called standard deviation rather
than the variance and standard deviation is essentially the square root of the
variance and the variances in square units so it's good to use the standard
deviation because it's is exactly the same units as the variable so during
standard deviation of age will also be measured in years rather than square
years which could be various so here you see we just taken the square root of
variance and that becomes standard deviation so we will return to our data
set and we will look at the variables then we've computed the mid the
the mean and the maximum before but now this time we'll add 10 deviation to it
we will return to our data set now and use another command called descriptive
statistics and click on
on descriptives and we select the
variables of choice and then example aid of the instructor and under options we
have the mean the standard deviation minimum and maximum and let's select the
other variables as well so we click on OK and we have the same variables
computed with the number of observations minimum maximum mean and standard
deviation now let me explain why mean and standard deviation have to go hand
in hand and I will revert to this example from basketball and talk about
the two giants of basketball Michael Jordan my favorite and Wilt Chamberlain
who preceded Michael Jordan now if you consider their average score per game
you would notice that they didn't differ much their average was around 30 points
for both Jordan and Chamberlain however when you look at the standard
deviation or variance with their performance and michael jordan was at
around four point seven six compared to chamberlain who was it around 10.59 and
here if you were to plot this distribution to see how a Michael Jordan
using the mean and standard deviation assuming that their scores are
normally distributed you would notice that Michael Jordan's consistency comes
out and it's a very different distribution very focused distribution
compared to the one for Wilt Chamberlain and same mean 30 but you see that the
distribution is different suggesting that Jordan was much more consistent in
his performance than Wilt Chamberlain we can also plot the variance using
histogram in SPSS so let's say we want to do this we return to our data set
click on crafts we have several choices let's say we use a legacy dialog box and
say histogram select the variable which is the teaching evaluations score and
then say ok and the output comes out here now you could see it's similar to
the one that we've got in our slide we can add titles and footnotes to our
chart and we can produce multiple histograms based on a variable let's say
some gender we click here and then to okay you see that we have got one drawn
for females and one got drawn for males so the the main argument in praise
of standard deviation or variance is that average will out the mean will
paint a partial picture if you really want to understand the complete and
picture for a variable or for your data set you have to compute both the average
ends and the standard deviation to understand the variance and notice that
all risk metrics and Finance have much to do with with standard deviation so a
mean with the standard deviation means something more useful than the mean
itself