馃攳
Statistics 101: Geometric Mean and Standard Deviation - YouTube
Channel: Brandon Foltz
[0]
hello my name is Brandon and welcome to
[2]
the next video in my series on basic
[4]
statistics if you are new to the channel
[7]
welcome it is great to have you if you
[9]
are a returning viewer it is great to
[11]
have you back if you liked the video
[13]
please give it a thumbs up share it with
[15]
classmates colleagues or friends or
[17]
anyone else you think might benefit from
[19]
watching so now that we are introduced
[21]
let's go ahead and get started so this
[25]
video is the next in our series on
[27]
descriptive statistics and it is about
[30]
the geometric mean and standard
[32]
deviation now in my experience the
[35]
geometric mean is often skipped over in
[37]
many stats classes and I think that is
[40]
very unfortunate for several reasons one
[42]
it's actually pretty cool and
[44]
interesting there are some insights
[45]
we're gonna learn about as we go but
[47]
also it is extremely useful especially
[50]
in business when you're dealing with
[52]
rates of return on investments or other
[54]
types of financial instruments but it's
[56]
also useful in other disciplines like
[58]
biology medicine agriculture or in any
[62]
other discipline where you're dealing
[64]
with growth rates over periods of time
[66]
so let's go ahead and learn about the
[68]
geometric mean and standard deviation so
[73]
let's start with what I call some
[74]
geometric insight so here on the Left we
[77]
have a perfect square so the square has
[80]
sides of 4 so the area inside this
[83]
square is just 4 times 4 or 16 what if
[86]
we want to know the average length of
[88]
each side so in this case we're dealing
[91]
with the two sides to find the area so 4
[93]
plus 4 equals 8 then we take 8 divided
[96]
by 2 that gives us 4 and of course 4
[98]
squared equals 16 we get our original
[101]
area of 16 units back however what if
[105]
we're not dealing with a perfect square
[106]
let's say we have a rectangle like we
[109]
have here so one of the sides is 1.3
[112]
units the other side is 2.9 units so
[115]
that gives us an area of 3.77
[118]
what about the average length of the
[121]
sides so we take the average of those
[124]
sides we get 2.1 now to find the area we
[128]
square 2.1 and we get 4.4 1
[132]
so now we have an area of a perfect
[134]
square that is 4.41
[137]
when we use the average side lengths
[140]
however that is not the same as the
[143]
original rectangle the area of that was
[145]
3.77 over here in the middle but on the
[148]
right when we use the average side
[151]
lengths we get a perfect square of 4.41
[155]
units so what's going on so here is our
[160]
perfect square again so again 4 times 4
[162]
is 16 in this case what we're going to
[165]
do is multiply them together to find the
[167]
area but instead of dividing that we're
[171]
going to take the square root of that
[173]
product so 4 times 4 is 16 the square
[176]
root of 16 equals 4 but what about our
[180]
rectangle that we had so here we have
[183]
1.3 times 2.9 so the area inside this
[187]
rectangle is 3 point 7 7 so now let's do
[189]
the same thing 2.9 times 1.3 that gives
[194]
us 3 point 7 7 we take the square root
[197]
we don't divide by 2 we take the square
[200]
root of 3 point 7 7 we get a value of 1
[203]
point nine four one six five now if we
[207]
create a perfect square with sides of
[210]
one point nine four one six five now we
[214]
get the area we expect a three point
[217]
seven seven so what we are saying is
[220]
that if we want to create a perfect
[221]
square of average side length that is
[225]
the same area as the rectangle in the
[227]
middle we have to multiply those sides
[229]
together then take the square root not
[232]
the average the square root to get that
[235]
side length square it and then we get
[237]
the area in return so let's take this to
[242]
three dimensions so here we have a three
[245]
dimensional box with sides of four six
[247]
and nine so four times six times 9
[251]
equals 216 so the volume of this
[254]
rectangle this three dimensional
[255]
rectangular box is 216 units let's take
[260]
four plus six plus nine equals nineteen
[262]
we take the average so 19 divided by
[265]
three
[266]
is six point three three three three
[268]
three and we multiply that times itself
[271]
three times where six point three three
[272]
three three cubed and we get a volume of
[275]
254 point zero three seven that does not
[279]
equal 216 so taking the average of all
[284]
three sides multiplying that by itself
[287]
three times or cubing it does not give
[290]
us the same volume inside as we found
[293]
using the actual three sides here on the
[295]
left so now let's do the same thing here
[301]
is our original three-dimensional box of
[302]
size four six and nine four times six
[305]
times nine equals sixteen that's the
[306]
volume inside this box so now what we're
[309]
going to do is take the product of those
[311]
three sides and then take the cubed root
[314]
of that product because we're dealing
[316]
with three sides or three dimensions so
[319]
the cube root of 216 equals six now if
[323]
we take that six and create a perfect
[325]
three-dimensional cube so six times six
[328]
times six what we get is a volume of 216
[332]
so what we have done is taken the
[334]
average of the three sides in the left
[337]
box to create a perfect cube over on the
[340]
right so all three sides are the same
[342]
they're identical a perfect cube and we
[345]
get the same volume in return that we
[347]
had over here on the left so here is the
[351]
formula for the geometric mean and it's
[353]
not anything complicated or scary
[355]
because we just did it on the previous
[357]
four slides so the geometric mean which
[359]
is X bar sub G remember X bars or a mean
[362]
sub G means geometric it's simply the
[365]
nth root of n values multiplied together
[367]
so the product of n values over on the
[371]
right we just have the exponential
[373]
notation for root so remember we can
[375]
rewrite for example the square root as
[378]
something to the one-half power or 0.5
[382]
power so we take a number like 16 where
[384]
the square root of 16 equals 4 but also
[387]
16 to the power of 1/2 or 0.5 also
[392]
equals 4 a different way of writing root
[394]
using exponents so let's you
[399]
the geometric mean in a real-world
[400]
problem the geometric mean is often used
[403]
in Business and Finance because finance
[406]
and business deal with a lot of rates of
[408]
return or growth over periods of time
[411]
now it's not limited to financial
[413]
applications like I said at the
[415]
beginning any discipline that uses
[417]
growth rates can use the geometric mean
[420]
biology medicine agriculture
[423]
what-have-you but in this case since
[425]
finance is the most often used
[427]
application we'll use something along
[429]
those lines so here we have $1,000 and
[433]
we're to look at its change over periods
[435]
of time we're gonna track its percentage
[437]
change and what's called its growth
[439]
factor which is just a different way of
[441]
saying percentage so over this first
[444]
period we had a percentage change of 5%
[447]
that is the same thing as a growth
[449]
factor of one point zero five to keep in
[452]
mind that a growth factor of one would
[454]
mean that our investment did not grow at
[456]
all it remained the same we didn't gain
[458]
anything we didn't lose anything so a 5%
[463]
change is a growth factor of one point
[464]
zero five we just take one plus point
[467]
zero five which is five percent so now
[470]
we have $1,050
[473]
and of course that is based off our
[475]
original investment from where we
[477]
started now in the second period we have
[479]
a percentage change of two percent so
[481]
that equates to a growth factor of one
[483]
point zero two so now we have one
[486]
thousand seventy one dollars and of
[488]
course that is based off our previous
[490]
value of $1,050 it is not based upon our
[494]
original investment we're going back to
[497]
the previous period the previous period
[500]
is where we start in this case to get to
[502]
our 1071 dollars let's say the next
[506]
period
[506]
unfortunately we lose three percent that
[509]
equates to a growth factor of 0.9 seven
[513]
and you can see how this works what is
[515]
taking one minus point zero three that
[519]
relates to a growth factor of zero point
[521]
nine seven and this is important to
[523]
convert to a growth factor because the
[526]
geometric mean can only use positive
[529]
values so now our investment is worth
[532]
one
[532]
thirty-eight dollars and 87 cents and
[536]
again that three percent loss is
[538]
referring back to our previous balance
[540]
of 1071 dollars
[542]
so our one thousand dollar investment
[545]
changed to one thousand thirty-eight
[547]
dollars and 87 cents sir relates to a
[550]
growth rate of three point eight eight
[552]
seven percent and what we often ask
[555]
ourself is what is the average growth
[558]
rate over those periods of time though
[561]
we could do is take the average of those
[563]
growth factors so one point zero five
[566]
plus one point zero two plus zero point
[569]
nine seven then we divide that by three
[571]
that's three point zero four divided by
[573]
three that equates to an average growth
[576]
factor of one point zero one three three
[578]
three or percentage average growth of
[581]
one point three three three however when
[585]
we go ahead and do the math with that we
[586]
have 1,000 times one point zero one
[589]
three three three to the third power
[591]
because we're doing it over three
[593]
periods we have a value of one thousand
[595]
forty dollars and 53 cents that is not
[598]
what we calculated above so what did we
[601]
do wrong so we'll start here now this
[607]
time we're gonna do it the proper way
[608]
now we will multiply our growth factors
[611]
together and then take in this case the
[613]
cube root because we have three periods
[615]
so one point zero five times one point
[618]
zero two times 0.9 seven we take the
[621]
cube root of that product we end up with
[623]
one point zero one two eight which is
[625]
one point two eight percent now when we
[628]
take our 1,000 dollars and we multiply
[631]
that by one point zero one two eight
[633]
cubed because of our three periods now
[636]
we get the correct value of one thousand
[639]
thirty-eight dollars and eighty seven
[641]
cents and that is how and why we use the
[645]
geometric mean when using growth rates
[647]
if we use the arithmetic mean that we're
[650]
typically used to we're going to get a
[652]
wrong answer because growth rates are
[655]
dependent on multiplication not addition
[659]
now one thing you might have heard of is
[661]
this concept called kegger in my day job
[664]
we actually use the term kegger all the
[666]
time
[666]
to track our growth rates over periods
[668]
of time in this case kegger stands for a
[670]
compound annual growth rate so you could
[674]
take your growth rate each year based on
[676]
the previous year and then use the
[678]
geometric mean to find the average
[681]
growth over that period of time in this
[683]
case annually now not to go too far into
[688]
the weeds on this but there is a way to
[690]
use natural logarithms and some of you
[692]
may have thought this already when doing
[693]
the geometric mean so here is our
[696]
geometric mean formula now what we could
[699]
do is take the natural log of each side
[701]
to the left hand side with the natural
[703]
log of our geometric mean equals the
[706]
natural log of each of our values we sum
[709]
those up and then divide by the number
[711]
of observations we have so in our case
[715]
we had the geometric mean using our
[717]
growth rates look like this well we
[719]
could take the natural log of that
[721]
geometric mean so the natural log of
[723]
1.05 plus natural log of one point zero
[725]
two plus natural log of zero point nine
[729]
seven add those up divide by three and
[732]
look at the natural log of the geometric
[733]
mean equals zero point zero one two
[736]
seven one one two now we don't want
[738]
natural log there on the left
[740]
so we have to think algebraically what
[742]
is the inverse of natural log and of
[745]
course that is using E so we take E
[747]
raise it to that power on both sides and
[751]
then we go ahead and do that out we end
[753]
up with a geometric mean of one point
[755]
zero one two eight which is exactly what
[758]
we calculated using the geometric mean
[760]
in the previous slide so again the
[762]
natural logarithm method is just a
[764]
different way of doing the same type of
[766]
problem and actually we can quickly
[770]
verify this using our three-dimensional
[772]
box so we have four six and nine as our
[774]
sides we can go into excel or whatever
[777]
else would take our three lengths four
[779]
six nine take the natural log of each of
[781]
those lengths then we can find the
[783]
average of those three lengths then we
[786]
can find the average of those three
[787]
lengths and we have one point seven nine
[789]
one seven five nine four six nine no
[793]
what we can do is take E and raise it to
[795]
that power so if you see you're in lower
[798]
right the exp Fung
[800]
and Excel is e so take erased ^ one
[804]
point seven nine one seven five nine
[806]
four six nine and then we get the value
[809]
of six which is the exact same thing we
[811]
found out before when we wanted to
[813]
create a cube that is uniform length on
[816]
each side of six six and six so finally
[822]
we have the geometric standard deviation
[823]
and it's not that bad on the left-hand
[826]
side we have the natural log of the
[828]
geometric standard deviation equals a
[831]
square root of the sum of the squared
[833]
deviations to the natural log X I which
[837]
the natural log of each observation
[838]
minus the natural log of the geometric
[841]
mean we square those differences sum up
[844]
and then divide by the number of
[846]
observations we have in and then take
[848]
the square root so yes I admit that was
[850]
a mouthful but let's see what this looks
[852]
like in our example using our financial
[855]
data so underneath the square root sign
[858]
we can see what's going on here
[859]
so our first growth factor was one point
[861]
zero five our geometric mean was one
[864]
point zero one two eight so natural log
[867]
of one point zero five minus natural log
[869]
of one point zero one two eight we take
[872]
that difference in square it plus the
[874]
next data value minus the geometric mean
[877]
squared etc etc so we go ahead and do
[880]
all that math out and then we end up
[882]
with the natural log of the geometric
[884]
standard deviation is zero point zero
[886]
three two seven four which is one point
[891]
zero three three two eight two or three
[895]
point three two eight two percent so
[897]
that is our geometric standard deviation
[899]
using our financial data okay so final
[904]
points the sample mean is only suitable
[907]
for additive processes of which this was
[911]
not one the geometric mean is suitable
[913]
for multiplicative processes so when
[916]
we're multiplying things in sequence all
[919]
values for the geometric mean must be
[921]
positive that's what we used growth
[923]
factors while the geometric mean is
[926]
often used in business for financial
[929]
growth investment performance it is also
[931]
useful for any measure that were
[933]
quartz growth so in biology we can think
[936]
of the growth rates of maybe like
[937]
bacteria for agriculture we can think of
[940]
growth rates of crops medicine same
[943]
thing for medical studies etc so any
[946]
rate of change over sequential periods
[948]
of any length is suitable
[950]
now while mathematically valid unequal
[954]
periods should not be used want to keep
[956]
the periods standard whether that's days
[959]
weeks years trials or whatever so we
[963]
want to make sure the measure of time
[965]
between each period is uniform otherwise
[968]
the results while you could do it math
[970]
wise would be misleading and of course
[974]
Microsoft Excel has the Geo mean
[976]
function that will automate this math
[978]
and I'm sure Google sheets has the same
[980]
thing and other software programs have
[982]
similar functions so in Excel you can
[985]
actually put in your growth rates and
[986]
then use the Geo mean function to go
[988]
ahead and have it do the math properly
[990]
for you okay so that wraps up this video
[995]
on the geometric mean and standard
[996]
deviation like I said at the beginning
[999]
the geometric mean is often skipped over
[1001]
or neglected in many stats classes
[1003]
assembly because of time and I think
[1005]
that is extremely unfortunate and
[1007]
somewhat problematic many of the
[1009]
students I deal with here on YouTube
[1011]
were in my day job or what have you
[1013]
often go on to careers in finance in
[1017]
business in accounting or even into
[1020]
other disciplines like medicine biology
[1021]
agriculture etc that all involve growth
[1025]
rates and there are many other
[1026]
disciplines that involve growth rates as
[1028]
well so think about it if you are a
[1030]
finance professional and you have a
[1032]
series of growth rates and you're
[1034]
working with a client and you take those
[1036]
growth rates add them together and
[1037]
divide with the number of periods and
[1039]
tell that client hey this is the average
[1042]
growth rate
[1042]
you are literally absolutely wrong you
[1046]
are telling your client the wrong
[1048]
information and that could cause serious
[1050]
problems both professionally and
[1051]
potentially legally however if you
[1054]
understand the geometric main and center
[1056]
deviation you can give that client the
[1058]
proper information and the accurate
[1060]
information which is obviously very
[1063]
important so thank you very much for
[1065]
watching I appreciate you spending some
[1067]
of you
[1067]
valuable time with me I hope you learn
[1069]
something new and look forward to seeing
[1071]
you again in our next video take care
[1074]
bye bye
Most Recent Videos:
You can go back to the homepage right here: Homepage