Area under Standard Normal Curve, Business Mathematics and Statistics Video Lecture | Business Mathematics and Statistics - B Com

all right today we're gonna take a look
at how to find the area under the normal
the standard normal curve and what I
have drawn here is a normal curve and
you have mu in the middle there your
medians also in the middle so you know
50% of your data lies to the right and
50% of your data lies to the left and
here I have marked off one standard
deviation two standard deviations three
two standard deviations and likewise on
the flip side now within 60 sorry within
one standard deviation left or right of
the mean you have 68 percent of your
data all right
and within two standard deviations
ninety-five percent of your data will
lie and within three standard deviations
ninety-nine point seven percent of your
data will lie no question is that what
if you're like one point five standard
deviations from the mean this is no good
the empirical rule uh becomes not as
useful as you would like it to be so
what we're going to do is we're going to
use a chart to look up these values in
the area under the curve
alright so suppose I was giving Z is
less than or equal to two point four
three now keep in mind if you don't have
Z you're going to have to find it it's
just x minus mu divided by your standard
deviation the raw score whatever it was
divided by the population mean divided
by the population standard deviation but
I'm giving you Z so I want to find the Z
values less than two point four three
all right if we draw a picture two point
four three you remember zero is right in
the middle so two point four three we'll
just borrow ballpark it's right around
here and I'm with the area left to that
so I want this shaded region what you're
going to have to do is flip to your
z-score charts all right and for me it's
called a table a and I have one chart
with negative Z's on the left here you
can see these are all negative values
and the next page is all positive values
all right and what we're going to do is
given this z-score two point four three
these charts give us the area to the
left of a specific z value so if you
look at the picture right here given
this specific Z value these charts does
give us the area to the left of it and
you can see Z is over on the right half
of the data because these are all
positive z-scores z-scores lie to the
right of MU which would be zero
and here are the negative ones so Z is
to the left half of this graph alright
and so we had Z was two point four three
so what we're going to do is on this
graph you're going to go over to two
point four so we're in this column right
here and on the top line it up with
point zero three two point four three is
right here point nine nine two five so
we could say that
ninety nine point two five percent of
the data lies to the left of two point
four three now in this next example
suppose we wanted the area between let's
say this is negative one point six seven
I'm just ballparking these negative one
point six seven I know it since it's
negative it's going to be to the left
half and one point four four I'll set
this positive is going to be on the
right half I want the area in between
this alright and what we're going to do
is we're actually going to find the area
from here and down and then we're going
to find the area from here and down and
then we we want this left over so what
we'll do is we'll just subtract the two
so we're going to look up negative one
point six seven on our chart
all right and negative 1.67 so here's
negative 1.6 I'm going to scroll over to
seven so I have point zero four seven
five
alright next we going to look up 1.44
and here's 1.4 scroll over to 1.44
alright this is just a hundredth spot up
here so that's point nine two five one
alright and we're just going to subtract
the two and when we subtract and we get
0.87 seven six so eighty seven point
seven six percent of our data will lie
between negative one point six seven and
one point four four standard deviations
from the mean and the last one here we
have Z is greater than equal to one
point nine and so suppose one point nine
is oh right around here we want the area
greater than this all right now before I
told you the charts only give you the
area to the left of the curve all right
so when I look up one point nine on my
charts it's going to give me this area
well since we know the area under the
curve is one it's always one if I found
this section I could actually just
subtract that from one and I'll be left
with what's left over so we're going to
look up 1.9 as usual in our chart all
right that's one point nine zero so it's
right here point nine seven one three
and we're going to subtract that from
one then when we subtract that we get a
point zero two eight seven and so you
could say that two point eight seven
percent lies to the right of one point
nine