As you may recall from your statistics class
to calculate the confidence interval you need to know
what is the number of observations you've been working with
and you need to specify the limits
of the probabilities you want to incorporate into your
confidence interval.
You might have also remember that for small
observation data sets we usually calculate
the confidence interval using the t-statistic, or the t-distribution
and for large datasets we use the z-statistic
therefore
when we go to calculate the upper limit that is calculated as the mean plus
the z times the standard deviation of
the sample divided by the square root of N the number of samples
and the lower limit is the same calculation accepted just
mean minus that quantity there.
The z is what we get from the normal distribution
so if we go to quick draw a quick graph of the normal distribution
our z values they were trying to capture
sort of define the limits of what are all the uncertainty in our data
a what we wanna find. And so the areas
outside ... the areas outside the z (you can see there highlighted in yellow)
is what we call our alpha value in other words the summation
of the area under the curve from negative infinity to
negative z plus the area under the curve from
z to positive infinity.
The summation of those two areas is our alpha and alpha represents
how much uncertainty we're willing to
ignore - that we are comfortable ignoring when we report this value.
To find out where that Z value is
along the probability distribution function we
do this little calculation one minus alpha alpha over 2
and that's going to give us the the
value the probability from negative infinity up to positive z
typically
we will use an alpha 0.05 that's a that's totally arbitrary
somebody back in the day said, "you know what we like to
we like to consider ninety-five percent of all the
uncertainty and that has kind of stuck so we ended up using alpha 0.05 all the
time
that corresponds to a probability of 0.975 I know this may sound kinda
confusing because we just said you
didn't we just say the alpha is 0.05, so wouldn't it be 0.95?
But as you can see there if in the probability table the probability
distribution table that you have already
the integral goes from negative infinity
to z and because
the confidence interval chops off the upper and lower tails
and the summation of the upper lower tales as 5 percent
then that means each other tales is only two and a half percent so it's actually
one minus two and a half percent. That gets us 0.975.
so the way bulletin 17 be talks about this now remember we're talking about
the log Pearson tight
redistribution they have similar ways
so there's an upper limit that has a function
that is this calculation where the mean
%uh the logs plus the case statistic
the upward up found upward limit
k statistics and the standard deviation that'll get you the upper limit of
the the
projections a 100 year flood or whatever it is you're trying to do with
and then you still have to remember to calculate X
up alright then the lower limit is same thing
%uh but you have to change the key statistic value so there's a
K up Anna K lower and that's the
the key statistic is a statistic that corresponds to the log Pearson type3
distribution
so que up is calculated using this formula
I notice the plus sign and the kale oo is calculated same plus the same formula
but with
negative sign they're so we don't know here's these
A's and B's is work and I'm familiar to you all
on the case if T is in this equation is
corresponds to do the key-value
over return period t so the key statistic
for a return period ft so for two hundred years storms the case statistic
for the hundred-year storm
so in this case a is just this quantity one miner Z squared over two times the
quantity and -1
and be is the key statistics squared minus the disease statistics cleared
over N
so both as you can see there this confidence interval
in its basic form is essentially a function I love
the number of observations and the probability that we're dealing with is
Easter to suggest like
the other one but we're kinda looking at it little bit differently because we got
us
a different probability distribution working with you
so that's the value like I said is this the statistically working with so
do quick example the sort of illustrate how this works
so let's say we've calculated that the 100 year flood forgiven river is 3800
in a CFS or three thousand eight hundred CFS
and that the me know the log stream flows is 3.1
the standard deviation to the log %uh the stream flows
standard deviation above the law gives the stream flows was gon come a mouthful
is point 289 and the skew is negative points 7
and of course the number of observations is 41 years
so for we want to do is find a 95 percent confidence limits
that means our alpha is point 05 Serena take the typical
Road the difficult path here when this so that means
we define what is the Z value and
the probability equal the point 9750
so before I tell you that answer once you get out you probability distribution
table
and father video inglenook that up right now
tell me what is dizzy value you know get that ready to go
all right did you happen to get 1.96
I hope so because that's the right one
so the Z value we're out at 1.96 then there's only
two and a half percent remaining to the right tail and a man were at negative
1.96
there's two and a half percent lower than that XO
so we're trying to find here is the
k statistic corresponding to the hundreds your event
so we go back to the log Pearson 3 table
we look up the skew as at negative points 7
and we try to find where the probability is point 01 so we find the key statistic
that corresponds to skew have negative points 7
and a probability of point 01 now to give us the hundred-year
k statistic so on she could do that go to the table
that's also part of the year when your handouts and look at that value
alright so did you find this value 1.80 621
I hope so hopefully I found the right one hopefully you found the same one
and you not cursing my name right now
so let's go in calculate eighty and be
a is
positive you can you do yourself first
point 952 that's what I got so be is equal to
plug in all the numbers and we have
place where you can pause the video and calculate the number yourself
3.16 now that's what I got
so just to save time calculating I'm gonna start by calculating the key
statistic squared minus AB
to the 1/2 there and I get a value
I love point 4955
alright so now we know we can substitute that into the
into the key up in kilo calculation so let's do that now
k of equal to the key statistic
that the 100 year mark plus
that quantity which is calculated divided by a
and I do the same thing for kale oo ok so for unity the
case artistic minus that quantity above their
and so these are the answers that I get here
a 2.4 18 and
1.37 seven so good move over to the left a little bit in saving up through the
bottom there who took over the left calculate the Y up
and the while oh so I go back to the beginning my why bars 3.1
my k up is 2.4 18 in my
standard deviations point 289 the equation now because the same thing
there
except the case that his dick is changed for the lower bound
and that gives me three-point 8
for the upper why up in for the Y lows 3.5 but we're not done here because
they're talking about
these are in log units so we need to take the anti log
those values and get
the ex up is six excels in 310
and the X lo
is what you do this when are you and what do you think
last number go ahead press the buttons do it yourself
alright like X Lo is
3,160 to CFS
so that's the spread on the data so what we're saying here
what these with this result tells us is that we
we project that the your project but we
estimate that the 100-year flood is three thousand eight hundred CFS
we her excuse me if this calculation were Ron
an infinite number of times it's a lot of times we expect that
ninety-five percent of these calculations
will give us a value between 6,000 310
and 3,000 162 so as it is really tells us is that if we were to repeat the
this scenario this calculation over and over again using
different observed data but along that same river
different observed data along the same river may be different intervals along
the same place
if we did that over and over and over and over and over again we would expect
that ninety-five percent of the time we would be between
6310 on the top and 3162 unabomb
so hopefully that helps the fairly easy thing to do
when you project your hundred-year streamflow to just plug these numbers in
honest you know especially if you take the time to
to get him ready and put them into Excel so
over that helps you sort out how to do confidence intervals