Uniform Distribution

And finance Allah shmoop What is the standard normal distribution

Senate Normal distribution is the destruction of the Z Scores

of the data points from a normal distribution Okay but

why do we need to create a new normal distribution

like the new normal Isn't that a thing Wasn't the

normal distribution we already had good enough before We explain

why the standard normal distribution is such a huge improvement

on the plain old normal distribution but we need a

quick recap of the original A normal distribution or normal

curve is a continuous bell shaped distribution that follows the

empirical rule which says that sixty eight percent of the

data is between negative one and one Standard deviations on

either side of the mean ninety five percent of the

data is between negative two and two Standard deviations on

either side of the mean and ninety nine point seven

percent of the data is between negative three and three

Standard deviations on either side of the mean well the

regular normal curve has its peak located at the mean

Ex Bar and is marked off in units of the

standard deviation s right there That's what it looks like

Adding the standard deviation over and over to the right

and subtracting the standard deviation over and over to the

left But what makes it normal The fact that sixty

eight percent of all the data is between one standard

deviation on each side of the means that makes it

normal It's that sixty eight percent truism that makes it

a normal distribution Then ninety five percent of the data

is between two standard deviations on either side of the

mean That's another test for normalcy And ninety nine point

seven percent of the data is between the three Senate

aviation's on either side Another test That's a third test

You passed all three your normal well tons of things

in nature and from manufacturing and lots of other scenarios

are normally distributed like heights of adult males or weights

of snicker bars or the diameter of drink cup lids

or eleventy million other things Okay fun size Snickers have

a mean weight of twenty point Oh five grams of

the standard deviation of point seven two grams and the

weights are normally distributed What that gives us this distribution

of fun size Snickers Wait it's the height of the

graph At any point it's the likelihood of us getting

a candy bar of that specific weight dire the curve

at a point the greater the chance we get the

exact weight This means that the fun size snickers wait

we'll get the most often is that twenty point Oh

five grams size that is smack dab in the middle

Right there waits larger and smaller than that will be

less common in our Halloween candy haul Waits like seventeen

point eight nine grams are twenty two point two one

grams will be extremely rare because there's shofar from the

middle and are at a part of the curve where

we have a very small likelihood of getting those weights

So why should we even mess with the normal distribution

we already have by calculating Z scores to create a

standard normal distribution And well what the heck is a

Z score Anyway We'll answer the first question in just

a sec but a Z scores of value we calculate

that tells us exactly how far a specific data point

is from the mean measured in units of standard deviation

Z scores were a way to get an idea for

how larger small a data point is compared to all

the other data points in the distribution It's like getting

a measure of how fast a Formula One racecar is

compared not to regular beaters on the road but two

other Formula One race cars the Formula One cars obviously

faster than the Shmoop mobile here But is it faster

than other Formula One cars That's what really matters A

Z score will tell us effectively where that one Formula

One car ranks compared to all the other ones we

can speed test If it's got a large positive Z

score it's faster than many if not most of the

cars It has a Z score close to zero Well

then it's right in the middle The pack speed wise

If it's got a small negative Z score well it's

the turtle to the other cars Hairs Why would we

plot the Z scores instead of the scores themselves Well

because the process of standardizing or calculating the plotting of

the Z scores of the data points makes any work

we need to do with the distribution about ten thousand

times easier When we calculated plot the Z scores we

create a distribution that doesn't care anything about the context

of the problem or about the individual means or standard

deviations or whatever Effectively we create one single distribution that

works equally well for heights of people or weights of

candy bars or diameters of drink lids or lengths of

ring tailed Leamer taels If we don't standardize by working

with Z scores we must create a normal curve that

has different numbers for each different scenario And we have

to do new calculations for each scenario for each different

set of values So let's explore the important features of

the standard normal distribution and how it differs from all

the other regular normal distributions The standard normal curve and

the regular normal curve look identical in shape They just

differ in how the X axis this thing right here

is divided Let's walk through an example where we compare

how the normal distribution of the actual data and the

standard normal distribution for the sea Scores of the data

are created at the same time Okay What are we

gonna pick here Well let's pick narwhal tusks They're very

close to normal in their distribution with a mean length

of two point seven five meters and standard deviation of

point to three meters The regular normal distribution of Narwhal

Tusk links are narwhal distribution is that I think we'll

have the peak located above the mean of two point

seven five meters We'll need the Z score of a

data point representing the length of two point seven five

to start labeling the standard normal distribution the same way

we'll Z scores were found by subtracting the mean from

a data point and dividing that value by the standard

deviation of the data To find a Z score we

subtract the mean two point seven five from our data

point also two point seven five to get zero And

then we divide that by the standard deviation of point

two three while we get a Z score for that

middle value of zero Here's the same normal curve of

the Tusk clanks paired with the standard normal curve of

the Z scores Now for the tick marks on the

straight up Tusk link distribution Right there we add the

standard deviation of point two three three times to the

mean of two point seven five to get the tick

marks to the right of the meanwhile we just get

was that two point nine eight and then three point

two ones were adding point to three to it And

then another point that gets us three point four four

There we go and we repeat that procedure on the

left but subtracted three times So we get to point

five to two point two nine And then what is

that two point Oh six on the left Well to

get these same values on our standard normal curve we

need to find some more Z scores The first score

of the right of the mean is that a value

two point nine eight meters It Z score will be

found by taking two point nine eight and subtracting the

mean of two point seven five to get that point

to three and then dividing that by the standard deviation

of point two three while we get one See that's

kind of a little mini proof there The second take

mark to the right will be for data points at

three point two one meters Well when we subtract the

mean we get point four six which we divide by

point two three and get Z equals two and the

third take mark their works out similarly gets a C

equals three See there it is Things will work out

similarly but negatively on the other side on the laughed

when we do the same thing for tick marks Negative

one negative too And then there we go Negative three

Well let's look at the two curves together One is

specific to the data of narwhal Tusk flanks while the

other is standardized to represent the perfect normal curve usable

for all normal data regardless of context or the values

of the means or standard deviations So after standardizing does

the standard normal curve follow the empirical rule Yeah it's

a normal curve After all it's even in the name

standard normal curve See they kind of tipped me off

to those things They're still sixty eight percent of data

points between Negative one and one on the standard normal

curve There's still ninety five percent of the data pretty

negative two and two on the standard normal curve And

there's still ninety nine point seven ten of the day

to pretty negative three and three on standard normal curve

so getting back to the ten thousand times easier thing

Well it comes in when we try to answer questions

like how many of the gummy coded pretzel logs weigh

between twelve and fifteen grams So here's the set up

Gummy coated pretzel log weights are normally distributed with a

mean of thirteen point two grams and a Sarah deviation

of point seven eight grams We want to know what

percentage of pretzel logs that come out of the gummy

bear coding machine way between twelve and fifteen grams which

the company considers their ideal weight range and likely that

customers wouldn't complain and send them back for being too

little or too big If we don't standardize things by

finding the Z scores of our boundary values of twelve

and fifteen grand we'll need some kind of technology to

interpret our mean standard deviation and boundary values in terms

of the normal curve specific to this situation If we

change anything about the problem like the boundary values or

mean or standard deviation well then we'll have to re

input all the new data and start completely over And

that would suck On the other hand since we know

that data are already normally distributed While we can simply

standardize the two boundary values by calculating their Z scores

and use the majesty of the Z table this thing

to answer our questions which is a table telling us

what percentage of data lies to the left or right

of an easy score across the whole standard normal distribution

Many lives were lost and billions of dollars were spent

Teo build this thing so you know you gotta respect

it not to put too fine a point on it

but if we don't standardize dizzy scores we need to

use a unique normal curve and unique calculations every single

time we work with those situations But if we do

standardized to Z scores we just need to check the

one table for every situation It's like choosing to go

to a different store every time we need a different

product or going toe one store that has all of

them in one place like you'd rather go to Safeway

than just the broccoli store and then the egg store

and then the milk store right So let's calculate our

two Z scores for our boundary values and then check

the Z Table to get our percentage of pretzel logs

in the sweet spot that twelve to fifteen range thing

What will take first data point twelve and subtract the

mean weight of thirteen point to giving us negative one

point two grams and then divide that by the standard

deviation of point seven eight which gives us a Z

score there of negative one point five three eight Then

we'll take the second data point fifteen subtract that mean

of thirteen point two to get one point eight then

divide that value by our standard deviation of point seven

eight to get his E score of two point three

eight Well there are two different kinds of ze tables

One shows the area to the left of a specific

Z score The other shows the area to the right

They both give the same info just so we'll use

a left ze table A Siri's of Z scores accurate

to the tense place runs down the left hand side

and the hundreds place for each of those e scores

runs across the top Well the percentage of data to

the left of a specific Z score can be found

at the intersection of a row and a column bullied

around both our Z scores to the hundreds Place negative

one point five four and then two point three one

respectively in order to locate a percentage of data to

the left of each one Well we'll go down to

the negative one point five row then across to the

column here headed by the negative zero point zero four

where negative one point five Avenue intersects with negative zero

point zero four street and we find a percentage of

data to the left of Z equals negative one point

five four of zero point zero six one seven eight

This thing Well well then head way down to the

two point three boulevard then across to the point zero

one road they cross at point nine eight nine five

six So now what What do we do with these

Two percentage is well glad you asked We know the

percentage of data to the left of our fifteen grand

upper boundary Which is that a Z score of two

point three one We also know the area to the

left of our twelve Graham lower boundary at a Z

score of negative one point five four announced time to

merge those two areas Check the area to the left

of the Z score of two point three one on

the standard normal curve This is the percentage of data

to the left of that value Now check the area

to the left of it Z score of negative one

point five four on the same standard normal curve Well

this is the percentage of data to the left of

that value If we cut away the area to the

left of Z equals negative one point five four or

left with the area here between Z equals negative one

point five for ends e equals two point three one

This is the percentage of data between these two values

and you're looking at this really heavily to be sure

that you got enough in that general sweet spot range

They don't get a whole lot of returns from angry

customers Well we just need to subtract the point Oh

six one seven eight from the point nine eight nine

five six to get the percentage of data between those

two values which is yes about ninety three percent so

What does that mean Well that means ninety three percent

of the gum encoded pretzel logs produced will be between

twelve and fifteen grams in weight And that's either good

news or not Well a couple of important safety tips

though Before you all head out to the store for

some more gumming coded pretzel log We should on Lee

try to standardize I'ii do things with Z scores if

the data are normal in shape to begin with If

they're not the data Maki nations here will be useless

to you Make sure you're paying attention to what kind

of ze table you have again Some show areas to

the left while others give areas to the right and

specific Z scores Every time you've got a set of

normally distributed data you should standardize the situation by finding

Z scores And while you'll save yourself a ton of

work in the long run what least tons of stats

work if we can't help you Sorry I do