Finance: What is the standard normal distribution?

What is the standard normal distribution? Standard Normal Distribution refers to statistical data in technical analysis and the level of standard deviation discrepancy from the mathematical mean. A normal distribution’s data is +/- 1 standard deviation 68% and 95% within +/- 2 standard deviations.

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Transcript

00:21

normal distribution we already had good enough before We explain

00:24

why the standard normal distribution is such a huge improvement

00:27

on the plain old normal distribution but we need a

00:30

quick recap of the original A normal distribution or normal

00:34

curve is a continuous bell shaped distribution that follows the

00:37

empirical rule which says that sixty eight percent of the

00:40

data is between negative one and one Standard deviations on

00:43

either side of the mean ninety five percent of the

00:45

data is between negative two and two Standard deviations on

00:48

either side of the mean and ninety nine point seven

00:51

percent of the data is between negative three and three

00:53

Standard deviations on either side of the mean well the

00:56

regular normal curve has its peak located at the mean

00:59

Ex Bar and is marked off in units of the

01:01

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

01:04

Adding the standard deviation over and over to the right

01:06

and subtracting the standard deviation over and over to the

01:09

left But what makes it normal The fact that sixty

01:12

eight percent of all the data is between one standard

01:14

deviation on each side of the means that makes it

01:17

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

01:20

a normal distribution Then ninety five percent of the data

01:23

is between two standard deviations on either side of the

01:25

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

01:28

seven percent of the data is between the three Senate

01:30

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

01:33

You passed all three your normal well tons of things

01:35

in nature and from manufacturing and lots of other scenarios

01:38

are normally distributed like heights of adult males or weights

01:42

of snicker bars or the diameter of drink cup lids

01:46

or eleventy million other things Okay fun size Snickers have

01:50

a mean weight of twenty point Oh five grams of

01:52

the standard deviation of point seven two grams and the

01:55

weights are normally distributed What that gives us this distribution

01:58

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

02:00

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

02:02

a candy bar of that specific weight dire the curve

02:04

at a point the greater the chance we get the

02:06

exact weight This means that the fun size snickers wait

02:09

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

02:12

five grams size that is smack dab in the middle

02:14

Right there waits larger and smaller than that will be

02:17

less common in our Halloween candy haul Waits like seventeen

02:21

point eight nine grams are twenty two point two one

02:24

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

02:27

middle and are at a part of the curve where

02:29

we have a very small likelihood of getting those weights

02:32

So why should we even mess with the normal distribution

02:34

we already have by calculating Z scores to create a

02:37

standard normal distribution And well what the heck is a

02:39

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

02:42

a sec but a Z scores of value we calculate

02:45

that tells us exactly how far a specific data point

02:48

is from the mean measured in units of standard deviation

02:51

Z scores were a way to get an idea for

02:53

how larger small a data point is compared to all

02:56

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

02:59

a measure of how fast a Formula One racecar is

03:02

compared not to regular beaters on the road but two

03:05

other Formula One race cars the Formula One cars obviously

03:08

faster than the Shmoop mobile here But is it faster

03:12

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

03:15

Z score will tell us effectively where that one Formula

03:18

One car ranks compared to all the other ones we

03:20

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

03:23

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

03:26

cars It has a Z score close to zero Well

03:28

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

03:30

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

03:32

the turtle to the other cars Hairs Why would we

03:35

plot the Z scores instead of the scores themselves Well

03:38

because the process of standardizing or calculating the plotting of

03:41

the Z scores of the data points makes any work

03:44

we need to do with the distribution about ten thousand

03:46

times easier When we calculated plot the Z scores we

03:50

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

03:53

of the problem or about the individual means or standard

03:56

deviations or whatever Effectively we create one single distribution that

04:01

works equally well for heights of people or weights of

04:04

candy bars or diameters of drink lids or lengths of

04:08

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

04:12

with Z scores we must create a normal curve that

04:14

has different numbers for each different scenario And we have

04:17

to do new calculations for each scenario for each different

04:21

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

04:24

the standard normal distribution and how it differs from all

04:27

the other regular normal distributions The standard normal curve and

04:31

the regular normal curve look identical in shape They just

04:36

differ in how the X axis this thing right here

04:38

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

04:41

how the normal distribution of the actual data and the

04:43

standard normal distribution for the sea Scores of the data

04:46

are created at the same time Okay What are we

04:48

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

04:52

close to normal in their distribution with a mean length

04:55

of two point seven five meters and standard deviation of

04:57

point to three meters The regular normal distribution of Narwhal

05:01

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

05:05

have the peak located above the mean of two point

05:07

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

05:09

data point representing the length of two point seven five

05:12

to start labeling the standard normal distribution the same way

05:15

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

05:18

a data point and dividing that value by the standard

05:20

deviation of the data To find a Z score we

05:23

subtract the mean two point seven five from our data

05:25

point also two point seven five to get zero And

05:28

then we divide that by the standard deviation of point

05:30

two three while we get a Z score for that

05:32

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

05:35

the Tusk clanks paired with the standard normal curve of

05:38

the Z scores Now for the tick marks on the

05:40

straight up Tusk link distribution Right there we add the

05:43

standard deviation of point two three three times to the

05:46

mean of two point seven five to get the tick

05:49

marks to the right of the meanwhile we just get

05:51

was that two point nine eight and then three point

05:53

two ones were adding point to three to it And

05:55

then another point that gets us three point four four

05:57

There we go and we repeat that procedure on the

06:00

left but subtracted three times So we get to point

06:02

five to two point two nine And then what is

06:05

that two point Oh six on the left Well to

06:07

get these same values on our standard normal curve we

06:10

need to find some more Z scores The first score

06:13

of the right of the mean is that a value

06:14

two point nine eight meters It Z score will be

06:16

found by taking two point nine eight and subtracting the

06:19

mean of two point seven five to get that point

06:20

to three and then dividing that by the standard deviation

06:23

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

06:25

kind of a little mini proof there The second take

06:28

mark to the right will be for data points at

06:30

three point two one meters Well when we subtract the

06:32

mean we get point four six which we divide by

06:35

point two three and get Z equals two and the

06:37

third take mark their works out similarly gets a C

06:40

equals three See there it is Things will work out

06:42

similarly but negatively on the other side on the laughed

06:44

when we do the same thing for tick marks Negative

06:47

one negative too And then there we go Negative three

06:50

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

06:52

specific to the data of narwhal Tusk flanks while the

06:55

other is standardized to represent the perfect normal curve usable

06:59

for all normal data regardless of context or the values

07:02

of the means or standard deviations So after standardizing does

07:07

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

07:11

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

07:14

standard normal curve See they kind of tipped me off

07:17

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

07:19

points between Negative one and one on the standard normal

07:21

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

07:23

negative two and two on the standard normal curve And

07:26

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

07:27

to pretty negative three and three on standard normal curve

07:30

so getting back to the ten thousand times easier thing

07:33

Well it comes in when we try to answer questions

07:36

like how many of the gummy coded pretzel logs weigh

07:40

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

07:43

Gummy coated pretzel log weights are normally distributed with a

07:47

mean of thirteen point two grams and a Sarah deviation

07:50

of point seven eight grams We want to know what

07:52

percentage of pretzel logs that come out of the gummy

07:55

bear coding machine way between twelve and fifteen grams which

07:58

the company considers their ideal weight range and likely that

08:01

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

08:04

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

08:06

finding the Z scores of our boundary values of twelve

08:09

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

08:11

interpret our mean standard deviation and boundary values in terms

08:15

of the normal curve specific to this situation If we

08:17

change anything about the problem like the boundary values or

08:21

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

08:24

input all the new data and start completely over And

08:27

that would suck On the other hand since we know

08:29

that data are already normally distributed While we can simply

08:33

standardize the two boundary values by calculating their Z scores

08:36

and use the majesty of the Z table this thing

08:39

to answer our questions which is a table telling us

08:42

what percentage of data lies to the left or right

08:45

of an easy score across the whole standard normal distribution

08:49

Many lives were lost and billions of dollars were spent

08:52

Teo build this thing so you know you gotta respect

08:54

it not to put too fine a point on it

08:56

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

08:58

use a unique normal curve and unique calculations every single

09:02

time we work with those situations But if we do

09:05

standardized to Z scores we just need to check the

09:07

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

09:10

to a different store every time we need a different

09:13

product or going toe one store that has all of

09:15

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

09:18

than just the broccoli store and then the egg store

09:21

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

09:23

two Z scores for our boundary values and then check

09:26

the Z Table to get our percentage of pretzel logs

09:28

in the sweet spot that twelve to fifteen range thing

09:31

What will take first data point twelve and subtract the

09:33

mean weight of thirteen point to giving us negative one

09:36

point two grams and then divide that by the standard

09:38

deviation of point seven eight which gives us a Z

09:40

score there of negative one point five three eight Then

09:42

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

09:45

of thirteen point two to get one point eight then

09:47

divide that value by our standard deviation of point seven

09:50

eight to get his E score of two point three

09:51

eight Well there are two different kinds of ze tables

09:54

One shows the area to the left of a specific

09:57

Z score The other shows the area to the right

10:00

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

10:03

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

10:07

to the tense place runs down the left hand side

10:09

and the hundreds place for each of those e scores

10:11

runs across the top Well the percentage of data to

10:14

the left of a specific Z score can be found

10:16

at the intersection of a row and a column bullied

10:18

around both our Z scores to the hundreds Place negative

10:21

one point five four and then two point three one

10:24

respectively in order to locate a percentage of data to

10:27

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

10:29

the negative one point five row then across to the

10:32

column here headed by the negative zero point zero four

10:35

where negative one point five Avenue intersects with negative zero

10:38

point zero four street and we find a percentage of

10:41

data to the left of Z equals negative one point

10:44

five four of zero point zero six one seven eight

10:48

This thing Well well then head way down to the

10:51

two point three boulevard then across to the point zero

10:53

one road they cross at point nine eight nine five

10:57

six So now what What do we do with these

10:59

Two percentage is well glad you asked We know the

11:01

percentage of data to the left of our fifteen grand

11:03

upper boundary Which is that a Z score of two

11:06

point three one We also know the area to the

11:08

left of our twelve Graham lower boundary at a Z

11:10

score of negative one point five four announced time to

11:13

merge those two areas Check the area to the left

11:16

of the Z score of two point three one on

11:18

the standard normal curve This is the percentage of data

11:20

to the left of that value Now check the area

11:23

to the left of it Z score of negative one

11:25

point five four on the same standard normal curve Well

11:28

this is the percentage of data to the left of

11:30

that value If we cut away the area to the

11:32

left of Z equals negative one point five four or

11:35

left with the area here between Z equals negative one

11:38

point five for ends e equals two point three one

11:40

This is the percentage of data between these two values

11:44

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

11:46

that you got enough in that general sweet spot range

11:49

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

11:50

customers Well we just need to subtract the point Oh

11:53

six one seven eight from the point nine eight nine

11:55

five six to get the percentage of data between those

11:57

two values which is yes about ninety three percent so

12:01

What does that mean Well that means ninety three percent

12:03

of the gum encoded pretzel logs produced will be between

12:06

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

12:08

news or not Well a couple of important safety tips

12:12

though Before you all head out to the store for

12:14

some more gumming coded pretzel log We should on Lee

12:16

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

12:19

the data are normal in shape to begin with If

12:22

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

12:24

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

12:26

of ze table you have again Some show areas to

12:29

the left while others give areas to the right and

12:32

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

12:35

normally distributed data you should standardize the situation by finding

12:39

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

12:42

work in the long run what least tons of stats

12:44

work if we can't help you Sorry I do