Z-Score

Finance Allah Shmoop What are Z scores A Z score

tells us the distance that a data point is away

from the mean using units of the standard deviation always

wanted a career in the high profile field of comparing

apples and oranges Well all you need is a couple

of Z scores and all of a sudden those apples

and oranges don't seem quite so different After all what

Z scores do best is allow you to take data

points from two entirely different sets of data like your

grade in your section of applied psychology of tinder and

your best friends grade in his section of the same

course taught by a different instructor and compare them as

if they came from the same data set to see

which is objectively larger or smaller Or while maybe you

want to do that Teo See who did better in

the course Nothing wrong with a little healthy competition particularly

when you're trying to figure out your tender score Well

Z scores have one job and that's to tell us

how far a single data point is from the mean

But the measuring stick we use isn't in inches or

meters or even Egyptian cubits Well we use the standard

deviation of the data set as the ruler That is

the standard deviation tells us the standard distance or change

from the mean or middle of the data set or

set another way Standard deviations Tell us how far on

average a data point is from the mean of the

data set Well using that standard deviation is a standard

of measure means that will be checking to see how

far our point of interest is away from the mean

in comparison to other points in the same data set

This whole process allows us to compare apples and apples

inside the same data set so we can compare apples

and oranges later on or set another way It's about

how deviant the set of data points is inside of

whatever collection of data we're doing like a very closely

aligned tinder set with low Z scores Might be this

guy in this guy and this guy gather all probably

models and a desperate one with high standard deviation might

be this guy And uh this guy and well this

guy assuming that's a human So that'd be a high

standard deviation on the scores right Okay So let's say

we took the average daily high temperatures last six days

and got now seventy three seventy four seventy five seventy

five seventy six and seventy seven degrees The mean or

average of this data set turns out to be seventy

five degrees Strangely at least one of the days actually

had a temp of seventy five degrees So seventy five

is both an individual data point and the mean of

all the data data points that are both in the

Davis said and equal to the mean of the data

set have a Z score of zero Z Scores can

also take on positive values when the data point of

interest is larger than the Meanwhile take the day when

the temp with seventy seven degrees right well that's definitely

larger than the mean temp of seventy five degrees The

data point of seventy seven will have a positive Z

score A negative Z score happens when the point of

interest is smaller than the mean Like if we picked

the day when the temple is seventy three degrees we're

picking a data point smaller than that mean of seventy

five Any time a data point has a value smaller

than the mean It will also have a negative Z

score So let's look at a different scenario We all

know what perfectly sized pineapple is Yes if asked every

single person in the world envisions exactly the same size

Pineapple must be some kind of weird collective consciousness thing

Let's pretend for the sake of the example that the

perfect size is the mean size of every pineapple that

ever existed Well imagine a pineapple It is quite a

bit smaller than our perfect pineapple The Z score for

this pineapple size will be negative because its size is

smaller than the mean size But as we imagine smaller

and smaller pineapples we get Z scores that gets smaller

and smaller negative values like from negative one two negative

tude and negative three and so on The farther we

get from the mean going left the smaller and smaller

negative Z scores we get right there More negative Now

imagine a pineapple larger than the perfectly sized pineapple This

larger pineapple will have a positive Z score because its

size is larger than the mean size and as that

larger pineapple gets larger and larger will see it have

larger and larger positive Z scores like one two three

and so on Right So how do we actually calculate

this mythical Z score Well the formula's pretty simple We

take the data point X subtract the mean ex bar

and divide the result by the standard deviation s looks

like that You and your friend are both taking the

class physics of quantum neutrino fields but with different instructors

who use different methods and give different assignments but cover

exactly the same material You got eighty seven percent Your

friend got eighty nine percent on the exam Well things

are looking grim for you in the eternal battle of

who's better But how can we really compare scores if

the teachers used different methods of instruction and assessment Well

if we calculate a Z score for each of you

will be able to see how each of you did

relative to or compared to others in your own class

You're class had an average of seventy eight point one

percent with the standard deviation of five point four percent

Well what would your Z score be then What will

take your eighty seven and subtract the mean of seventy

eight point one to get eight point nine Then we'll

divide that by the standard deviation of five point for

giving us a Z score of one point six for

eight you scored one point six four eight standard deviations

above the class average Good for you Now we'll find

your friends Z score Her class had an average of

seventy five point four percent with a standard deviation of

eight point eight percent Well that's interesting so her average

was lower but the deviation higher So what's her Z

score Well her average eighty nine months The class average

of seventy five point four gives us thirteen point six

We divide that by the standard deviation of eight point

eight to get a Z score of one point five

four five Yeah you actually did better relatively in your

course compared to your friend because your score in your

courses farther from the mean in a positive direction where

larger and larger positive Z scores live You over there

then her score is from the mean in her course

While Z scores are literally eveything we should use to

compare apples and oranges We need a little context for

what very large or very small Z scores mean well

Z scores above four and below negative for our pretty

uncommon These kinds of Z scores generally mean that the

data is genuinely very large or very small and it

talks like that compared to the rest of the data

Z scores also have another use their used to create

the standard normal distribution which is like any other normal

day distribution But you know more standard the process of

creating a Z scores often called standardizing ah score or

indexing it if we take a previously existing normal distribution

of heights of adults are lengths of rainbow bass or

weights of Gummi there covered pretzel rods and calculate the

Z scores for every data point and plot them while

we create Then a data set of standardized scores which

is still normal and shape And it's called the standard

normal distribution So yeah just remember that Z scores are

the best way to compare values in two different data

sets We take the data point subtract the mean from

it and then divide that difference by the standard deviation

of the data set positive Z scores indicated data point

larger than the mean The farther a point is above

the mean the larger the Z score Negative Z scores

indicated data point smaller than the mean The farther a

data point is below the mean the smaller busy score

a Z score of zero means the data point is

equal to the mean right is the mean So when

Mom drops an apple and an orange on the table

and demands you compare them well you just grab a

couple of means and standard deviations and calculate yourself some

good old fashioned low calorie Z scores She'll be so 00:07:05.673 --> [endTime] impressed Mmm Tasty