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Finance Allah Shmoop What are Z scores A Z score

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tells us the distance that a data point is away

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from the mean using units of the standard deviation always

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wanted a career in the high profile field of comparing

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apples and oranges Well all you need is a couple

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of Z scores and all of a sudden those apples

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

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Z scores do best is allow you to take data

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points from two entirely different sets of data like your

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grade in your section of applied psychology of tinder and

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your best friends grade in his section of the same

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course taught by a different instructor and compare them as

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if they came from the same data set to see

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which is objectively larger or smaller Or while maybe you

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want to do that Teo See who did better in

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the course Nothing wrong with a little healthy competition particularly

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

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

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how far a single data point is from the mean

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

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meters or even Egyptian cubits Well we use the standard

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deviation of the data set as the ruler That is

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the standard deviation tells us the standard distance or change

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from the mean or middle of the data set or

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set another way Standard deviations Tell us how far on

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average a data point is from the mean of the

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data set Well using that standard deviation is a standard

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of measure means that will be checking to see how

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far our point of interest is away from the mean

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in comparison to other points in the same data set

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This whole process allows us to compare apples and apples

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inside the same data set so we can compare apples

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

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how deviant the set of data points is inside of

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

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aligned tinder set with low Z scores Might be this

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guy in this guy and this guy gather all probably

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models and a desperate one with high standard deviation might

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be this guy And uh this guy and well this

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

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

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we took the average daily high temperatures last six days

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and got now seventy three seventy four seventy five seventy

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five seventy six and seventy seven degrees The mean or

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average of this data set turns out to be seventy

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five degrees Strangely at least one of the days actually

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had a temp of seventy five degrees So seventy five

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is both an individual data point and the mean of

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all the data data points that are both in the

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Davis said and equal to the mean of the data

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set have a Z score of zero Z Scores can

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also take on positive values when the data point of

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interest is larger than the Meanwhile take the day when

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

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larger than the mean temp of seventy five degrees The

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data point of seventy seven will have a positive Z

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score A negative Z score happens when the point of

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interest is smaller than the mean Like if we picked

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

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picking a data point smaller than that mean of seventy

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five Any time a data point has a value smaller

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than the mean It will also have a negative Z

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

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know what perfectly sized pineapple is Yes if asked every

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single person in the world envisions exactly the same size

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Pineapple must be some kind of weird collective consciousness thing

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

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perfect size is the mean size of every pineapple that

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ever existed Well imagine a pineapple It is quite a

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bit smaller than our perfect pineapple The Z score for

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this pineapple size will be negative because its size is

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smaller than the mean size But as we imagine smaller

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and smaller pineapples we get Z scores that gets smaller

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and smaller negative values like from negative one two negative

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tude and negative three and so on The farther we

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get from the mean going left the smaller and smaller

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negative Z scores we get right there More negative Now

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imagine a pineapple larger than the perfectly sized pineapple This

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larger pineapple will have a positive Z score because its

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size is larger than the mean size and as that

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larger pineapple gets larger and larger will see it have

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larger and larger positive Z scores like one two three

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and so on Right So how do we actually calculate

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

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take the data point X subtract the mean ex bar

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and divide the result by the standard deviation s looks

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like that You and your friend are both taking the

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class physics of quantum neutrino fields but with different instructors

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who use different methods and give different assignments but cover

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exactly the same material You got eighty seven percent Your

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friend got eighty nine percent on the exam Well things

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are looking grim for you in the eternal battle of

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

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the teachers used different methods of instruction and assessment Well

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if we calculate a Z score for each of you

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will be able to see how each of you did

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relative to or compared to others in your own class

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

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percent with the standard deviation of five point four percent

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Well what would your Z score be then What will

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take your eighty seven and subtract the mean of seventy

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

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divide that by the standard deviation of five point for

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giving us a Z score of one point six for

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eight you scored one point six four eight standard deviations

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

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your friends Z score Her class had an average of

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seventy five point four percent with a standard deviation of

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

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

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score Well her average eighty nine months The class average

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of seventy five point four gives us thirteen point six

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We divide that by the standard deviation of eight point

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eight to get a Z score of one point five

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four five Yeah you actually did better relatively in your

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course compared to your friend because your score in your

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courses farther from the mean in a positive direction where

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larger and larger positive Z scores live You over there

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then her score is from the mean in her course

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While Z scores are literally eveything we should use to

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compare apples and oranges We need a little context for

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what very large or very small Z scores mean well

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Z scores above four and below negative for our pretty

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uncommon These kinds of Z scores generally mean that the

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data is genuinely very large or very small and it

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talks like that compared to the rest of the data

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Z scores also have another use their used to create

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the standard normal distribution which is like any other normal

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day distribution But you know more standard the process of

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creating a Z scores often called standardizing ah score or

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indexing it if we take a previously existing normal distribution

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of heights of adults are lengths of rainbow bass or

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weights of Gummi there covered pretzel rods and calculate the

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Z scores for every data point and plot them while

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we create Then a data set of standardized scores which

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

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normal distribution So yeah just remember that Z scores are

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the best way to compare values in two different data

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sets We take the data point subtract the mean from

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it and then divide that difference by the standard deviation

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of the data set positive Z scores indicated data point

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larger than the mean The farther a point is above

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the mean the larger the Z score Negative Z scores

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indicated data point smaller than the mean The farther a

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data point is below the mean the smaller busy score

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a Z score of zero means the data point is

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equal to the mean right is the mean So when

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Mom drops an apple and an orange on the table

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and demands you compare them well you just grab a

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couple of means and standard deviations and calculate yourself some

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

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