Understanding Z-Scores and their Applications in Statistics

Introduction

In the world of statistics, understanding the concept of z-scores is crucial for interpreting data and its distribution. This article delves into section 5.3, where we explore the process of finding z-scores based on probabilities, specifically when trying to determine cutoffs for various scenarios. We will look at practical examples such as university admission scores and medical research, wrapping it all up with detailed illustrations and calculations.

What is a Z-Score?

A z-score indicates how many standard deviations an element is from the mean of a data set. The z-score formula is:

[ z = \frac{x - \mu}{\sigma} ]

Where:

( z ) is the z-score
( x ) is the value in the dataset
( \mu ) is the mean of the dataset
( \sigma ) is the standard deviation

This standardization process allows researchers to understand how far a particular data point is from the average, which is vital for further analysis, particularly when examining probabilities.

Finding Z-Scores from Probabilities

In section 5.3, we reverse-engineer the process explored in the previous section (5.2), where we calculate probabilities from z-scores. Here, we start with probabilities to calculate corresponding z-scores. This reversal is especially useful in applications like:

Determining admission criteria for universities based on test scores
Setting benchmarks in medical studies based on patient age or conditions

Example 1: Finding Z-Score for a Cumulative Area

Let's say we want to find the z-score that corresponds to a cumulative area of 0.3632. We assess whether the z-score should be positive or negative. Since this area is below the mean (0.5000), we look for a negative z-score. According to our z-table, we identify the closest area, and we find:

For area 0.3632, the corresponding z-score is -0.35.

Example 2: Z-Score with Area to the Right

Next, consider the problem of finding a z-score for which 10.75% of the distribution's area is to its right (equivalent to finding an area of 0.1075 to the right). This requires us to convert to the left side:

Area to the left = 1 - 0.1075 = 0.8925.
Using the positive z-table, we find the closest area is 0.8925, which corresponds to a z-score of 1.24.

Example 3: Finding Z-Score for the Middle 95%

Next, let’s examine the scenario of wanting to find a z-score such that 95% of the distribution's area lies between negative z and positive z. We know that this means:

Remaining area (2.5% on each side) would have to be calculated, leading to the area we find in the z-table.
Using our z-table, z = 1.96 is the appropriate score for our purposes.

Percentiles and Their Applications

Percentiles provide additional insight into how a particular score compares to the rest of the distribution. If we want to find the z-score corresponding to the 83rd percentile, it means that 83% of observations lie below this point. This translates to an area of 0.83.

Using the z-table, we look for the area that’s closest to 0.8300. This would translate to a z-score approximately equal to 0.95.

Converting Z-Scores Back to Data

Understanding how to revert from z-scores to the original data is equally important. Once we calculate the z-score, we can use the rearranged formula:

[ x = \mu + z \cdot \sigma ]

For example, for a set of cat weights:

Mean (( 9 )) and Standard Deviation (( 2 ))
For z = 1.96:
[ x = 9 + (2 \cdot 1.96) = 12.92 ] and rounding would give approximately 13 pounds.

Example Conversion

If we had a z-score of -0.44, we could determine the weight as follows:

[ x = 9 + (2 \cdot -0.44) = 8.12 ] leading to approximately 8 pounds.

Conclusion

The relationship between z-scores and data probabilities is fundamental in statistical analysis. Understanding how to navigate between these two forms and utilize z-scores effectively is crucial for making informed decisions in academic, medical, and various research scenarios. By mastering these calculations, you not only enhance your analytical capabilities but also ensure accurate interpretations and applications of statistical data in real-world situations.

so today we're looking at section 5.3 and um we're going to kind of do the opposite

of what we did in section 5 2. in section 5 2 we took a piece of data we changed it to

a z-score so that we could find the probability that um that piece of data did or didn't

occur or what were the chances that it would occur and in section five three what we're

going to do is we're actually going to start with the probability and work our way back to the z-score and then from

the z-score work our way back to a piece of data so that if you wanted to ask a question like um

for instance reading from the paragraph here for instance a university might want to know the lowest test score a

student can have on an entrance exam and still be in the top 10 percent um and so

we're kind of looking at what piece of data would we find acceptable given this situation so the lowest test score

student can have on an entrance exam and still be considered in the top 10 percent

or a medical researcher might want to know the cutoff values for selecting the middle 90 percent of patients by age

so sometimes we have a probability in mind and we'd like to apply it back to the data

and so we're going to kind of undo the process that we have done in section five one and five two or work it

backwards so for example one example one is telling us to find a z-score given an

area and one thing i should mention is that sometimes the area we're looking for is

not found on our chart um the chart that we have doesn't cover every single four digit number that there is between zero

and one so sometimes if we can't find the exact match on the chart we go with the number that is closest on the chart

so number one says find the z-score that corresponds to a cumulative area of 0.3632

so the first thing i kind of have to think to myself is whether i think my z-score should be a positive z-score or

a negative z-score if i'm looking for an area of 0.3632 that means that we haven't made it to

the mean yet because when we get to the mean we have gotten to an area of 0.5000

and so since it's 0.3632 i would be looking for a negative

z-score and so if i bring my little chart over here

we are looking for a z-score of 0.3632 that goes along with an area of 0.3632 um so let me grab my highlighter here

um 0.36 i'm looking down towards the bottom like when i get to negative point four i'm noticing i've got uh point

threes um i'm moving on down point three six three

i actually find that specific value on the chart here and again sometimes you won't find the

specific value sometimes we'll find the closest one this one actually happens to be there

and that one corresponds with a z score of negative point three five

so my z score let me slide that over here so my z score then

would be negative 0.35 my next one says find the z score that

has 10.75 of the distributions area to its right

um so some things to think about there they are saying 10.75 of the distribution's area to the right

so if i would draw a standard normal curve that would look something like this

we want 10.75 to be on the right side

and so um 10.75 that's the same as 0.1075 and so we want this space right here

to be 0.1075 the thing is when i go over to my charts my charts give me area to the left of

the z-score not to the right of the z-score so my first step here would be to take 1 and subtract

0.1075 so that i can have the area to the left when i take

1 minus 0.1075 i get .8925 so .8925

and again that would be this area over here and that's the area i need um because

again when i go to my charts my charts give me the left side not the right side um this time my number is bigger than .5

so this time i'm looking for a positive z score so

if i get the positive chart up here i'm looking for 0.8925

so 0.8925 and i think on this one again it's actually going to be a number on my

chart um 0.8925 i saw it just a minute ago is right there

and that is a z-score of 1.24 so my z-score

would be 1.24 so you try to find the value if you can

find the exact four digit number in the chart that's great if you can't then we go through and we

try and find the number that is closest and sometimes you might have it be exactly in between and we'll look at

that in a future example so for this number two um this is kind of an interesting problem

we are trying to find the positive z-score for which 95 of the distribution's area

lies between negative z and positive z um so what's a little bit tricky about this

one is when we look up a z-score we always get the area to the left of it in the chart

and for this one we are looking for two scores and we want the area between whatever

this z over here this would be the negative and this z over here this would be the

positive we want um 95 percent to be between those two values

so in the center here this area we want this area to be 0.95 so this area is supposed to be 0.9500

the thing is if i go look at 0.9500 my chart is assuming that i started on the very left side and worked my way all

the way across to the positive z and right now i don't have the area of this little space added in here

um it the 0.95 isn't complete enough of a number i need that area as well and so if 95 percent

is between negative c and z that leaves me five percent on the outsides

and so if i take and i um divide five percent by two since i have two outsides um i get two point five percent

and so the area right here is zero point zero two five zero and so if i want to know this z score

this z score is going to be made up of both of those pieces of area so i would need to take the 0.0250

and i would need to add the 0.9500 to get the area starting on the left side working its way all the way over to

my z-score so i would take 0.95 and i would add 0.02

and i would get an area of 0.9750 um 0.9750 that is definitely

a positive z-score because it's above the mean and they told us we were looking for a positive z-score

so if i bring that chart over we are looking for 0.9750 and so if i go looking in there 0.9750

right about here i believe .9750 and that is uh 1.96 so our z score would be 1.96

they didn't ask us for the negative z score but technically this would be 1.96 and this one would be negative 1.96

i know that started to look like a 7 there negative 1.96 so they only asked us for

the positive value but the idea when you want to find the um z-score between two values um is first of all

you got to think about the fact that the percentage they gave you is the number in between those

and unfortunately for your chart you need all the area from the left over to that final z-score so you are going to

have to figure out how much area is in that little space that we don't have add those two areas together and then that

total is what you would go look for in your chart so the next thing we want to look at

we're going to be doing percentiles and a lot of times when you do the percentiles this is where the areas in

our chart don't exactly match so the things we're looking up aren't exactly there and so i just want to talk through

a couple scenarios there i'm going to actually start with the blue scenario and let's say that the area we are

trying to look up in our chart is 0.8773 so that's a positive or that's a bigger

than 0.5 so i'm looking at positive z scores and i go to my chart and i cannot find

0.8773 um i can find a z-score of 1.16 and that has an area of 0.8770

and i can find a z-score of 1.17 that has an area of 0.8790 and so this number is somewhere in

between these two areas and so when that happens

um you pick the number that it is closest to

so um 0.8773 is closer to 0.8770 than it is to 0.8790

and so on this problem we would go with a z-score of 1.16

that would be our z-score that we would use so if you can't find the area exactly you go look for

the one that's closest if i go over to the black example here uh this time our area is .0521

and so again i went to my chart this time i went to the negatives because this area

is less than 0.5 and i found two z-scores that are closest negative 1.62 has an area of

0.0526 and negative 1.63 has an area of 0.0516 the problem that i run into on this one

when i try and see which score i am closer to my area that i've been given is exactly

in the middle of these two is not closer to one or the other it's exactly in the middle of those two

and so if your area is exactly in the middle of two z-scores what you're gonna do is take the two z-scores and add them

together and divide by two and when i add these together and divide

by two i'm going to get negative 1.625 my z score is actually going to have three decimal places in this case so if

you can find the number that you're closer to then you just go with that z-score

if you are exactly in the middle of two z-scores you're going to average them and use that as your z-score

um the second thing we want to look at before we go to the next problem is again sometimes they're going to use

percentiles and so if they tell you something like x

represents the 83rd percentile and sometimes we write that with a piece of 83.

please remember that that means 83 percent of the data is below that x value and 17 of the data is above that x

value and so when we think in terms of area if we think about a z score

when we look at the 83rd percentile what we are kind of trying to say is that the area

to the left of z should be 0.8300 and then the area to the right of z should be 0.1700

because again the 83rd percentile means 83 percent of people or data or whatever

were below the x value and 17 were above so we will be using that on

the next couple examples so for um example

two here we are supposed to find the z-score given a percentile and um so i'm actually going to start

with number two because number two is going to be the fastest easiest one here um for number two if you are looking for

the 50th percentile that means 50 below and 50 above what that means is that you are actually at the mean

and if you are at the mean it means our z score should be

zero because um on our standard normal curve distribution um the mean is in the center and the mean and z equals zero

are in the same spot so we don't need any charts or anything extra for that one um now i'm gonna go back to number

one number one says to find p sub five so the fifth percentile and so again if i think about that

what that is saying is that um we are looking for five percent to be below and 95 to be above so my z score is

going to be somewhere here in the negatives and again this area right here should be

the 0.0500 and the area over here should be 0.9500

and so i'm going to use the negative chart and i'm gonna go look up point five zero zero

um so let me pull the negative chart over here and so we are looking for um

point zero five zero zero so point zero five

um i'm looking and okay here's what i notice

uh .0495 and .0505

my number of point 0.50500 so just writing that bigger i am looking for 0.0500

and when i look at the chart i am in between those two numbers and i am not closer to one or the other i am exactly

in the middle of those and so my z scores there are negative 1.65

and negative point six four and so i add those two numbers together

and i divide by two and my z scores should end up being negative one point six four five

so if your area is not in the cur or not in the chart you are going to look for the closest value if your

area is exactly in the middle of two values you write down the two z scores you add them up you divide by two that

will give you your z-score um for number three um

for number three we are looking for the 90th percentile so we are looking to find p sub 90

and again what that means if i look at the curve is that we want 90 percent to be below

and 10 to be above so this area in here is going to be 0.9000

this area over here is going to be 0.1000 and i don't really need that information

the 0.1000 i'm just trying to show you how you would know which area you're looking for we always take the area to

the left so that would be the point nine zero zero zero so um that's a number bigger than point

five so i'm going to go get my positive z chart

and we are looking for an area of point nine zero zero zero um and so if i look at this

um i'm looking at right here we have 0.8997 and right here we have 0.9015

um so my point 9000 is closer to the 0.8996 seven

so i'm going to go with that z-score that z-score is 1.28 so my answer for my z-score my z-score

is 1.28 so the next thing we need to talk about is once you have the z-score so you go

to your chart you look up the area you find the z-score um a reminder we're trying to get back

to the piece of data and so the next thing we need to talk about is how do we find the data

if we have a z-score we've spent a lot of time going the other direction here's your piece of

data go find a z-score now we want to go backwards and so what you'll notice here is they just kind of undo our formula

for a z-score i will do it off to the side here to find a z-score we usually take our

piece of data we subtract the mean and then we divide by standard deviation so we need to undo this process we would

like to get the x by itself so i'm going to get rid of dividing by standard deviation by multiplying both

sides by standard deviation that gives me its standard deviation times z should equal x minus mu

and then to get x by itself to get rid of subtracting mu i'm going to add mu to the other side and so i get standard

deviation times the plus the mean or mu and so they write the mu part first but

basically our piece of data is found by taking the mean and adding the standard deviation times z

so we're going to do what we've been doing on the next example except then they are going to ask us to actually

find the piece of data so once we find the z-score we will take the mean and the standard deviation and the z-score

we will plug into this formula and we will be able to find the piece of data so example three says a veterinarian

records the weights of cats treated at a clinic the weights are normally distributed with a mean of nine pounds

and a standard deviation of two pounds find the weight x corresponding to each z score interpret the results

um so for number one my first z is 1.96 and looking at the paragraph i know that

mu is nine and i know that standard deviation is two

and so our formula to figure out a piece of data is to take the mean and add standard deviation times our

z-score and so on this problem i'm going to take 9

and i'm going to add 2 times our 1.96 and so if i take 9 on my calculator plus

2 times 1.96 i get that the piece of data is 12.92 um if i do number two same idea

number two my z-score is negative 0.44 and so

my data is going to be 9 plus 2 times my negative 0.44

so i'm going to take 9 plus 2 times negative 0.44 and i get a piece of data 8.12

and number three technically hopefully in your head you know the answer here hopefully you're thinking to yourself i

know this answer the answer is nine if my z score is zero then my piece of data is the mean and so the mean is nine

but just to demonstrate my z is zero and if i plug into the formula i take

nine plus two times zero and we get a piece of data of nine notice a couple things about our answers

here um a z z-score of 1.96 that is going to be to the right of the mean and so i would expect my piece of

data for 1.96 um to be larger than the mean so if the mean is 9 and my z-score is 1.96 i

expect to get an answer that's bigger than nine and we did um for number two my z-score is negative

which means that i am sitting to the left of the mean and so i expect that my piece of data that i find is going to be

smaller than the mean which it was and again for number three my z-score is zero which means i expect to be exactly

at the mean and when we went through and plugged into the formula we got exactly the mean

okay so for number 31 here it says in a survey of women in the united states ages 20 to 29 the mean height was 64.2

inches so our mean is 64.2 inches and our standard deviation is 2.9 inches

part a says what height represents the 95th percentile part b says what height represents the 43rd percentile and part

c says what height represents the first quartile so for part a

we need to start by figuring out the z-score that goes along with p of 95 so the 95th

percentile and so what i'm looking for is the area under the curve

should be 0.95 so my z score is going to be somewhere over here this is what we're looking for

and this area in here should be 0.95

so 0.9500 so that's a number bigger than 0.5 so i'm going to go to the positive

z chart and i'm looking for .95 and when i find the area closest to 0.95

i feel like we already did this one um i get 0.9495 and

0.9505 and i my my number is exactly in the middle of those and so we've got 1.64 we've got 1.65

if you take 1.64 and add it with 1.65 and divide by 2

you will get that your z score is going to equal 1.644

and so then moving that back over here um once i have the z score then i can

use that formula that we were just using and so my formula is that my piece of data is equal to the mean

plus the standard deviation times my z-score and so the mean is 64.2

and i'm going to add to that 2.9 times the z-score that i got which was 1.645 and so if i take 64.2 and i add 2.9

times 1.645 i get a piece of data that is equal to 68.9705

and since they've got other values up there rounded pieces of datas uh

we would probably round this to a whole amount usually go back if the mean is one decimal place then your data was

typically a whole amount because usually the mean has one more decimal place past the data and so our piece of data would

be approximately 69 inches so a height of 69 inches would put you at the 95th percentile

i'm going to skip over b and i'm just going to do c because it's the same kind of situation again

um so i'm going to erase what i have here and one thing that's a little different

on c they don't really tell you the percentile they do they just don't quite tell it to

you um as obviously as they do in a and b um a reminder that the first quartile

represents 25 percent and so when we talk about the area under the

curve the first quartile would be 0.25 so we are looking for a z-score it will

be a negative z-score such that this area under the curve is going to be 0.2500

so the first quartile represents the first 25 percent of data or 0.2500 of our area um so this time i'm going to

need the negative chart so i'm going to bring that over here and

we are looking for 0.2500 so 0.2500 um i've got 0.2483

and i've got point two five one four so point two four eight three

that is point zero zero one seven away from our our area and point two .2514

that is .0014 away from our area so technically what i'm doing there is i'm taking

.25 0 0 our area and i'm subtracting those two numbers to figure out which one is

closer so i subtract .2483 and i also subtract um and i know this is going to end up

negative but we kind of ignore that um i'm going to subtract .251 and this one is a distance of .0017

away and this one again i know your calculator is going to tell you it's a negative but you're looking at distance

wise which value are you closer to and this one is a smaller amount and so we would say we are closer to the 0.2514

and so my z-score is going to be negative 0.6 and so i'll slide that back off to the

side and so if my z score is negative 0.67 my piece of data is going to be the mean

64.2 plus the standard deviation 2.9 times my

z-score of negative 0.67 and if i take 64.2 and i add 2.9 times negative 0.67

um [Music] i get 62

and so if we were rounding data to a whole number again we're kind of going backwards so the mean has one decimal so

our data would have a whole amount our answer would be 64. so for number 35

the undergraduate grade point average of students taking the law school admission test in a recent year can be

approximated by a normal distribution as shown in the figure what is the minimum ugpa that would

still place a student in the top five percent and between what two values does the

middle fifty percent of ugpas lie um so let's start with a um notice in the figure there

they give us um the mean and the standard deviation so the mean grade point average for the undergraduates is

3.36 and the standard deviation is 0.18 we want to know what the minimum gpa

would be that would still allow you to be placed in the top five percent so top five percent means that you need

to be in the 95th percentile and so again that means we are looking for a z-score

where 95 of our area is below the z-score so the z-score is going to be over here

and this area is going to be point nine five zero zero

and i'm pretty certain we've looked this one up a couple times but let's go ahead and find it again

um point nine five zero zero um if i look up .9500 i get

.9495 and i get .9505 and so we are in the middle of those two z or those two areas and so my z-score

on this one is going to be z is equal to 1.645 we would add 1.64 plus 1.65 divide by

two we will get a z score of 1.645 um all right once i have the z-score

then i gotta find the piece of data and so i'm using this formula x equals the mean plus the standard deviation

times our z-score and so our mean is 3.36 and our standard deviation is 0.18

and since those numbers are two decimal places our piece of data should be one decimal place

and i'm going to multiply by the z score of 1.645 so if i take 3.36 and i add 0.18

times 1.64 i get x is equal to 3.6561 and i would tell you definitely make

sure you read what my lab math says if it says round to two decimal places round to one decimal place follow the

directions there but in general data tends to be one decimal place shorter than the mean or the standard deviation

and so since this is 3.65 we would round that to 3.7 so your

lowest gpa that you could have that will keep you in the top five percent or p sub 95 um

would be a 3.7 all right part

2 says between what two values does the middle

50 of the ugpas lie

so what are the two middle values um that you can have that will uh the gpas that will basically put you in the

middle 50 percent so again let's start by kind of thinking about what that looks like graphically

reminder that the standard normal curve is symmetric so we are going to be looking for two z-scores

and for those two z-scores we want to make sure that we are in the middle 50 percent

and so if i try and look up a z-score right now i'm not going to get a correct answer um if the area there is 50

that means there's 50 percent of the area left in those two ends and so that means that the area down

here would be 0.2500 and the area over here

would be 0.250 and so we want the area that gets us all the way over to

that farthest z-score and so the area that we would want to look up is the 0.2500

plus the 0.500 and so if we add those we are going to get

0.7500 so that's the area that we would want to look up so i'm going to bring back my positive z

chart again and we are looking for 0.7500 and so let me grab a highlighter here

0.75 um looks like 0.75 is either going to be

this one or this one 0.7486 and point seven five one seven um point seven four eight six is point

one zero four is point zero zero one four away and point seven five one

is .0017 away so the closer value is going to be the 0.7486

and so we would use a z score of 0.67 um so then let me move my chart out of

the way here um all right so we have a z score of 0.67 now a reminder on this one you're

supposed to have two z scores and so this z score is going to be 0.67 and this z score since we were in the

middle of those two with our 50 percent this one's going to be negative 0.67 and so we need to find two pieces of

data we need to find one piece of data for z being negative 0.67 and we need to find a second piece of

data for z being positive 0.67 so the data is going to be found by taking

our mean of 3.36 and adding our standard deviation of 0.18

times our negative 0.67 so if i take point three six

plus point one eight times negative point zero six seven not zero six seven just six seven

i get um a piece of data of 3.2394 and so we would say approximately 3.2 and for our other z-score we would take

3.36 we would add 0.18 times 0.67

and if we did that we should get 3.4806 so our piece of data is 3.4806

and that would be approximately 3.5 so if your gpa is between 3.2 and 3.5 you would be in the middle 50 percent of

where the ugpas lie you

Heads up!

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