Z-Scores and Percentiles: Crash Course Statistics #18

Today we’re going to talk about how we compare things that aren’t exactly the same - or aren’t measured in the same way. For example, if you wanted to know if a 1200 on the SAT is better than the 25 on the ACT. For this, we need to standardize our data using z-scores - which allow us to make comparisons between two sets of data as long as they’re normally distributed. We’ll also talk about converting these scores to percentiles and discuss how percentiles, though valuable, don’t actually tell us how “extreme” our data really is.
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[@ElizabethAlmey]

it boggles my mind how something that caused a mental breakdown after class could be so clear and simple after this video. thank you

[@DillonScottDVM]

When she didn't say apples to oranges.. I died a little inside.

[@RavynSymone]

same here!!!

[@cezarydudek6156]

Can you say... that was "annoying"? Nyahahaha

[@sudeepjoseph69]

@@cezarydudek6156 Can you say...... that was "coronavirus"? Nyahahaha

[@maniam5460]

So I actually went and did an analysis on Brady vs LeBron based on z-scores. Being an analytics nerd i decided to do it based on more accurate measures of value/skill other than points/touchdowns per game as they are not really that useful. For Brady, I used Total QBR. This takes situational data from every play a given QB is involved in and effectively mixes a whole bunch of other advanced stats and spews out a rating between 0 and 100. The mean of TQBR scores is, by definition, 50 and the standard deviation is 12.2. Tom Brady;s best ever TQBR was 87 in 2007. His z-score is (87-50)/3.03 which is roughly 3.03.This is insane. putting him in the theoretical 99.88th percentile, meaning there will probably not be anyone better than him for quite a while to say the least. For LeBron I used Win Shares per 48 minutes (WS/48) which, as the name suggests, estimates how much extra wins a player gave his team for every 48 minutes he was on the floor. LeBron's best year was in 2012/13 when he had 0.332 WS/48, compared to the league average: 0.071 and standard deviation: 0.154. LeBron's z-score is (.332-.071)/.154 which works out to 1.63. This is still incredible, placing him in the 90th percentile of all time, but is blown out of the water by Brady and his deflated balls.

[@aabhinnav]

The Stattactician My man, you're going places! Great job!

[@GuidoPerdomo]

It seems to me like this stat you used for Lebron is not as associated to the actual performance of him, compared to TQBR. Maybe there's another stat you can use with Lebron, one as sophisticated as that other one.

[@mikes4937]

I'm not certain your analysis is accurate. I did a little research and TQBR is already a percentile stat, not an expected value stat. Therefor it would already be normally distributed, i.e. Tom Brady's best season would put him in the best 87% of QB seasons. If Lebron's z-score really is 1.63, then that puts his best season in the top 89%.

[@maniam5460]

Guido Perdomo Mike S thanks for pointing out the mistake on LeBron. Edited original comment to fit. Concerning TQBR, it’s not a percentile stat in that, although the max is 100, minimum 0 and mean 50, the distribution does not follow a straight line, it is in fact, normally distributed. This means that a rating of 87 doesn’t correspond to the 87 percentile even though a rating of 50 does correspond to the 50th percentile. Regarding my choice of stats, it’s always a pain and there’s never a right answer because in reality there is no one stat that is completely perfect. I might do this again with other advanced stats and find their average z-scores but anyway, I originally when I wanted to use Wins Produced as I think it is the most reliable/ least unreliable stat we have available currently, but wages of wins don’t publish all time lists of players’ Wins Produced which made it difficult. I ended up settling for WS/48 because it is reasonably accurate and best suited to what I was doing

[@mikes4937]

Hmm, if that's the case then maybe the Wikipedia article on TQBR needs an update... "Lastly, the resulting adjusted EPA per play is transformed to a 0 to 100 scale, where 50 is average. The result can be though of as a percentile."

[@jedcartledge6888]

Crash course makes a mockery of the UK educational system. In each 10 minute video, I swear I learn more than in an hour-long class, with all its cutting and sticking and dithering about. Thanks Adrienne Hill, another great vid with a great presenter.

[@dude4742]

crash course helping me in college.. there's genuinely tears in my eyes.

[@AlipashaSadri]

"Call of Civic Duty"... I want that game :D

[@ankushmenat]

Alipasha Sadri make it happen game devs.

[@danielmclaughlin5573]

Yes! Make this happen !

[@jmorrow22]

I'd find a way out of playing.

[@rhoharane]

This happened in the alternate universe where Popcap acquired EA and started making spinoff games.

[@alantam4579]

same

[@FineAndAndy]

I'm just watching this video now, so I'm a little late. But the most dominant player in their sport of all time is almost certainly Esther Vergeer, a wheelchair tennis player who retired on a 10-year, 470-match win streak. Her career singles record was 695-25 for a win percentage of 96.5% (by comparison, Roger Federer's career win percentage is about 82%). I can't think of anybody that even comes close to that level of dominance in their sport, but a possible runner-up would be the pair of Kerri Walsh and Misty May-Treanor in volleyball.

[@rjmacready9690]

Oh my god i finished stats this semester and it killed a part of me.

[@Scerttle]

Bad teachers will do that. I've taken 3 stat courses. Hated it after the first one. Now I love it. It really depends on how it's explained.

[@aaron-n]

Yeah, I had to take a stats course for each of my majors. One was awful and one was great and they covered the same material.

[@nutellalatte7522]

Jordan Armstrong sameeee wanna go to therapy together?? 😭

[@theodorechandra8450]

I got a combined course on statistics and stochastic I believe I died for a full hour

[@epiren]

And you haven't yet met BIOstatistics...

[@ashishgautam3679]

I saw many videos on Z score, but none were close this this one in explaining why in the world do we use Z scores. Most other videos concentrate more on calculations and formula rather than how Z score can be used in real world. After watching this video my understanding is much better on the uses of Z scores.

[@faithlindahl8484]

ngl I think the likelihood of being on a flight with Chadwick Boseman IS way smaller than hitting 4 red lights. RIP

[@sydznl]

I've already passed my statistics last year but only when I look at these video I truly understand statistics. Thank you.

[@murphygreen8484]

Please never stop making these videos!

[@rparl]

Dan Murphy All of the Crash Course vid series are limited length. But after this one ends, she'll be in another, later.

[@murphygreen8484]

Ross Parlette I know. But I will be sad when this one ends. I think this has been my favorite series so far.

[@TonyWesley]

In the Z-value example for ACT and SAT, she picks the value for 0 by taking the mean of the scale of the scores. She gets the mean for the SAT by taking the midpoint between 1600 and 400, i.e., 1000. And the mean for the ACT by taking its midpoint between 36 and 6, 21. But that's not necessarily the population mean. To calculate the Z-score, she should start with the population mean.

[@jeffreybernath6627]

Sir Donald Bradman. GOAT. 4.4 SD above the mean. Incredible. No complete sentences.

[@Fr00stee]

Perfect timing I was just reviewing for my math sat subject test

[@chuckfan50]

Just passed my stats class... thank god!!

[@aymanestitou8510]

please I have an exam in 2 week on stat and I have no clue about it , give me advice !!

[@AVEdrums]

Thanks, good to hear these things again cause I need to redo my statistics exam, yay

[@neekasworld6695]

Crystal clear! Thanks for sharing!

[@JohnSmith-nc9ep]

Stats understanding +1, thanks :D

[@kujmous]

We're so close to kurtosis! So close!! I was talking with a friend who is trying to make a DnD stats-rolling schema with randomness, but not too much. Love this series, btw. ~thumbs up~

[@jacquelinelabovitz4613]

really incredible and spoken so well

[@iamtechboy3298]

Madam you're the one who gave me push to learn statistics. Thanks have long healthy life

[@emmariesoriano1945]

I learned a lot! Thanks for this!:)

[@research1917]

Crash Course is helping very much!! Thank you so much :)

[@saikumartadi8494]

probably the best explanation i ever saw .. thanks a lot :D

[@graysonnipper2647]

Amazing. Please keep making these.

[@will4not]

Watched every video so far and this topic (outside the intro video) is the only topic I remember from my last stats class.

[@quinius173]

Great video!

[@claushelge1366]

That was very helpful in understanding z-scores. Also: That sweater looks comfy.

[@moriahmilner6435]

This video would've been so much more helpful like 2 weeks ago! I took my stat test then😭😭😭. This did reaffirm some information though.

[@andrewmwinda216]

yeah now you preach too much

[@reagan_reads_]

Hopefully this will stick with me tomorrow during my stats midterm!

[@bcnicholas123]

How’d you do?

[@rexbarr_5783]

THANK YOU.

[@ThePositiveTarot]

The Big Apple joke gets me everytime 😂😂😂😂😂

[@victornoagbodji]

great series : )

[@kajansanmugasundram6218]

Thanks so much

[@andypeterson2126]

Where was this video while I was in stats class ?!?!?

[@noahmidgette3943]

I just took stats and passed but I never understood what the z score actually meant. I knew how to do the work but didn't understand it. Thanks CC! I'll have to binge this series sometime.

[@alantam4579]

Please make a video on hypergeometric distribution

[@pre-eminencenkomo3698]

Writing a stats test today. Wish me luck🤞🏽

[@valor36az]

Lady you’re hilarious, you make stats interesting

[@ianrbuck]

"Call of Civic Duty" killed me!

[@akshay.poddar]

CrashCourse is the GOAT!

[@sbombfitness]

The greatest athlete in their respective sport was me in high school playing basketball at my nephew’s preschool. I averaged 100% of the points per game and won every game

[@jamesbaio9327]

The Brady vs LeBron would be more comparable if you were to look at times LeBron passed the ball to someone and THE RECEIVER scored, not LeBron simply due to the fact that Brady does not score directly...he must pass off to someone else. Also, another point to consider would be time available to score. The amount of time that Lebron gets opportunities to be part of the scoring of a game vs amount of time Brady does is different...and under different circumstances. Finding a common ground on which to compare the two evenly becomes exceedingly complex when you consider the differences in how each comes in contact with an ability to be part of a scoring situation. EDIT: that said...THIS WAS AN AWESOME VIDEO and i've not understood z-scores as clearly as I do now! Thanks!

[@knights_limit]

Hi Ms. Hill, I was interested in your challenge, so I decided to do some research on the average points per game and the average # of touch downs per game. All the websites I found listed the stats but not the mean or S.D, so I thought about how these distributions would look, which probably isn’t normal. Wouldn’t that mean we couldn’t do a z test? I also agree that Lebron would be the goat. In basketball; every player has the chance to score in B-ball, but in football only the QB can throw a touchdown. Basketball players’ aren’t all good scorers, so this would weigh the mean downward and create a right skewed graph because of Lebron and other stars. Where as in football, a QB is given the position specifically because of his talent in passing. This would make that position much more competitive and harder to be ahead in. Hope this counts as an analysis!

[@Kilroxx]

Although I'm not sure of the best analytics for either sport, I took a stab at comparing Lebron James and Tom Brady on two of their respective sports' stats. I'm sure there are some basketball statisticians who have worked points, rebounds, assists, and blocks into some super-score of basketball players, and there are just too many arguments of how to rate football players that I have decided that Passer Rating is the best for QBs and no one can stop me (because I already did the calculations). I used points per game for the NBA/ABA, taking the top 250 scorers in this regard, and passer rating for QBs, taking the top 121 players in this stat. Lebron James has scored 27.15 points per game over the course of his NBA career, putting him 4th on the all-time list of players. This ppg is 2.89 standard deviations above the mean of 18.81 ppg, at a standard deviation of 2.88. Tom Brady has a career passer rating of 97.6, putting him 3rd on the all-time list of QBs. This passer rating is 2.41 standard deviations above the mean of 82.23, at a standard deviation of 6.34. So, in this very singular test of these very multi-talented athletes, Lebron is the GOAT GOAT.

[@Vicbmo]

If only this video was published 2 week ago x"c

[@BeatriceGane]

Ffs my exam was at 2pm today I KZbind this today and nothing good came up !!

[@sy-vf4js]

for the apple or not apple example @8:08 , shouldn't it be 0.01% instead of 0.001%? 100%-99.9%

[@undergroundskeptic2916]

I MISS YOU ADRIAN HILL!!!!!! BACK THEN IN SOCIOLOGY GREAT TO HAVE OYU BACK SHAKE HANDS

[@Noob___Noob]

I passed 1st and 2nd year Statistic courses, but never understood what the hell they mean. LOL Good to know now.

[@MoeZeppelin]

good explanation of z-scores, but comparing the SAT and ACT is more complex than just standardizing results statistically. the standardization assumes the tests are equally valuable in determining one's actual fitness for legitimate college level work, and that is categorically untrue. one of the tests is much less challenging and has much less utility in determining academic fitness for legitimate college level academics. but that's not surprising given the general decline of academic standards over the past 40 or so years, lol. so, if you have a choice, always choose the ACT. it's an easier test, and the scoring methodology makes it easier for you to make the cutoff at most institutions which accept it :-)

[@alphameetpatel]

Good.

[@kylehenderson9489]

What is it with apples when learning about mathematics? Is it something about Newton or is it just the fact that Apple states with the first letter we all learn growing up, A?

[@carinakarina5966]

Are the z scores per game 3.7 for Labron and 1,19 for Tom?

[@ActiveAdvocate1]

Me: "AH! NOT Z-SCORES!!!" My Evil Statistics Professor From Hell: "Muahahahahaha!!!" Actually Z-Scores weren't that bad, not compared to Sum of Squares. Have you covered those yet?

[@supermairo987]

my stats exam is literally today and i haven’t studied at all :))))

[@caitlinsmith5075]

Statistically Don Bradman is actually the GOAT

[@caitlinsmith5075]

His batting average of 99.94 (runs per dismissal) has a z-score of 4.4, which is just nuts. (also of note is that the next highest batting average is 61.87 by Adam Voges)

[@fatsquirrel75]

I think Walter Lindrum (billiards player) might be up there. He was so good they had to change the rules of the game at least twice just to give others a chance. At the world championships he had to give a 7000 point headstart to all competitors (when average score was 20000) and he still won it.

[@markkavanagh9635]

For the hockey fans: probably Wayne Gretzky (so many records)

[@Mewzyque]

Z-SCORES are basically the standard deviation

[@imofage3947]

I heard video game. What is Call of Civic Duty and where can I buy it?! Is it any good?

[@josh3658edwards]

Are z-scores and their corresponding percentiles independent of degrees of freedom?

[@snapple5950]

I mean this in the most positive way possible but you remind me of my mom

[@yourtypicalkamfosfor1314]

i was waiting for dftbaq at the endcard😅

[@brittanydickerson1726]

she had so many opportunities to say apples to oranges, but she said apples and then said grapefruit... that broke my heart 😂😂love these vids tho

[@awaisafridiyt7736]

wow script writer is Zulaiha Razak !! MashaAllah. guys your channel is so helpful. Thank you everyone.

[@CesarTrombeta]

what is ACT and SAT?

[@MicrowaveOvenVideo]

I think my chances of meeting Chadwick on a flight has significantly gone down since 2020.

[@danhillyer33]

4:31 I thought they were zombies. You can't convince me otherwise

[@goxdie000]

Wow, Wayne Gretzky gets a Z-score of 4.6 SD!!!!!!!

[@adriancaceres1472]

He's a freaking outlier! GOAT

[@ir6aish]

In order for us to find "how much higher above the average score," each player is, we need the stats of all players that shows touchdowns/points per game/per player. Where can I find those stats? or am I misunderstanding things ?

[@rparl]

Sa'ed Abu-Haltam I think that's an exercise for the reader (viewer). Fortunately, there's a big wide internet out there. Play around.

[@samaysoni4015]

Did anyone notice the boxplot decorations in the back?

[@julienf2301]

Many different examples, NONE of those had any reason to be described by a normal distribution ! When statistics are applied to science it's most of the time Gaussian, because we're aiming for reproducibility and therefore averaging independent draws from the same distribution (in which case the Central Limit Theorem applies). Then z-score makes sense. But there's nothing like this in those examples (in a university entrance exam for instance every student is just a different individual), and using p-values would give terrible results.

[@0MoTheG]

You are right, at 2:25 she slipped in the normal distribution. As usual without giving any reason nor making clear that this is an assumption.

[@SammyNewman]

What about t-scores?

[@nicogloria5378]

When is “call of civic duty 2” coming out ?😂

[@AdamPetersonRP]

4:29 THAT FACE

[@EdEddnEddyonline1]

ACT composite of 15

[@ArtingFromScratch]

@crashcourse did someone actually win something? if i guess what it is can I also have one? is it the angler fish plush?

[@donnysandley4649]

I'm totally with yeah darlin 😏👍

[@nutellalatte7522]

Why would you relese this 4 weeks after my stats exams 😭😭😭😭

[@aka_pcfx]

The most intruiging plotpoint this decade: is it an apple?

[@Joeviocoe]

Call or Civic Duty 😄

[@KyloPrime]

The GOAT is Kobe Bryant “RIH”

[@-fantini84]

the GOAT is Gretzky

[@urbanlegendsandtrivia2023]

The sports analogy might be flawed. The NFL and NBA (and many other sports) have clearly changed their rules since the 1980s to please fans who love offense and increase television ratings and make more money. Many players from other decades would put up huge numbers against the helpless defenses of today. In addition, in the NBA (even up to the 1990s) players would often get called for "palming" if they put their hands to the side of the ball when dribbling. Thus, the crossover dribble which benefits the offense was pretty rare. In addition, the various balls for different sports and technology have improved over the years in sports. In golf, the balls are so round now the players rarely bogey anymore. These modern numbers are pretty "padded" compared to older numbers.

[@rkpetry]

*_...did we miss something here-being in the 'x'-percentile is only a probability-of-being..._*

[@chrishillswrites6442]

I wanna play Call of Civic Duty now

[@iefe65]

You should have done the normal distribution video before this video not after this video.

[@SkylosSobaka]

I win a goat! How, might you ask, do I know the goat is so light? Well, it was on her scale, and it didn't even move perceptibly! This is a smol goat and will be good for trimming my wheat grass.

[@Sono_1337]

wheres hank .

[@trentshaffermidwestronin5419]

Of course I found this video a semester too late lol

[@davidsweeney111]

Why should test resulots be normally distributed? All students should be getting very similar high marks if the teaching is any good!

[@oldasyouromens]

David the sampling distribution should be normally distributed (centered around the population mean) because of the central limit theorem. The sampling distribution is what happens when you do a whole bunch of samples of the same size (a thousand, ten thousand, a million, infinity) and make a distribution of the mean of all those samples. It gets you a reasonable estimate of the population mean, and due to stuff I can't explain, it's almost always normally distributed regardless of sample data if the sample size is sufficiently large. If the sample size is, say, greater than 30, there is no evidence of non-normality. That's the reason you can do all this stuff.

[@andrewmwinda216]

there is no possing

[@rkpetry]

*_...sort of like statistical-tensors..._*

[@KatiePrivate-jl4fi]

I don't think Jury Duty is in any adult's top three picks of games we play

[@neomashego]

"Call of Civic duty"

[@lordofbuns5636]

Where tf was this video when i took my stats test today

[@ChristopherKalafarski]

Semantically, doesn't being in the 95th percentile mean you're *not* in the top 5 percent? Or at least not always? The way I learned it was that being in a percentile means your value is at or below (ie, everyone is in the 100th percentile), so being in the 95th percentile could mean you're values is at the 1st, or 2nd, or 3rd…or 93rd or 94th. Being *at* the 96th (not 95th) percentile would put you in the top 5%). The way it's phrased her makes me think that that's wrong, and it feels like an important distinction.

[@massimookissed1023]

Christopher Kalafarski , I thought the same thing.

[@fatsquirrel75]

You're confusing being on the 95th percentile and being in the 95th percentile. On a continuous distribution you're not going to find your value at precisely the point where 95% are below you and 95% are above you. It is a dividing line and probabilistically there's 0% chance you'll end up exactly on that line. So if you end up in the 95th percentile then it realistically means that at least 95% of people are below you, meaning you'll be within the top 5%. Sure if you were exactly on the line then you're better than 95, and worse than 5, but typically if you're in the 95th percentile then you are within the top 5%.

[@andrespalacios6540]

@Christopher I think you're right. I prefer to use, "lower", "higher", "two highest" etc. to avoid confusion.

[@nitingaglani5743]

8:25 thats not how humans walk

[@omayrag4]

Always doing too much 😂

[@MegaSteven011]

Call of civic duty 😂