The Perils of Averaging Averages

all right thank you all all right welcome everyone hope everyone's doing well on this afternoon for those of you who don't know me U I'm John Berto I'm the associate probos for faculty Affairs and it really is my great pleasure to welcome you to the 2023 2024 distinguished scholar teacher lecture series for those of you who don't know the distinguished scholar teacher award was established in 19 1978 to recognize tenur faculty members who are committed to and have demonstrated excellence in instruction and in the research efforts the award is sponsored and administered by the office of Faculty Affairs on behalf of pro uh provos Jennifer King rice and recipients are are selected by prior DST um um nominees and recipients I'm very pleased and honored on behalf of provos rice to recognize Dr Chris lowski as one of our new newest distinguished scholar teachers as we will hear more shortly um his research his scholarship focused primarily in model Theory which is a branch of mathematical logic in addition to his research um Chris has contributed significantly to the University's learning environment uh engaging students in courses he teaches and through his mentorship provos rice and I congratulate um Chris on his this much-deserved award in recognition it's my pleasure now to introduce Don Levy chair of the Department of mathematics who will formally introduce Dr lowski thank you thanks John and uh thank you all for coming um I'm Doran Levy I'm the chair of the math department and it really is a great pleasure and honor to introduce Chris laskovski our new distinguished scholar teacher um you know these introductions sometimes start with you know the the usual stuff of like you know all this boring information you can probably just read on the backs side of the pamphlet that you received uh but nevertheless I'll still mention a little bit of that just to uh you know honor Chris yet again so pH uh Chris received his PhD from Berkeley in 1987 after which he was a more instructor at mat and then he joined the University of Maryland in 1989 as an assistant professor doing mathematical logic we heard that uh ma mathematical logic has been a big there's a big tradition in our department for mathematical logic and at the time that Chris joined us how many logicians were there four five okay Miss by one uh gradually that number went down and over the years as uh people gradually retired and were not replaced by new logicians uh not that long ago our logic group numbered one um so if you look at all departmental reviews uh the report of the logic group was we need more people right uh which actually happened as we recently hired two logicians uh Christian rosenal and artam chernikov and now from one we have three and probably one of the best or certainly one of the best uh logic groups in the country if not in the world so small numbers doesn't mean low quality actually when it it's concentrated really really well uh you can do wonders okay um so I mentioned something about the trajectory of Chris uh coming to Maryland uh but Chris has been nothing short of a wonderful colleague so being a wonderful colleague is a combination of many things it's uh a great researcher so a wonderful Mentor teacher and someone that's in charge and uh involved greatly involved with our Outreach activities but really a great person to have around he's here pretty much every day not many people are and just seeing Chris in the corridors and welcoming him and greeting him and him greeting everyone it's I mean how should I put it differently Chris is a fixture of this department okay um so what Chris does for research is something I'm not going to attempt to tell you about uh which sounds mostly like my letters for the AP uh committees so uh and and I think that actually Chris will also not try to make this attempt and he promised to give us a very uh accessible talk uh so I think all of us know what averages mean and we learn about averages of averages uh but let me just say again that I'm super pleased that Chris has been bestowed this award of a distinguished scholar teacher by the provest office and by the University of Maryland congratulation Chris and with that Chris laskovski okay so hopefully people on Zoom can hear as well so thank you Dr BAU and thank you Doran for the wonderful uh introductions I'm very honored to be here speaking in front of all of you and all everybody on Zoom um I'd like to also just thank um both uh Doran Levy and Larry Washington who were both previous DST winners and they nominated me for this so none of this would have happened and uh in particular I'd like to thank my wife Carol def Francis uh quite frankly without her constant love and support most of much much of my career would not have uh transpired so thank you okay so before launching into things uh just in there was discussion right at the beginning uh today is mold day happy mold day to everybody um again since most of you are mathematicians you might not have heard about this uh you can view this as being a chemist knockoff of Pi Day uh to think for mole think 6.02 time 10^ the 23rd and I sort of cryptically wrote this today is 1023 uh uh the 2023 is redundant or whatever um so uh unlike just reciting lots of digits of pi uh one tends to sit around and tell mle jokes uh this this will play A Part uh as we go along so um as to what I'm going to talk about well uh can read from the thing my main field is model Theory which is a branch of mathematical logic which is a branch of mathematics great but this is really a lousy thing say at cocktail parties or elevator speech it's really so number one someone says what do you do I teach uh what do you teach mathematics the usual thing majority oh I really hate math many times they'll say I got up to level X and then I had a really bad teacher and then I can't imagine what life's beyond that okay um but then there are a few people say math okay yeah what kind of math and then you say mathematical logic and you can see that they tense up oh no and like look down at their hands they're sure that the next words out of my mouth are going to be this statement is false or some sort of a paradoxical thing they're they're looking and trying to remember what's the symbol the the Vulcan greeting from Star Trek or something like this uh uh because for whatever reason pop po press mathematical logic uh certainly discusses paradoxes or uh in and around girdles incompleteness theorem or the collection of all sets is not a set uh or Bertrand's Russell a barber can shave everyone's head except his own just all sorts of things um and when I teach set theory I try to make a really special thing I think the term Paradox is very poorly aimed it isn't a proof that 0 equals one or that things are all falling apart but rather when you see something Paradox it means you need to be careful it means warning D something non-intuitive is going to be happening and rather than bore you with my research I want to give an instance of something in popular press which goes is a a paradox but one should really View and try to see what is going on because it really as we'll demonstrate has a lot of real world applications so with that as a preamble let's start out and let's talk about baseball I said real world but we'll get to more things beyond that um but there let's take two uh players David Justice played for a while uh many years with Atlanta Braves and then moved uh moved on to um to other teams whereas Derek Jeter always played with the New York Yankees what about them well let's look at the years 1995 and 1996 say start with David Justice and uh many of you know about uh statistics in baseball but one of the most popular things is batting average so you take the number of hits so in this case 104 and div by the number of times he comes up to the plate the number of at bats this is just a ratio and uh in this case comes out to it really should be 0.253 but no one says the zero so he say he bats 253 so clearly the larger the number the better off every uh you're doing so in 1995 David Justice had a uh higher batting average than Derek Jeter now you can note here that U for Derek Jeter uh the number of at bats was somewhat lower this was in fact his rookie season and he got called up during the year um so he didn't have that many at bats but then now we passed to 1996 and once again David Justice had a higher batting average than Derek Jeter so in both years years Justice's performance dominated that of Derek Jeter but if you then take the two years combined if you take uh so the combined thing say for David Justice uh he had 104 hits in uh 1995 plus um uh 45 hits um and then divide by the number of at bats is um sorry is uh 411 plus uh 140 so this comes out to the 149 over 551 or in other words they bad it combined batting average of 270 but as you can clearly see Derek Jeter's combined batting average is 310 now at first this should seem kind of odd how could be that David Justice dominated uh Derek Jeter in both 1995 and in 1996 but when you combine these guys it switches okay well this is sort of what we want to be discussing with this now why does this feel strange well suppose we just have numbers so we have a big number A1 and a smaller number B1 and another so A1 is bigger than B2 A2 is bigger than B2 then from that if you add the two big numbers together you get something bigger than the sum divide by two so the average of uh A1 and A2 is bigger than the average of B1 and B2 whenever um A1 and B is bigger than B1 and so A2 is bigger than B2 great so but this is not what you're doing with batting averages remember batting averages are hits versus at bats so when you're getting this combined thing same computation that's here uh you're taking the hits Plus in 1995 plus the hits in 696 divided by the sum of their at bats and in general this is certainly not equal to the lad think about the right hand side is as in my title the average of of averages but this leftand thing is not now I guess uh when I gave a talk like this before Larry Washington pointed out this is very close to we spend a the math faculty math faculty spends a lot of time teaching freshmen that this is not how you add fractions there's a half issue but even without that okay so um keep that in mind so the the the thing that I want you to go for going forward is that care must be taken when averaging averages so uh we're going to get to more sophisticated things than this I promise uh but this whole thing this flipping that's occurring with this goes by the name of Simpsons Paradox and from my general comments I'm really kind of unhappy with the word paradox here because it's simply that we need to be aware of what's going on so now I guess this being the new fangled thing what do I mean Simpsons Paradox I do not mean Homer it is not named after Homer Simpson much as he might like it to be but rather much earlier Edward H Simpson of 1951 from um uh from the UK a rather noted statistician at the time uh always whenever you name something in mathematics there's some ambiguity and some people uh call this the Simpson Ule effect from much earlier Ule in 1903 was sort of aware of this kind of thing and certainly I prefer the word effect to Paradox that just saying something is happening here okay now so this is just something you should be aware of to start off that just when you're looking at data sets uh baseball certainly provides lots and lots of data sets then stuff like this can happen so in the wild which to ISA Chas just means that just uh just in nature without any sort of contriving things uh it's relatively rare but it does occur and uh just other baseball pairings the most recent I could find was actually two Red Sox players Ellsbury and LEL who had the same phenomenon for two years in a row one dominating the other I forget which was which uh you can always check these just if you just literally Google your favorite baseball player stats you're just going to get a listing of these and you can do these on your own somebody painstakingly went through cases of Allstar uh players and found these just among allars very familiar names to Baseball fans fans I guess the last one's a little remarkable with uh Babe Ruth and Lou garri these were the first three years of garri season uh career and he garri actually beat uh Babe Ruth in batting in three years 1923 24 25 but if you add up all of those together babe Bruth dominates uh um l garri in in the sum of the three okay so there's more to it than just baseball this can happen in the wild and it can have some real world significance so let's just imagine now that you are a doctor Circa um 1990 you're a small town and uh your specialty is kidney stone treatments so now a patient comes in woman just huge amount of pain and clear clearly has a case of kidney stones and you want to know what to do what should what procedure should you do to uh to to help uh ease her kidney stones well you're in 1990 and you're on top of your game and you've read the following thing this is at the time it really was the gold standard for what you do with uh uh treating kidney stones there was a long multi-year thing all in the UK uh published in the British medical journal and it compared uh again there are a lot of words renal Cal calculi is of course kidney stones uh and now you could uh gets a little Grizzly but you can either open surgery which is you go in and get the things uh this other percutaneous I'm not going to embarrass myself but let's call this PN for the second method it's a kind of putting in a straw a very thin thing and trying to suck out the the the um the stones and now this third newer thing was this extra corporeal so in other words outside of the body the idea is you get a machine and you just hit it just with a shock wave and the idea is to try to jiggle things around enough that you'll just pass the stone uh but that requires extra equipment and uh you're in this small town remember so uh that's out of the picture so you really don't have the equipment so you're either going to treat this patient by open surgery or this PN meth straw like method what do you do well you look at this paper and it's very clear uh the data says that open surgery succeeds 78% of the time whereas this PN straw treatment uh succeed needs 83% of the time done deal right accept that if you go in and ask does the patient have small kidney stones then uh uh this treatment o the open surgery actually beats the going in for a straw 93% to 87% uh if on the other hand if the patient has large it's more problematic so the probabilities drop but still once again the OS treatment the open surgery dominates that of um of of the PN so in either subcase if the stone is small you should use open surgery if the stones are large you should use open surgery but if you don't know the size of the stones then that then that that then you should use the straw so now seriously so imagine you are this doctor what do you do so which treatment do you use and then even a more basic question to you is should you even bother to check whether the stones are big or small because if so it'll maybe confuse you about what to do okay so it it's curious that in this paper this this this really gold stand P gold standard paper they don't discuss this issue at all they just have just the facts here's the table uh here they are now admittedly they were rooting for this extra corpal uh thing uh and much of the paper is discussing that that was the new fangled thing and they're discussing the pros and cons of it but I just in reading the paper uh this Simpsons uh idea just isn't mentioned at all question yeah the difference both 78% and 83% they're both about four out of five yeah and that point one could look at two treatment and say which one is is dangerous well but but but then then if you're going to go to the danger route then then certainly open surgery would presume be worse but in either case it's dominating that the success rate is dominating that okay well anyway so so I can imagine that that each of you being a doctor can have different opinions about how to answer this but at least it's an issue uh okay so let's continue on uh but sometimes and this is going to be the bulk of the talk uh what appears to be simpsons's effect can really be a hint at some missing causality in the data set and if I've learned one thing in preparing this talk and just thinking about things for a number of years uh at some level statisticians really don't understand causality and even even what the definitions should be for it this isn't necessarily a failing it's just a really uh involved problem and there are a lot of traps uh okay great so let's first talk with a completely toy example that will get things across so question should students study for a test yes or no well let's do a scatter plot uh we're just going to randomly so the number of hours study is the x-axis and the score on the test is the Y AIS and we're just going to look at these various thoughts and and what do you conclude here's all of the data you get a best fit line and clearly things are not good with studying the best fit line is decidedly slopes down another words if you're a student you should not study for a test now you can almost anticipate from the general shape of of of what I had here suppose I tell you that in this that all of these top guys are graduate students and all of these guys are undergrads in many cases it's the other way true true but let's okay so now Within These subpopulations let's try to get uh um the the best fit whoops oops oops oops oops oops oops oops sorry uh let's let's let's let's get the best fit lines uh for this and and clearly if you are a graduate student you should be studying for the test if you are an undergraduate then you should be studying for the test but scrolling back if you're a student you should not be studying for the test okay so one could easily look at this data set this toy and write two different contradictory compelling papers about the conclusion and in fact the thesis of this by means of subdividing what's going on the same data set can be used to justify two different contradictory conclusions okay so this was all a toy uh got a good laugh out of it but this really came up probably the by far the most famous example was uh the issue of graduate admissions to UC Berkeley uh in 1973 so start off with the guts of this fact in Fall 1973 44% of all male applicants uh to to graduate school were accepted but only 35 % of female applicants were accepted and this is a pretty huge data set 12,000 these are the precise numbers so overall 41% were admitted uh uh but eight like this uh this if you do any sort of analysis This is highly statistically significant so just on a Kai Square value of this this is 110 it's huge the probability that this could be happening by by chance is is very very small so this certainly would be in the realm of of the legal system or something that uh should should Berkeley be sued here say for uh for sex discrimination on this on on on how they get in uh and as you're going through this certainly there there exists many parallel situations where uh in in today's world uh where it has entered the legal system but given this uh Berkeley was quite alarmed with what's going on uh so just for a thought think to yourself what could be happening to cause this uh and uh the Provost commissioned a report which uh actually turned out uh the result of it it became a seminal p uh paper in statistics uh it was published in science and and it was a big noise when it came out uh Peter Bickle all three of these were professors at at Berkeley uh Peter Bickle was uh a young at the time uh statistician and he wrote one of the standard textbooks for uh introductory statistics uh Hamill was uh an anthropologist I couldn't find the affil the field of of okano uh but they really studied what was going on and they went one by one at the admissions data for each of the 85 departments on campus to see what was happening with this and this this all here is is accurate data with it um now rather than go through all 85 the top six or the six largest departments uh on on campus uh I'll fill in some of these but but a through F uh you can see at first blush that there is some wild things so so first of all in a uh 82% women uh 3734 uh this guy is very close to even very close to even so uh say really the only big place where there's a big difference in admission rate is is in a uh but if you look at this a little bit more closely one thing you're going to observe is that there is a huge difference by Department in the number of applicants say number one or the the a is is the engineering department and there were 825 uh people uh males and only 108 females that were uh admitted to it uh but the admission rate into engineering was Sky High it was uh uh well even I guess they favored women somewhat with this but uh at about 70% uh on the other hand say in English this is either English or a compendium of English comp complet or something uh that there there were almost twice as many women that that applied rather than men but note the Stark difference the admission rate was only 34 35% uh as opposed to up here in the 60s so if you look at item C there uh there were 560 men versus only 25 women but yet this was a relatively easy department to get into uh and and so on so the the there are big big swings in the male to female ratio of applicants and the admission rate is by far from from from being uh uniform okay and this is really what's explaining if you go down to the 85 the departmental level uh go I'll now quote literally from from the summary uh of uh the summary paragraph of their paper so first of all EX examination of the aggregate data take everything all together on graduate admissions to Berkeley uh shows a clear but misleading pattern of bias against female applicants we have this huge Ki Square score of 110 however when you break it down to the disaggregated data Department by Department uh they there were few decisionmaking units that show statistically significant departures from expected frequencies in either direction and about as many units appear to favor women as opposed to favoring men so what's happening is is that women are were uh were getting tracked or were applying to highly competitive departments whereas men on the other hand were applying to departments which accepted many more people the ratio was uh was different one thing that I found surprising just a a couple uh sentences down from this The Graduate departments that are easier to enter tend to be ones that require more mathematics in the introductory Preparatory curriculum and I'm curious is that true at Maryland that again this was in 1973 and enough just really to State it that like engineering I know here admits an awful lot of people but is the engineering acceptance rate uh much lower than uh much much higher than then for English and how do English and math compare there's a lot of things that we can ask our associate provos here this might again so so just does this does this final sentence still or hold at Maryland in 2023 okay so let's finish with all of that and have to throw this in what was avagadro's favorite Olympic event the mo Vault none of these are any good but you have to throw them in every once in a while okay onward um so now that was all 1973 let's skip ahead almost 50 years to a really odd thing uh at first blush with uh coid 19 in the early days of this and uh this this is brought put together by a blog post of Dana McKenzie at UCLA and uh Jordan Ellenberg who is uh a professor of math at University of Wisconsin but you can almost view him as being a HomeTown guy he was uh went to high school in in Maryland and was one of the winners of the uh his of of the high school mathematics competition very bright guy um guess we don't want to put those updates but uh but also I also had the that was before I came here but I had the pleasure of actually teaching him uh while I was at MIT I was teaching a graduate course and he would walk over he was an undergraduate at Harvard and came over to uh to take this so anyway good guy uh but let's give a couple of slides of data from the CDC from early on in um uh in in in the coid thing so things started well really really started going in March 20120 and now um uh so this was only just the first four or five months and I know the slide is hard to see but we're just going to be concentrating here on the white non-hispanic cases roughly a third uh 35% of the cases were uh were white non-hispanic you can't read that up here is the Hispanic uh white thing almost the same and the black is here uh those are the the main bulk ones but by contrast if you look at the deaths that happened uh between this among in the white non-hispanic category it's gone from onethird roughly up to a half now if you just take a look at this this goes completely against what you the what everyone was saying in the newspapers about uh about that somehow that it the the pandemic especially early on was really hitting all of the uh uh uh the minority communities really hard uh uh and and it was having a profound effect there but why is it that uh among the white non-hispanic you could think privileged people they had a third of the cases but a half of the deaths so what is going on well let's try to to answer that by looking at a breakdown of things again for this march to June CDC data and here it's it's lots and lots of cases a million plus cases and 100,000 plus deaths this is not a small data set by any means but we're going to break things down by age so say among people in the 30 to 49 range 26.5 so all of this is for whites not non-hispanic whites so 26.5 of the people percent of the people 30 to 49 uh had the coid um 26.5% of the 30 49s who uh had Co had coid were white um non non-h Hispanic but in that range even though at 26.5% of the cases the deaths were only 16.4% so if you look at this a little bit uh uh a little bit more closely you'll see well first of all in the zero to four thing thankfully there were very very few cases especially early on I read somewhere that among the 100,000 deaths 13 were in this category early so we could just sort of ignore this thing it was really microscopic but then just going uh line by line um again if you were were white then your probability of uh of dying was less than your compatriots in every one of these categories this you can say okay the better outcomes are due to the privilege and the better health care and better diagnosis and these various finger things to see about your your blood oxygen levels uh um but still how does that explain if uh the whites are doing better in every one of these things how does this explain the 35 to 49% and the key thing here is the the key is it's right sort of not written with this but uh white people are old and this can really be seen not in the the assembled room but of of uh of in the general population 9% of of whites are 75 or older but uh of nonwhites uh only three% are 75% are 75 years or or older so in these two problematic categories and these were really where I now need to put better outcomes in quotes because quite honestly the death rate was absolutely huge it really uh uh uh again the Lion Share of all of the deaths were really just in these two categories so what was happening was that um uh just there were just so many more very elderly whites that that dominating what's happening throughout this and that can be just explaining this but but you one needs to be really careful just if you looked at that first graph uh about in the explanation is somewhat deeper than um than than than what might expect okay so now continuing on what do you get if you cut an avocado into a large number of pieces guacamole yes everyone you're good good okay well we needed that after the talking about this and especially since there's going to be another maybe not completely cheery topic coming up this is something known as the low birth weight Paradox that was studied excessive uh extensively by this Allan Wilcox uh um who is a a researcher at U NIH and Research Triangle uh so in order to get to this I need to Define three things first of all the median weight of a newborn is 3.6 kilograms uh and for Years Gone by uh a baby was labeled lbw low birth weight if uh his or her birth weight was less than or equal to 2.5 kilog and for the chart that's going to come the mortality rate is of of these babies is the number per 1,000 that do not survive their first year so uh a lot of data was collected on this and here it was split by WEA or not the mother smoked during pregnancy so the mortality rate uh for a thousand so in the general population so just among the non lbw uh uh babies uh for maternal non-smokers the death rate was 11.1 so enough roughly a 1% chance of or 99% let's be positive a 99% chance that that the baby would survived the first year uh but among maternal smokers it's it's slightly worse than that but if you go to the low birth weight things then 210 out of a thousand low birth weight babies die to maternal nonsmokers but if the if the mother smoked during pregnancy then this rate would drop to 114 so roughly get cut in half okay and this was published by this guy Yosi um and I I'll talk about this this was actually I would say an important paper in a in an odd way but at this moment you might want to think what is going on and hint I'm not advocating that mothers smoke during pregnancy yeah yeah yeah that's that that's that's going to be the that's going to be the kicker and I'll be able to illustrate this by by a series of pictures but good uh great so what is going on with this well let's start off just straight uh this I couldn't get anything else but this this is for birth weights uh in Norway but I think it's basically Universal um uh so the average birth weight is like 3.6 kilograms and this is very roughly a normal distribution however to the left there's a tail the tail to the left is much longer than the tail to the right but fundamentally it's it's a normal distribution as one would expect the numbers are large uh what else could it be okay so on this uh doctors have arbitrarily determined that uh the low birth weight cut off is uh at 2.5 kg which in the normal in the standard population cuts off this tail now here just the Grim things to think about this includes almost all pre-term babies but also ones with genetic problems with uh uh just something wasn't right in this uh so here I hesitate to write everything in red it sounds like things are doomed when in fact the good news is really 80% of these did survive for at least the first year but uh but it's really the flattened part of this tail okay now another thing that is known and has been tested many times throughout is the effect of mother smoking through pregnancy this decreases the average birth weight but it still happens a lot and uh this this this can be quantized so it's known that mother smoking during pregnancy decreases the birth weight by approximately 200 grams so forget the units on this but roughly what's happening is we're getting this normal distribution but we're shifting it to the left by 200 gram so Peak is going to be uh 200 grams off and the same general shape but then what really the problem is in my mind the definition of low birth weight is not changed depending on whether or not the mother smokes so as a result if we cut off this thing now there's going to be in the blue a much bigger tail this is exactly what you were saying that um a among mothers that smoke there are many many more low birth weight babies but also many many of them are fundamentally healthy I'm not saying that smoking is a good thing but fundamentally there's no uh there there's not much wrong with them so if you're looking at a ratio think you're just dumping in lots of fundamentally health healthy babies and you're putting them to the left of this divide as a opposed to the right so to to summarize this the point is uh because of the shift in birth weights well at not cor making a corresponding change in the definition of low birth weight many more fundamentally Healthy Babies of smoking moms are put into this category uh but among the low birth weight babies so so thus uh there's a greater share of fundamentally healthy babies and that's what's going to cause that ratio to go down so just to be clear this does not mean that the mother's smoking is good for the baby causing this shift to the left however this Yoli who uh was biostatistician is trained at John's Hopkins uh uh moved up uh through the ranks and then actually in the 1950s he he created the bio statistics Lab at at Berkeley so he really had uh a a national following with this he was also a smoker and this was right at the time 68 to 71 was when uh there were just discussions about smoking that everyone smoked all the time but but the uh CDC others were trying to cut back and the FDA warning labels were were coming on but he used this data this fact uh scroll back here uh this this is his data he was using this as an argument in favor of mother smoking or saying it wasn't bad because look here's actually a benefit um sight I find this quite shocking uh but worse this paper that he submitted got into the general press and two titles that I managed to find in the Boston record American I don't think it still exists the the title mothers needn't worry smoking of little risk to baby from 1971 worse in defense of smoking moms in this Family Health magazine that was still a I remember that from being a kid uh that uh uh this really got into to to the thing um so to see that this effect doesn't all have to be this smoking versus non-smoking if you want say a more a cheerier example of this whole thing uh so this Willcox who's been studying this effect uh if you just concentrate on babies that are born in Colorado then maybe it's because of the elevation who knows but uh the percentage of babies born that are low birth weight is significantly above that of the US population but uh for any other purpose they're just as healthy as in other states so I don't mean this to be a a smoker versus non-smoker th um this is this I view as being a a cheery cheery thing I would say if anything among epidemiologist that the big problem might be that uh just absolutely fixing this low birth weight thing at 2.5 kilograms come hell or high water is what's causing these these seeming um seeming paradoxes but in any event um just the takeaways if you want to look at all of these examples taken together the main takeaway I would have is that we're looking at a single data set but in by doing various means of subdividing the same data set can be used to justify contradictory conclusions so I'd say beware of headlines or or sound bites uh with this if there are any journalism Majors here uh uh that that one should really be careful about how to uh uh how to interpret this and then finally to link it back to my title in short averaging averages can be a perilous undertaking so uh with that thank you very much for listening and I hope you'll enjoy the snacks in the crossing the hall