hi there today we're looking at activity in class activity 4E what is unusual and so what we're going to be doing is we're going to be looking at the empirical rule and we're going to be looking at z-scores and we're actually going to be looking we're going to take a little step away from politics and go into clinical trials instead it's a little break from the political turmoil of the day and we'll be looking at toxicity of it's clinical trials on poor little mice and we you we look at mice because it's really hard to experiment on people um and we're going to be looking at their spleens which is little organ a little tiny organ even smaller and mice and we're going to be looking at livers um also in mice bigger organ both of these organs are filters in the body so if you expose the mice to some toxins that in the form of chemicals that maybe are going to help with a drug uh drug treatments for illnesses but how about you know how's it doing on this lien in the liver how's it doing on toxicity so so the mice are going to be a vehicle for us understanding two concepts z-score or standard score and the empirical rule so I know you looked at the empirical rule in the preview activity but I actually want to just review it a tiny bit before we jump into the activity so um if you you should probably have printed out this worksheet or you have it downloaded I'd like you to skip to the back because I always leave a few blank pages at the back and I just want you to write a few little notes about what it means to be the empirical rule and standard score so it's you don't want to assume that all data follows the empirical rule and that's something that I don't think has been emphasized enough so I'm going to go ahead and skip to the back here and we're going to do some review from the uh preview activity so review uh thank you for e um if and it is a big if if the data distribution so the picture of the variable that you're interested in is bell-shaped or normally distributed then we can assume the empirical rule so not all data follows a nice spell Shake but a lot of them do so bell-shaped is synonymous with normally distributed normal we say the data is normal because it's common it's a very common shape um so an example of this mems Heights in the U.S are bell-shaped with a mean of 70 inches and a standard deviation and that's actually going to be Sigma because we're talking about all men in the U.S and so this mean would actually be you because we're talking about the entire population of men so with the standard deviation of sandwiches so what that means is the average man is 70 inches tall give or take three so most men are between 67 inches and 73 inches and so I'm going to draw a picture of that just so I understand it because I'm a very visual person as are most people in the U.S I mean in the world oh so that wasn't good so I really want to try to get this bell shaped so to a statistician this is a thing of beauty in this beautiful curvy um well-proportioned shape and I'm going to label it so I'm going to say X equals Heights uh individual no so all every so there's this it's really if we look at this here oh maybe there's ten thousand men here there's two little tiny dots and they they stack up and they make all so in this one is these do you see those little tiny dots there that's all the men that are exactly 70 inches tall have the average height so really it's not a lot actually that are exactly 70 inches and this three inches right here this means give or take this is the variability so this right here that's three that's the give or take so most men are not exactly 70 inches tall most men are between 67 inches and 73 inches so if you want to have a precise so now who are all the men that are ex that are the most average so I'm filling in little this is this is I don't know a million men in the U.S little time dots oops they're getting a little I'm getting a little sloppy here but these are all the men who are exactly 73 and all those dots represent all the men who fall in that range um so this really is mu minus Sigma and this is Mu plus signal so but we're focusing on and I'll give it a title um height of men in U.S probably if you go to men in Ethiopia Ethiopians are known to be very very tall so maybe the the average in that case would be 73. or if you go to another part of Africa like I don't know where the pygmy people live but um they're very very short if you go to China they're going to tend to be shorter uh though that's changing because as people with the global economy people across the world are getting taller and taller and you notice that people who emigrated from Europe to the United States and got access to Better Health Care for or better food particularly the Irish they got really tall once they came to the U.S the Next Generation so it's not just nature it's nurture and nurture changes but certain groups of people tend to be smaller so um we're just gonna We're not gonna we're gonna focus on the U.S because that's what I know so those are the heights of the men and um and what the empirical rule says is that um 68 percent of all men in the U.S fall in that range and then if you go out a little bit further if you go out another standard deviation so you go out an equal distance and it's that give or take three so now we're going to be at 64. and we're going to be at 76. if you go to this boundary 95 of all men fall between 64 inches and 676 inches so that's that give or take and what's the Bell shape means is exactly this that you trap almost all men if you go out to standard deviations now if you go out three standard deviations and I should have made my boxes a little bit longer here I'm here if you go out three standard deviations so I'm trying to do equal distance so not one not to but three standard deviations all the way to here so you've taken another three inches away so this is going to be 61 inches and then you're going to go one standard deviation above not two standard deviations above but actually three standard equations above so that'll be 79 inches you have now trapped 99.7 percent um between 61 inches and 79 inches so this statement men's Heights in the U.S are bell-shaped with a mean of this and a standard deviation of that bam it gives you the whole picture it gives you this distribution in a very succinct way and a lot of data follows this nice Bell shape but not all of them so not all day to follow Bell shape not all data follow the empirical rule so but that I did want to review the empirical rule um and the empirical rule is this idea of 68 95. 99.7 so 68 95 99.7 bam you've got almost everyone 99.7 0.3 of the population are outside of those bounds that's three out of a thousand people might be outside of those bounds um but we're not really interested in that and for example the military military don't really want people that are Beyond those two boundaries uh if you're more than two standard deviations from the mean in either direction going to be problems getting new boots helmets clothing um fitting in the fitting in the um the transportation Vehicles if you're super tall you're not going to fit well and if you're super small you might not be able to run the machines well because your body is outside the realm so we say that anything outside of two standard deviations is not typical so um rule of thumb and it could change because it really depends but rule of thumb so it's not written in stone but values outside two standard deviations are unusual foreign rules some other time but this is going to be our rule of thumb right now and then what's a quick way to see how far something is away from the um Center um standard scores or Z scores and that's observation minus Center over spread so this is the center and this is the spread of your distribution so for this distribution up above the center is 70. and for this distribution above um the spread is three so it's it's describing the center of your bell shape and that and this right here that's a hump that hump do you see where it changes its big frown here and then it starts to smile and that hump is that 68 percent so if if it's got if it's bell-shaped and 68 is trapped in the middle so kind of like the box plot 50 is trapped in the Middle with a bell-shaped 68 is trapped in the middle um so this is called a z-score and it is a mathematical formula and as you saw in the preview activity it measures how far and observation or data Flame is from if meaning so you standardize it basically so my son is um six foot six inches tall so if I go up here six foot six is six foot is 12 times 6 is 672 plus six more inches is 78 so Theo my son is right here that's where Theo is he's one of those dots so I can tell his z-score is going to be not one not two but almost two and a half standard deviations away so his z-score Z Theo well let's do it so it's uh I said 78 inches 78 minus 70. that's the center of his distribution over 3. foreign that's going to be 8 over 3 because you always want to work that out first and shoot I don't have a little calculator oh pause this and go get my calculator I'm gonna call us this and get my calculator I again got my trusty calculator um gotta get it back to sharing screen welcome back so that so what's your standards for 2.66666 repeating so usually with standard scores I just go to two places past the decimal so that's going to round to 2.67 so he's a little more than two and a half standard deviations from the center and you can physically see that up here so what does the standard score tell you it tells you two pieces of information how many standard deviations away the observation is and also whether that observation is bigger than the average or smaller than the average depending on if it's positive or negative so I'm going to interpret this one interpret EO my darling boy is more than two and a half let's do this 2.5 standard deviations yeah from his name is average and because it's positive he is above average in height okay so that was kind of a crash course in what is a standard score or a z-score tells you two things how far away and whether you're above or below average and then the if you're told that something that data is normally distributed or it has a bell shape then it follows this 68 95 99.7 rule which is called the empirical rule and that's a rule of thumb later on we'll get a more precise it's not exactly it's not exactly 68 but it's close enough so it's a rule of thumb so all rules of thumb okay so let's get started with this um with this activity so why do you think so around the world pharmaceutical companies conduct clinical trials to evaluate the safety and efficacy how well they work of new drugs um clinical trials are research studies performed on people and are aimed at evaluating if the drug is safe and effective people who participate in clinical trials are called volunteers I hope they're volunteers because it would be very bad to experiment on people who don't agree to do it so um the question here is why do you think some people volunteer to participate in clinical trials what are the pros and what are the cons why would someone do that um are there alternatives to having human volunteers participate in clinical trials so there's several pieces to that question so pause it and answer it for yourself no wrong answer here um and if we were in a classroom I'd take a poll and you give me a whole bunch of answers and I'd say those are all great so I'll try to predict what you said so why do people participate um well they could be curious or they could be desperate if you have an unusual disease and there is no standard cure for it then you might want to really get into a clinical trial because they might um offer you some breakthrough and then science is amazing so for example I have a friend she actually used to be um in charge of the math lab you know 20 years ago and she was diagnosed with melanoma and I had a friend 30 years ago who was diagnosed with melanoma and he died and so when she got it I was like I was ready to say goodbye to her I think I spelled desperate wrong desperate that's probably not how it's spelled it's like really you you're terrified so she should have been terrified because the statistics on melanoma was really bad but she got into a clinical trial that was doing stem cell research I think it was and she's alive today she retired all those years ago and she's been having a great life ever since so clinical trials might offer you a hope that you wouldn't have otherwise so um another reason is if you're curious um I had when I was younger and I was living in England I had friends who would come and stay with me and they would do clinical trials because um they would do clinical trials in Amsterdam because they wanted they loved being they love to experiment with drugs MDMA ecstasy all that stuff so they signed up for every kind of um it wasn't recreational drugs but they were kind of doing it recreationally so could be curious um another reason or money clinical trials pays you pretty good money now would I want my kid to be involved in a clinic a clinical trial probably not I because they are experimenting they don't know what the long-term side effects are and indeed a lot of the studies on MDMA ecstasy are showing that there is a depletion of spinal fluids or something I'm not I'm not I'm a statistician I'm not a medical research scientist but there's some evidence showing that if you did excessive ecstasy or ndma that it could really set you up for struggling with depression and other moods later on in life because uh it it sapped your serotonin and deplete other neurotransmitters in your brain that would keep you happier so but people get paid to do clinical trials so um are there pros and cons well the pros and cons are the pros are you could get uh Pro breakthrough research or money the cons are risky you have to sign all kinds of waivers risky so for the person those are the pros and cons it's for the researcher um the pro is if you're trying to study the effect of the effectiveness of a drug on people studying it on people is the best possible thing is people are more accurate in terms of results than animals and I would say the uh con all problems associated with a volunteer sample people who volunteer to be experimented on are probably not a good representation of everybody in the country so if we want to know how MDMA affects I think they use MDMA now Ecstasy with marriage counseling and also with people who are um with severe depression particularly if they're facing fatal or terminal illnesses uh so you're drawing conclusions from a group of people who were probably very adventurous probably younger uh volunteers they fit a profile that maybe is not everybody almost certainly isn't so um all that but anyway that's just a background for setting up clinical trials so we are going to look at standard spores otherwise known as z-scores and we're going to look at the empirical rule to determine if an observation is usual or unusual so we're going to use those two pieces of information and we'll utilize the standard scores and the empirical rule to determine if an observation is unusual and we'll compare two observations so it's really helpful for comparing if I wanted to compare how super tall my daughter is to have super tone my son is my daughter is six foot one and my son is six foot six and a half really he's a little more than six foot six and a half who is more extremely tall for their group and it's kind of surprising that actually Delia is taller for women than men of amphios for men um my daughter Delia six foot one that's more unusual and we'll sh and we'll see why maybe if if we have time okay so back to clinical trials we usually do not experiment on people we usually experiment on mice mice have some advantages and disadvantages one of the reasons that the scientists um like to study Mount mice as they know a lot about them and they can manipulate how they're raised so that they can get mice that are all almost identical they're all cousins or brothers and sisters of each other you rarely can do that with people so that you can all those little white mice they're all related so you're that isn't that is a form of control where you're controlling for the mice being all similar so when you expose them to chemicals how they respond is unique to probably what the chemicals are so um so they're bred to be identical probably not exactly identical but that Lim that limits some confounding variables is one of the confounding variables of how much you react to a certain drug could be how much you weigh how old you are how um what what race you are all those things so we're just going to make all we'll just study the white mice even though we may care about the dark mice just as well but we're going to keep that all the same and then the other thing is that their lifespan is very short so they metabolize really quickly so that you can see results you can follow that Mouse from the time it's born to the time it dies and I think they say that the average lifespan is 800 days so if you're going to do a long-term study on people you've got to devote like 80 years and it's not going to help me but it might help my great great grandchildren um so toxicity of a chemical um and its impact on the organs is an interesting um is is of interest when assessing the effects of chemical treatment a standard method used to measure level of toxicity so level of toxicity organ weight so those are kind of um if you want to know how how sick somebody is you look at their organs and you've probably heard that especially if you watch those um autopsy TV shows where they if somebody dies unexpectedly they want to know what the size of the heart is they want to know what the size of the liver is sounds great to have a big heart it's actually not a good thing it means that you have over pumped you were not healthy big heart is not healthy big liver is not healthy the bigger your liver is the the more unhealthy you are and so if you're exposing these mice to chemicals that's a way if you get up Rosie hold on my doggies sorry about that um so um in this episode class activity we're going to be looking at the weights of livers and I don't know why but I think of Louver as and I could be wrong but I think of it as a um as a organ that filters blood and then we're also going to be looking at spleens so I'm going to do that one in green um of so we're going to be looking at the livers and the spleens of mice who are 26 weeks old and female so we care about the male mice we care about the young mice and the old mice but by holding steady their age and their gender that's a confounding variable that's out of the way so all these mice are very very similar there's they're females they're not they're not very old um and we're going to expose them to chemicals so sad but true so what we're what we know is that the mean for liver is almost one gram just shy of one gram and the standard deviation is right here okay so I want to draw that distribution and I should have said I should have been explicit that this distribution is bell-shaped so I'm going to say that now so we're going to draw because I'm a visual person okay here we go and let's I'm going to try my best oh glue it make it that nice symmetrical now because it's symmetrical the mean and the median are more or less the same so the average and we like average is better because they're easier computationally so that's 0.999 and I'm going to label my Axis so that I don't forget what I'm doing I'm going to say x equals liver liver rates so on average these female mice have its liverweight for individual female mice who are 26 weeks old um so that's my Center and then I've got the oh give or take this so if I go out I know if I go out this much 0.0 8 7 grams in this direction bam what's this going to be that's going to be my average minus my standard deviation and when I work that out shoot I did not write this down what was I thinking right well I can do it now good thing I got up and got that calculator 0.999 minus 0.087 so this is going to be points nine one two okay so that's if I go out one standard deviation smaller now I'm going to go out one standard deviation bigger so this right here I'll grab my Center Plus my standard deviation so what's that going to be um so I get 1.086 grams so that tells me that about 68 of all female mice in this experiment should have liver weights between um 0.912 grams and 0.120 86 grams so that's for the 68 of all and then if we go out one if we go out to standard deviations not one but two standard deviations not one but two standard deviations bam we're gonna land here and then we can do it in that direction and we can do it in this direction so we're going to add point zero eight seven and add .087 and then we're going to land right here okay so um I'm going to cheat and just go off of what I see here I'm going to subtract 0.087 from that one because that'll save you just a little bit of time and I'm going to do it in the other direction too so cool zero eight seven so that's going to be 1.173 and in this direction I get points 0.8 25 and some Math teachers get really upset if you want to put a zero in front I don't care um so that was the liver so I want you to pause and I want you to try and draw the distribution of the spleen um the spleen distribution for these same mice and for the spleen we know that the mean spleen weight is 0.8068 0.086 my bad and the standard deviation is much smaller right there and I think that little two up in the air there that doesn't mean Square that's a footnote down here so I think that's a little misleading so watch out for that so draw the distribute before we answer any of these questions I'd like to see a visual of the distribution you won't have to do that after this class but um I think it's a good thing um so pause and do that okay so if you did it right um we're gonna have point zero six point zero eight seven so it's going to be a distribution I'm purposely making it over here because it's much smaller values and the center of this distribution is right here so I'll slap that in the center that's our average our mean it's our measure of center and although we bashed means a lot there are great measures of center when you have Bell shape or you have symmetrical so the standard deviation is uh much smaller .007 it's about it's a tenth of the other one so your data is going to be clustered around this Center a lot closer so I'm going to do I'm I'm doing this because the truth is that if you have a smaller standard deviation then it's telling you that 100 of your data points are much more clustered around the center so it's a taller thinner distribution because this is a much smaller number um so if I add 0.007 I'll end up getting um I don't know I can't do this in my head so thank God for calculators 0.08 6 plus .007 is going to give me and this is going to be tricky because it's all running together so let me try to make that as small as possible so that was Zero like that oh eight seven and if I add 0.007 I'll get 0.0 93 that's if I go out one standard deviation in this direction now if I go out two standard deviations to capture the next one um that's what I get 0.1 and 0 0 and if I go out three standard deviations so this value right here is going to be mu plus 3 Sigma or it's going to be point zero eight seven plus three times point zero zero seven I'll end up getting foreign okay so that's in that direction going the other direction if I subtract .007 that's going to capture 68 of all the data [Music] um you know imagine if your parents had your parents had to do this without calculators point zero eight you guys agree with that do I agree with that [Music] um it's not what I got before I'm going to try again one more time point zero eight seven minus 0.007 and zero eight all right and then if I go out about one but two standard deviations so this value right here is point zero eight seven minus two times .007 I'm going to cheat by just taking away point zero zero seven from point zero eight and I get almost impossible to fit in that box just write it out 0.073 and I'm like getting slightly different numbers okay just checking somebody [Music] um getting this wrong all this time and none of my students ever corrected me before and so if I go out another standard deviation I get points zero six six okay so I've got my distribution and I wish that I had a little bit of room to [Music] um label this so I am going to find a little bit of room to label this and I'm going to say x equals individual bleeding weights and I really should say in grams so those are our distributions and we know that because they said bell-shaped uh or they should have if they didn't and um so if we're using these pictures so the first graph is relevant to these questions and the second graph the green graph is relevant to these questions so I'm going to answer the first one and you're going to answer the second one so for the first one 68 of all data is in here within two standard deviations so it's going to fall between um points nine one two grams and points sorry 1.086 and I'm going to put grams it's really important that you have your units so people it's not pounds it's grams so that was 68 percent so the next one is 95 so for 95 you are interested in going out to standard deviations folding this way and this way so it's going to be all with this so it's going to be points 0.825 grams and one point one seven three grams oh they asked for three standard deviations which I did not do in this case so I've got to go out one more time I've got to go out one more time this direction and this direction so I'm going to be subtracting another 0.0087 from that other one um so 1.173 minus or no Plus 0.0 87 bam so that is going to land me at 1.26 um so I've got 1.26 and the other way if I'm going in this direction a third standard deviation um it's going to be if you work it out 0.7 three eight and that is your 99.9 I've gone out three standard deviations in both directions so I've trapped this one little bit on this bit and that right here is going to be three standard deviations you've captured everyone but the freakiest freakiest freaks in terms of uh liver there's going to be some people with some nice with it there will be maybe there's probably maybe one guys out there who has a huge liver and one Mouse it's a tiny tiny liver and the one with the tiny liver technically is probably healthier okay so that's how we do that so I want you to pause and do the green distribution and see how you do so 68 is going to be the inner part it's gonna be about 68 is the hump that's this one so that ranges from points zero eight and point zero nine three grams and then the next one so that was 68 of the data and then if we want to go out to standard deviations we have captured 95 of the data so we get this step and this bit of all of the stuff in the middle that's 95. that's pretty unusual so I got .073 grams and .093 grams were so different from the values I got before I hope I didn't make a mistake oh oh the spleen rate is right there it's a six this is a six and that's what's throwing off all my numbers ER so since that's a six I am not redoing this video sorry guys um I misread this it's .086 not a seven so I'm going to fix these numbers now and I apologize you guys so it's one point higher so this is going to be 9 4. yeah 0.086 so I'm fixing all this you probably were wondering about this clear standard deviation of 0.007 0 8 6 Plus .007 is point zero oh I did that one right but it's the other way right zero eight six minus 0.007 is so this one is point zero seven nine and the next one is .072 so is it only off by a tiny bit what 0.06 5. last one right here and I'll work this one out it's that times three of these and I end up getting 0.08 6. minus three zero zero seven foreign just kills you so this is six five point zero six five okay um so that should be a seven two well that's a little bit better than having gotten what's wrong last semester and no one telling me and then .065 .107 and these are all grams okay so those are the two distributions and just be careful don't rush it because you'll lose you'll drop a decimal point or something like that okay um what do you think are unusual unusual spleen and liver weights and why well looking at the distributions of 68 that's not very unusual 95 is getting kind of unusual but I redeem after two standard deviations that's when it's when you're really on the margins you're really on the fringes so I'm going to deem unusual to be Beyond um should we make it Beyond Two or Beyond three well in this class we're going to say Beyond Two so once you go beyond two um so that's one two one two those are the unusual okay so um so which one did they ask for first spleen so we'll look at the green one so for the green one anything um sleeves less than this one foreign zero seven two grams so that's the cut off right there or greater than so what's the cutout over on this side 0.100 grams so those are really unusual results because they're less than 95 percent of the observations fall outside those boundaries for the um for the spleen and um livers less than and you're going to go to standard deviations smaller or greater than you're going to want to choose standard deviations bigger so when I look here I'm looking at those boundaries in order to so anything beyond there so I kind of obliterated it but less than zero point age two five and the other direction was 1.173 . okay and again put your units or we'll get confused Rams so by using the empirical rule so it's a rule of thumb we're just eyeballing it 68.95 okay this is unusual um but what we haven't done is talked about z-scores at all so um higher organ weights is an indication of higher toxicity suppose a mouse liver has um 1.07 grams so I'm gonna help myself and rely on the color coding here so this is from the red distribution and that same poor little mouse has a spleen weight of this okay it's so um part a calculate and interpret the z-score for the liver and then later can't do the same thing for this thing okay so I'm gonna just to refresh my memory the z-score in general equals observation minus the center of your distribution over the spread of your distribution and since you're dealing with normal distribution or bell-shaped distribution that's going to be your data point or your observation minus mu because that is your best measure of center here's your distribution U and your spread is Sigma and so we've got this over Sigma so that's your z-score um and that's the way to calculate it um to interpret it it tells tells you how far how many standard deviations your observation is above or below the mean so if it's above it's going to be positive if it's below it's going to be negative okay so um so since this poor little mouse has been operated on probably both of the values are going to be above because that's when we're indicating that the mouse is not healthy so for calculating the first one observation Z liver is going to equal observation minus Center over spread and you I if you were in my class you'd know I really have trouble with memory so this these kind of tools help me so since I'm dealing with the red distribution let me go up there and I've got it visually that the center is 0.999 and the spread it's right here 0.087 and the observation that I'm interested in is right here 1.07 so oh looks like that Mouse has spent 1.0 7 doesn't seem that far now you've got to be careful when you run this through so pull out your calculator and calculate that and make sure that the answer you get is 0.816 and if you did not get that answer the chances are you did not hit enter after you want to figure this out order of operation says you figure this out first so you literally go 1.07 minus 0.99 and then you want to hit enter or equal and by doing that you're basically throwing parentheses around here because they're built in they're not there and then you get a lovely decimal and then you hit divide and um points zero eight seven and then you will get this okay so I just did the calculation I haven't done the interpretation yet so the interpretation is going to be this mouse is River foreign this is almost one so it's slightly less than one standard deviation is it above or below well it's a positive number so above I mean okay and if I were to put this on here this observation 1.07 sorry one point yeah 1.07 1.07 it's going to be right about there so it's not quite it didn't make the cutoff for one standard deviation so it's about right there it's a little bit less than okay so that all checked out um but boy it I know people don't like math that much maybe but it sure seems easier to calculate the z-score than to draw this distribution I mean I made a mistake on this one didn't I I dropped a decimal so these are pretty tedious so I think I'd rather calculate z-scores than draw these distributions which is something you'll have the freedom to do later on so for the next one calculate and interpret the spleen so go ahead and pause this and see how you do and work this one out okay welcome back I hope you did it so Z spleen so that means you are on a different distribution you're not looking at this red distribution anymore you're looking at the groom distribution and the center of this distribution is right here and that's where I screwed things up a little bit 0.0 86 not eight seven and the standard deviation is right there .007 so that's what the our distribution looks like so I'm going to go ahead and remember my formula which is observation minus Center over spread and I know that my center of my green distribution was .08 um six and I know the spread of my distribution was 0.007 and the particular observation that I was given for the poor little mouse is right here and I'm using orange for observation that's my data we want zero point one zero four so which is bigger the liver or the spleen well the spleen is definitely smaller than the liver but it should be anyway so that's why you want to standardize it we don't want to it's not relative to each other it's relative to other spleens and other livers so the z-score is going to give us how the spleen is measuring up with other spleens so if you run that through and remember there are invisible parentheses in here so you're going to want to hit enter after that and if you do it all right you should end up with two point five seven one so let's do the interpretation interpret don't forget that this analysis I don't think that's right either spleen is more than 2.5 standard deviations above the mean wait for other explains actually for all students so while his spleen or her her spleen is smaller than her liver relative to other swings this theme is in crisis so the liver wasn't that unusual but the spleen is outside that 68.95 boom it's outside of there the the liver is inside the 68 but this the screen is outside the 95. so the last question which organ shows higher levels of toxicity since weight and size is correlated with toxicity uh this right here is going to be worse so um the liver is less um the liver is less than one standard deviation away from its average so it is pretty typical nothing unusual there um the screen on the other hand is more than 2.5 standard deviations above it's I mean so this is considerably larger than other schools and therefore show signs uh reader toxicity size equals toxicity poor little mouse are there other so you can experiment on mice but maybe you don't think that's the right thing to do and you can therefore get deeper people volunteers instead one way or the other it seems like the science something's got to give all right um so that was the end of that so let's go back and make sure that I hit everything standard spores RZ scores and uh they tell you how far away observations are from the center and they also tell you whether they're above or below so we haven't had any examples of negative values but if your z-score is negative it's telling you you're below average and the empirical rule only apply it if I tell you it's bell-shaped and I have those two together help you determine things are unusual um so that's that one and we just went through an example of that and it allows us to compare observations from different distributions so it could compare a woman's height to a man's height in terms of unusual compared to other women and other men we didn't do the women though all right we're done and uh so take a break have a cookie go for a walk call your best friend or your mom or dad or your cousin and chill out for 20 minutes and then do the practice and that's the best way to allow yourself a little bit of time to forget and then you kind of build it back up but not too much time because otherwise then you're kind of Reinventing the wheel all right take care and happy practicing