Statistics – Standard Normal Distribution

I wish do what's my character's nameth mathletes gay mathletes y [Music] [Applause] athletes oh that winded me okay all right thanks for that that was fun now was a catch my breath so uh you like it you like the costume rock and roll yeah you got it I mean you know make sure you be able to return it after all right in the frame there head top of the head everything actually can you tell we're good okay great okay um today's lesson on statistics is on the standard normal distribution or standard normal curve it's actually pretty intuitive I think you'll like it just feel free to ask any questions you have I need that peanut butter after need protein oh I forgot my dumb at home that'd be fun just like little two Pounders anyway um standard normal curve it's pretty straightforward just ask questions because it's weird at first but then hopefully you'll get it okay so we've been introduced to the normal curve already so let's actually just draw one maybe on the side of your paper somewhere what goes in the middle of the normal curve zero okay good job here it's zero here it's zero but what in general happens in the middle of the normal mean the mean which is the average okay so here we have the mean and in the standard normal curve Cornelius that actually Betty rocks Dawn hey what is the mean in the standard normal curve you said it before zero zero right so when the mean is zero that's one marking of the standard normal curve okay the next one about the standard normal curve do you agree that normally and a normally in the normal curve each increment is one standard deviation is that remember that jump by standard deviation one standard deviation 2 3 4 now in standard normal curve besides the mean being zero standard deviation is it's on your paper you want to tell me one standard one Devi so you jump by standard deviation so what would this be it's not one negative one it's like Z exactly that's the next thing we're saying 2 3 standard deviation so -2 -3 so isn't it true that data and zcore have what relationship in this case they're the same they're the same so imagine a normal curve that had a mean of zero and a standard deviation of one that data actually equals the zcore so these are zcore and their data so that's the marking of a standard normal curve okay so we're going to actually go in and uh experiment with it so the mean is zero we talked about it standard deviation is one there it is marked for you and the total area under the curve is one do you agree that this under the curve all this stuff is 100% of the population 100% of the population what perc of the population is to the left of my beautifully gloved hand 50% right let me ask you this now leading question what is the probability or the chance that data will be at zero or less what percent chance is there that a data item will be at zero or less 50 50% right so not only is this the 50th percentile there's also a 50% probability that she'll actually be here or less okay and you remember that each of these is 34% on either side so then if this is 50% up to here 34% up to one what's the probability of being one or less 84 84 4 84 right so this is just getting us acquainted again actually later in the notes you'll see the notes are there it'll give you a chance to do that but it's going to get more intricate now so we have a table near the back of the book which relates zc scores with the area under the curve so basically zcore of zero we talked about it I'm going to match it to my finger there it is zcore of zero 50% area zcore of Z 50% area well what about zcore of like here let's do an example zcore 1.27 do you agree that zcore of 1.27 is right here right there what is the area to the left of my finger roughly yes well to the left like to the left is it the majority or the minority yes it is the majority do you most of the area under the curve is to the left of my finger yeah right so good guess what was one again one or less was about 88 8 848 oh so you're okay yeah this was 84 so then it would be 85.1 maybe 85 is good job it's around that it's around that so there's a table in the back of the book that helps us do that this is an example from your notes let's try an example what is the probability associated with with zcore of 1.27 note the standal normal curve shows cumulative areas from the left so let's talk about it here's your normal curve do a quick one here this is zero this is one so 1.27 will be right around here and we want the area to the left of this and we're going to look it up at a table okay 1.27 Sor do a probability uh if we go to the proportion bless you don't know I'm throwing something can we set up a proportion and say 1 is to 84 as 1.27 is to X that work good guess we can't because the amount of population in here is varying as you leave the as you approach the extremes so it's not a linear progression right like the difference in area between 0 to 0.5 is different in the area from 0.5 to 1 oh okay does that make sense so it's actually straight out of a table it's in your book but I have it here conveniently for us not Rocky thank you sir for that okay so at first this table looks intimidating but we'll teach you how it works okay this is the standard uh normal curve cumulative or total area from the left okay from the left we want this is all the area from the left measuring zores negative so what's our zcore that we want 1.27 so scrolling scrolling scrolling here it is positive Z scores originally you saw here negative zc scores so positive Z scores 1.27 let's see if we can find it one point here it is would you round to 1 no not because I'll show you what that means this is the tenth digits right here so this is 1.2 and this is the hundreds digits once you learn how to read it so this is this is the zcore of 1.20 oh this is the zcore of 1.21 this is the zcore of 1.22 oh it's just so they don't have a big chart of 1.2 one 1.2 going right yeah so they aggregated it into a table so 1.27 I'm going to Circle it 1.27 right here so zcore of 1.2 what is 1.2 plus7 1.27 good and that zcore is 8980 so in fact peaches and cream remember when you said 88 match my outfit when we said 88 actually pretty close it's 89 so 89% of the population resides to the left of zcore 1.27 can we actually have someone say that for us while I get this set up I'm going to format it okay so area equals 8980 volunteer to put this in a complete sentence I'll start it off there is an area of 8980 left of 1 left to the Z of 1.3 very good now someone else to do the same thing using the word probability there is an 89 try there's an 89.8 or 89% chance that you will like fall I guess fall nice fall within within the zcore of 1.27 or to the left of it or to the left of it right there's an 89% chance or 89% probability that you will be here or to the left Fair yes um even though we don't round the zcore cuz it's 898 would you accept rounding to 90% or would 89.8% yeah in the purpose fair question in the purpose of our class I'm not going to like mandate tenths and hundredths and thousands okay if you get 89 I'm I'm with you okay or 0.9 yeah you get 09 I'm with that's very close isn't it okay but calculator will give it to us exact so no sweat giving it away giving it away okay over herey that's where they took it from good job do you like my socks nice H step you do that in practice Yeah H H tired that water bottle is great you want some I'm good okay anyone anyone no just so hot man that was nice you can't return yeah right I don't want to anymore okay assuming oh I taste young vanilla okay assuming a normal distribution if the mean rating is 0° C so this is a thermometer okay we're talking about a thermometer do you agree that if you're testing here let's go back to Blue here please BL work no it didn't it didn't work okay all I tapped it again okay so suppose this is how thermometer manufacturers actually operate they take they build their thermometer and they verify that the temperature in the room is 0 de they they have it with thermometers that work they make they have a test room with thermometers at 0° then they take their new thermometers and they go in and they test the temperature what do they hope that their thermometer says zero zero they hope that their new thermometers read zero okay so on average on average those new thermometers when tested at 0° read 0° however there some thermometers need to be tweaked and worked and stuff so the mean of a thermometer tested at zero see the mean rating is zero but the standard deviation is one on average on average the thermometers are wrong by one degree in either direction standard deviation one standard deviation one we call it s before same thing these are both the same so on average they're wrong so this is how it works assuming a normal distribution you'll get into it okay you'll get into it if the mean reading is zero and the standard deviation of the readings is one what's the probability the freezing point of water will read 1.27 or less so here's 1.27 back to you Don 1.27 not only is the reading of the thermometer but it's the what in this case standard normal curve go ahead didn't you just get the answer 1.27 is not only degrees but it's also try again yeah 1.27 is not only the reading of the degrees it's also the what of this curve stand Z zore the zcore the the standard deviation is one standard deviation is one maybe I didn't phrase the question okay standard deviation one this is a zcore because if the standard deviation is one and the mean is zero this is a standard normal curve question please wouldn't it just be the same as the other one before it's the same answer yeah it's applied to real world now very good so now once I identifi that the zcore is 1.27 can you phrase the answer um there is a point there's an 89.8 percentage chance interest that you have a 1.27 or what one yeah so 1.27 1.27 at most 1.27 at most 1.27 there's an 89% probability that when you test the thermometer at freezing your thermometer will actually wrongly read 1.27 or something less than that okay so there's a lot of chance that it'll be 1.27 or less okay questions in process questions on that a lot of these standard normal curve problems involve temperature because it's very it's a very easy example so put it down zcore 1.23 what's the probability of cting a thermometer that shows the freezing point of water as above 1.23 or 1.23 here zero where's 1.23 to the left right all right to the left 1.23 and what area is it asking for which direction to the read it out yes to the right how do you know above above above right so what is this area it's asking same sorry the same uh good guess if it was 1.27 negative Oh I thought it was it would be good job so really close very close very close that's a really good point is it true that the normal curve is symmetric mhm right on so 1.27 to the right same area as it's going to be another point8 something yeah I'm with you so let's actually go ahead and look at the table we're going to look at there's a couple ways to do this actually so you want to do negative zcore 1.23 here it comes negative zcore 1 I know I passed it I'm going to try to get the top almost go a little more one more okay well what's the column we want this one so we want this column that'll help us that's great 1.23 do we see it what is the area to the left yes here it is oh that happens now here's the thing they're looking for the right they're looking for the right how do we go yes like I'm if you have 100 and then minus that from 100 very good so I know you mean 100% yeah 100% is equivalent to what number um one so you just have one minus I love when students use the word just yes it's just one minus this right on so this again the table it says very clearly throughout the table that every value that you're given is to the left see this oh cumulative area to the left of your zcore so they gave us a zcore of 1093 subtract it from one Bravo can I line it up again there one more whatever up I'm being too ambitious or 1.1 right here 93093 is this area so what is the area to the right then everything else yes 8907 89.7 1093 = 897 so it's 707 07 thank you9 07 BL your soul so the probability or percentage is still about the same cuz one was 89 yeah 89.8 she called it it was very it's very close definitely to be exact point 8907 this one's closer 89 but the other one's closer to 90 yes you're right this one's closer to 89 the other one's closer to 90 because it was further away wasn't it 1.27 questions on process we going to go to our next page okay doing great everyone okay assuming same parameters as above mean is zero standard deviation one what is the probability a random thermometer shows the freezing point of water is between -2 and 1.5 so for you guys to pick this up right away is pretty awesome it's a lot to do here's zero so where is -2 zcore here -2 and 1.5 here it is and what area are they looking for a probability between between may I shade them yes you may it's my shade dance or Shady dance I have to do that in the play little dance what through the metal detector it's a good idea you should do that that would be fun did you say at the airport no I said in the pl oh in the play on the plane oh you do that on the pl you do a whole dance yeah I walk with the pineapple like that oh cool I can't wait gen 26 through3 awesome be there yeah so there what is the area of the between these two Z scores how do we do it suggestions is my band like totally jacked up now is okay tag is just like the tag is just hanging out that's okay I'm just I love I love my Brands I'm a brand allic so um how do we get those areas using the zcore table better yeah okay what what do we do yes and you can calculate the the area to the right of -2 and subract it by the area to the to the right of 1.5 nice why not find the area to the left and left because that's what the table gives us oh yeah is that the same idea good job because because the table is associated with area to the left we can get both of those easily so we have 1.5 right these are negative let's go up here to positive no negatives first okay one or 1.5 1.50 1.50 is that it yeah 9332 is the area to the left of 93 of uh what' you say it's high it's high it's very high question please ask always so on that uh graph oh not sure um yeah um yeah so you see the top where goes 01 right how do you know these are the hundreds place but how do you know which one to look good job the question ask for zcore of 1.5 okay so 1.50 what the zcore that they're asking for is 1.5 exactly even says 1.50 right here yeah on that when you're looking how do you know oh good job yeah okay yeah we have to learn how to read it how do I know which like which place one this is 1.5 right and then you add this 100 spot thir the third de it's the third decimal yeah this is 1.50 according to what they give you yeah the problem has 1.5 but do you agree what's if I add 1.5 and 01 what's that number 1.51 1.51 Bravo we're not looking get it why it adds up to 1.51 how to read how to read the table like yeah like which column to look at well in this case good job before we looked at the telling me thousands of people have the same question is this table look at this table all together right yeah yay math no math right so they look at this and this is crazy okay but now we're asking for the area to the left of zcore 1.5 is that true yes so we want we're focusing on zcore 1.5 yes okay in this table zcore 1.5 is 1.50 okay if the zcore is 1.51 that would be the area if the zcore is 1.52 is it whatever it ends in yeah yeah this is the last digit I heard someone else say it's the last digit or something like that yeah this is 1.53 this is.3 right here13 cool yeah so what is this what does this zcore represent 1. 1.21 this is the area of zcore 1.21 very good in this case I'm glad you asked 1.5.9 332 putting it down so this area is 9332 you're going to give us a table on absolutely yeah yeah I thought about having you memorize it but I thought against it so want to be fair you know you could also squirt water on yourself in the final that'd be awesome like [Music] okay so negatives are above -2 area to the left of -2 wa just two that's it -2 yes question yeah can you reiterate what we're doing though good job we're finding the area between -2 and 1.5 and to do that first you need to what happens the good job according to our chart the area to the left of 1.5 is9 then we're going to get the area to the left of -2 and then we're going to subtract them Bravo then we're going to subtract them okay let's do it did someone get that before I turn the screen I did what was it um 0 or no was like 0 0228 I see it as well 0228 good job 0228 any questions about how to read the table all right 228 putting it down 0228 and like it says area probability so we say the probability you could still say 0228 they're the same okay good job this is the same answer the area is the probability the area is the probability and what's that number did someone get it for us 9104 good job 9332 minus 0228 say it again please 9104 9104 there okay you ready for the good news the calculator does it for us we don't have to look at the table don't have to look at the table singing now we don't have to look at the table okay our calculator lets us do it you're welcome to look at the table but you don't have to okay let's look at it in the graphing calculator the function actually we'll show right on your paper first it's called normal CDF it's under second Bears normal CDF and as for two inputs the two inputs are the left zcore and the right and the right zcore so we can solve this problem now ready to go right now okay second there normal CDF option two pause for everyone to get caught up normal CDF Bears normal CDF H okay now many of the newer calculators has this functionality if you don't have this functionality you have exactly what you want to see you'll just see normal CDF and then it'll say give it to me who who has this raise your hand yeah many of you have it good job who has what I have similar okay good so half pretty much split so normal CDF asks for left and right zcore left and right zcore so you could actually put it in what is our left zcore -2 doing it -2 enter what is our right zcore 1.5 1.5 you can do it as comma format for those of you that have the thing that I wrote it's -2 comma 1.5 is that already subtract them it does area between I know I feel the same way I feel it I'm like slogging through the table and then I look at this I'm like Hall so my socks fell no I'm ready lower and upper like greater wise like yeah left and right Z score yeah what is the mean in our case Zero isn't the mean zero for us yeah is standard deviation one for us so it says paste all that in so for you fancy calculator folks oh hit enter no no good job enter enter enter enter okay so just it put in some default values this is the mean this is the standard deviation don't worry about it don't worry about it just let it be as it is is that what we got yeah awesome awesome really good okay moving on almost done actually vandalism okay scroll sh problem we did finding a zcore from a known area Okay so zcore of zero what's the probability that you'll be to the left of zero 505 is that okay what's the probability no I understand you're good I okay what's the zcore to the or what's the probability uh or area to the left of1 around do you agree that this is 34% based on our studies of normal curve yeah so what's going to be here approximately 16 about 16 fear not good friends this is in the table too you can get it in the table it's right over here zcore of1 I see I see zcore of1 there's your 16 roughly roughly okay to the left of1 okay moving on zcore of one probability of that 84 84 very good it's really really crucial that you're able to not only read the table use the calculator but just know your basic normal curve increments yeah these are both 34 13.5 and 2.4 we talked about that okay now using the table what's the zcore of 095 this is a preview this a big preview of where we're heading okay we're talking about 95% of the population or most would you agree would you agree that if you said something with 95% accuracy then that's pretty accurate yeah yes that's why this is a crucial number in statistics what's the z for an area of 95 give me a estimate 95% 2 Something 2 point something very good very high very high 2 point something can you explain that again yeah if you're talking about an area of 0.95 that's 95% of the area as opposed to 100 which is the which is the full normal curve so that means that like that's one yeah the area of one would be 100% yes and area of what would be 0.95 oh you know what I'm going to do based on what you said I'm going to turn this to 095 so that we start to get used to the decimals 95 there it's area 0 95 and the zcore is going to be very high like two something now I challenge you to find it in the table where is an area of 95 uh you passed it's in here it's actually oh an area of an area of 09 it's oh it's 1.7 yeah you can get closer oh 1.6 yeah it's actually right here you see this star oh many of the books actually mark it it's such a common area it happens so much in the field of Statistics that they pointed out to you SFL that's an air yeah a little snowflake or like wait asterisk right so this is around 1.6 1.6 something let's Follow the arrow there it is 1.645 1.645 or did I not yeah 1.645 there it is see that I highlight it 1.645 is an area 0.95 putting it down 1.645 is the zcore questions okay what are the Z scores for these two areas okay now oh we did that already let's put that here in other words 95% of the thermometers will be less than or equal to what temperature can you answer that question 95% of thermometers will be less than be less than or equal to what temperature 1.6 1.645 1.645 questions you okay you all right how's the blue pencil working out have the blue pencil all the shavings like filled thing with blue was very cool did you know they Bend oh they Bend wow I didn't know that don't bend them don't too much yeah let's do this let's find the temperatures separating the bottom 2.5 and the top 2.5 the bottom 2.5 and the top here it comes 2.5% and 2.5% here it is let's go here this is 025 area and this is 025 area what are the Z scores that's the area it's the area right so yeah this will be a negative zcore this will be a positive Z score remember what you said before can you tell me the relationship between this zcore and this zcore it's not3 or whatever can you tell me the relationship because of the symmetry yes because of the Symmetry what's the relationship it's symmetrical right so I mean whatever is true on that side is true on so what if suppose this was two what would this zcore be -2 yes cuz the area is the same to the right and to the left yes I do you agree that the zcore here let's suppose a zcore here um is the zcore associated with 2.5% of the population to the right okay okay that's okay let's get it let's get it 2.5 to the right is in our table here we go no okay area of 2.5 actually give it to you it's coming up we'll find it where is an area of 2.5 to the left or 2.5 to the right 025 where where starus starus that's 0.005 let's try again 02 there it is see it so this is this is an area of what to which direction 22525 to the left yes an area of 025 to the left stay with us25 to the left area has what zcore with it Sherman Way go ahead what zcore is associated with this area yes 1.9 1.9 6 negative 1.96 negative 1.96 WR it down again oh it's actually on our board here 1.96 is the zcore so now back to your question if the zcore associated with an area of 2.5 or 2.5% 025 is 1.96 here what do you think the zcore positive what positive because of what we said before symmetry would you like to do this on calculator now too I see eyes like oh I can't handle more info wait can't what's the 975 oh wake you up 975 well if this is an area of 025 wouldn't everything to the left be 975 yeah so what's the zcore for also 1.96 is that what you were saying 1.96 well yeah the area to the left of 1.96 is 97.5 okay 9 7.5 it's everything that 2.5 is not let's put it down I'll give you the calculator but that's when they specifically say like um the zcore of 0.025 have to realize that it's to the left yes yeah let's talk about it when tested of freezing 2.5% of thermometer readings will be less than or equal to how many degrees can you tell us um 1.96 less than so it's on the left the left one um you want to see the picture it helps yeah the picture helps I'm glad you're trying um 2.5% of the thermometers will be less than or here less than this what temperature is that um it's like3 oh no it's negative 1.96 yeah negative 1.96 I realized I'm drinking sink water yum actually it tastes like vanilla yeah why taste so what you're like vanilla oh yeah vanilla let's flavor our water and 2.5% of thermometer is be greater than or equal to what temperature 1.9 yeah 1.96 2.5% of thermometers will be here or more here or more that's 1.96 you okay in the back JS yeah okay you sure yeah I'm good all right apprciate that putting it down one point .96 meaning 95% of the thermometer readings will be between what degrees do you agree back to the thing do you agree if that this is 2.5 and this is 2.5 how much of the area is between 95 95 you're good at this so between 0.025 and 0.2 well yeah I'm talking about these areas these are the are so we can put them down oh those are the so it's the same answer as before yeah so but how much is between so it's from a zcore of 1.96 so 1.9 how much area is that 95% good job so it's just 1.96 and 1.96 the area between them is .95 and remember we know how to do that on the calculator can do it right now calculating the area between 1.96 and 1.96 I recommend doing second enter you get the same thing back and then you just scroll through here do 1.96 to 1.96 we go 1.96 comma 1.96 95% 95 probably probability all right there's one more calculator function I want to show you and then we're done inverse Norm it's the last one okay total area to the left of the zcore so this in the calculator right here will give you the area to the left oh excuse me it'll give you the zcore yeah it'll give you the area sorry yeah no it'll give you the zcore sorry I was I was thinking a second I'll show you one more time the Cal in nor sorry want to get it perfect you know there you go inverse Norm area to the left gives you a zcore can you do two calculations with me and then we're through okay invers Norm area so if we did this one here we go what's the area to the left here 025 area equals 25 so if we did inverse Norm of this value second vs actually let's clear this second vs notice you see inverse Norm option three area to the left 025 what do we expect to see what zcore what zcore has an area 025 1.96 so this asks you for area to the left and it will give you the zcore associated with that again asks you for area gives you the zcore okay symmetry friend what would be positive 1.96 what area Associated here last question of the day you asking me yeah I'm asking you wait what I I got lost in this last part okay this calculator function is it just the opposite of what we did before sort of yes so before on the chart before we're looking we're given no no no not that the whole the number oh the number chart okay this thing the table the table we don't even need we don't need the table the table let's look at the let's look at the normal curve that's what we want look at the normal curve what's the area Associated to the left of the hand me everything to everything to the left of this how much area is that uh 95 plus 0 yes 9595 plus 025 what is that can someone help her 975 975 so let's put that in blue this area is 975 so if I wanted to get the zcore associated with area to the left of 975 M I use this calculator function called inverse Norm observe this gave us the zcore associated with area to the left of 025 now let's get the zcore associated the area to the left being tell us uh9 what do we expect this to be this is your question what do you mean what's the zcore associated with this over two can I look and show you again isn't it 1.96 it's 1.9 do you know why but what are we doing differently we already knew that we already knew that for proving how to get in the calculator yeah we already knew that this is 95 or 97.5 all the way through 1.96 this is 025 netive 1.96 symmetrically they're the same amount off the extremes they're the same amount off the extremes let's put that down and we should get 1.96 and we hit enter okay same thing what are the questions before we conclude any concerns can we do can you try another one of this next class this was like yeah you have a lot of homework to do there's a lot of this what's the zcore with this what's the zcore of that what's the area of this what's the area of that so actually try a lot of practice okay so the more you practice the better you'll get but I seriously thank you all everyone let's do it let's do like yeah W amen you can actually hit that bye [Music]