Hi, welcome. We are on section 8E, and we're still talking about the normal distribution. We're going to be talking about the normal distribution for quite some time probably now, because it's such a beautiful thing in statistics. All right, so let's get started.
So here is your guided notes. And we're now going to be talking about cholesterol. So this is something that will become more important to you as you get older.
And cholesterol in a person's blood plays an important role in the health of their heart and their blood vessels. So it's this kind of stuff that floats around in your blood. Most adults have their cholesterol levels checked at least every five years. I mean, after the pandemic, I'm wondering all the data that came out about who has access to health care. I'm wondering if that's actually true, but most adults should have their cholesterol checked.
That's for sure. To help guide treatment, medical organizations have established guidelines to determine which cholesterol levels are considered healthy. And so we'll be looking at that data in this worksheet.
What statistical tools or methods do you think would be useful for a medical organization when defining the guidelines for healthy levels of cholesterol? So go ahead and think about that, put a little pause and then come back. All right, back. I would hope that they would get access to a whole lot of data on people and what their cholesterol level is.
And then I would want to know. you know, kind of number crunch that to find. So use statistical software, oops, use statistical software to get the mean.
and standard deviation for our population. I imagine it's different than in Japan where they have a very different diet. So what's our general population so that we can get a distribution so we can see our distribution. That would be one of the first steps and then track to see.
how many of those people go on to develop some real serious heart conditions and draw a line as to what's healthy and what's not healthy. And that is what they do. So, but for me, I really want to know what's the distribution and where do I fit in there?
And then another thing is use Z-scores once you have that distribution and assuming that your data is normally distributed, which it is always. Use Z-scores. to study how far from the mean you are. So again, what's super useful is to know the distribution.
and to figure out where you're at in that distribution by looking at the z-scores. You know, am I in a healthy range or am I not healthy range? So here goes. Okay, what we're going to be doing in this section is we're going to be looking first at cholesterol and then later we'll look at blood pressure to understand these concepts.
So a lot of data turns out to be normally distributed. The idea of a random variable is that you're randomly selecting the people that you're studying. That's what's random about a random variable. And we're going to be looking at z-scores to try to understand the results. The normal probability distribution can be used to make probability statements.
So essentially, we're going to be calculating probabilities. as long as we get that beautiful piece of information that the data is normally distributed, which it almost always is. So then later we're going to use technology, we love technology, to calculate the probabilities and we're going to use technology to calculate the percentiles.
So to calculate the probabilities The probabilities are the areas, it's manifested as area under the curve. That's how it looks in our world of distributions. And the percentile, just given the probability, the percentile, if you're given the probability. you're going to look and see, you're going to want to find the cut points.
That's my term. Those are going to be the observations associated with the probability. So essentially, sometimes I'm going to give you an observation and ask you to find the area associated with it, that being the probability.
And then other times I'm going to give you the area and ask you to work backwards to find the observation associated with it. So and we'll use technology to do all that. And then we will. be sometimes looking at the empirical rule. And I call the empirical rule kind of a rule of thumb.
It's a way of quickly estimating where you think your data is. If you want a precise probability, you are going to use technology. If I say find the probability, blah, blah, blah. So when in doubt, use technology. But sometimes we just want to kind of be quick and dirty about it and see how far away are we.
If that's one standard deviation from the norm or from the mean, then maybe about 68% of observations fit there. Two standard deviations, about 95. Three standard deviations, I'm off the chart. That is basically 99.7% of all observations.
So everybody but the most extreme data. fall within three standard variations. That's a rule of thumb.
And when we don't want to go and just tap things into our computers and we use that rule of thumb. Okay, so let's start with cholesterol and then later we'll do blood pressure. So for cholesterol levels for an adult in the U.S., now notice they didn't say healthy adult, they just said adults in the U.S.
The good news is that it's normally distributed. Big surprise. And we know that because we put the data in and we see that it makes this normal distribution.
So the normal distribution is a useful tool. So we can use the data center normal distribution when we're answering these questions, the website there for evaluating cholesterol levels. According to the CDC, the mean cholesterol level.
for adults is 91. So I'm going to be using yellow because somehow I think maybe I'm wrong, but I'm thinking cholesterol is kind of fatty stuff. So I'm just going to draw a quick sketch here. So the mean, I know it's normally distributed. So I get that shape and the mean is 191. So that means it's right at the center.
It's right at the peak. And the standard deviation is 40 and we use SD for standard deviation, that to get to the edge of your hump, that's 40.7. So to figure out this score and this score, I would take 191 and I would add 40.7 and I would take 191 and subtract 40.7.
And I know that that would trap about 68% of it. the middle range so i could do that or i could so it's going to be tedious so i'm going to rely oh look at this i'm going to rely on my dana center my beautiful normal distribution tool because it's going to help me so much so i'll put right in the middle that my i know that from the from the page before that this is 191 And I think it's good to put this on there. point seven but I don't want to do the arithmetic I want to use the tools so I'm going to go over just probably do it here okay and I'll find out but I think I'll...
so Dana Center Map Pathways Space Shiny. That's all I remember. Oh, it's got a decimal in it for some reason. Shiny Tools. Let's see what it comes up with.
So there it is, hopefully. Oh, nope, that's not the right one. Let's see if they give it to me.
Not what I wanted. What's the first do I have? Come on, baby.
There we go. Hopefully once I get rid of that decimal point. There we go.
Okay, so I'm going to go to the normal distribution. I'm going to go to distributions, and there's normal. It's the most common one. So you could have probably done that a lot faster. But to get this picture over here, I'm going to, so explore, all nice and fun, but I'm going to go to find probabilities.
I'm going to put in the. 191 and then I put in the 40.7 and it did it for me and there look they do all that they do all the arithmetic for me so if I added 40.7 I would get 231.7 if I add again I get 272. 4 and the last one is 313.1. So I'm sorry the numbers are off a little bit, but it's hard for me to jam all that in. And going in the opposite direction 150.3.
So any of your former math teachers that told you you couldn't be successful without a good sense of arithmetic, well, they were wrong. we can use technology. So there's our distribution.
I really though want to make sure that we know what we're doing here. And so I'm going to say X equals cholesterol level in individuals. So individuals cholesterol level. So I'm just note to self that if I put a mark on here. I'm marking up somebody's results for their cholesterol.
And the, um, the units are pretty obnoxious. They're, um, where are the units? Something in deciliters.
Oh, here it is. Look at that. Um, milligrams per deciliter.
So it's just floating in your blood. So I'm not going to put the units down. I'm just going to keep that note to self. Okay. So I've used the technology to get the cut points, the observations.
So everybody gets a cholesterol score. And create a graph, shade the area under the curve for one standard deviation below and above the mean. So I think what I'll do is I will pick pretty green.
Mean one standard deviation above the mean. So that's going to be above the mean. These people, and there are little tiny dots there.
There's all the people. So I guess that was too light. Those are all little dots of people who got exactly 231.7. um, milligrams of, of cholesterol per deciliter of blood that's above. And then below is going to be these guys.
And if I want to now shade the area, um, area under the curve from one standard deviation below to one standard, I'll just do this. So that's it. That is 68% of your data.
You've got some histogram under there, but we're just ignoring it and calling it a nice swoopy curve. Okay, so if we tested the cholesterol of a random adult, what's the probability that they would be within one standard deviation of the mean? Well, if I use the empirical rule, I know that it's about 68%, but that's not what they're asking us.
I'm going to go out of my way to say use the empirical rule. If I don't say that, then I want you to calculate the exact percentage given our model. So I'm going to come over here and to play around with this, I've got to make sure that I've entered my data incorrectly. Incorrectly, not incorrectly. And it sure looks like I have.
There's my mean. There's my standard deviation. And I want to track that inner. percentage. So I'm going to need to change my tail to not be a tail at all.
So I'm going to find the interval, which is this one. And then I need to say what A and B are. And yours might look a little different because you won't be on an iPad.
But I'm going to have that my A is going to be and it always has to be in the right order. So my A is going to be 50. because that's the lower standard, one standard mediation away. So 150.3. And my B is going to be this one, 231.7. And then I...
I have an image of what the picture should look like. And I'm just going to make sure the best way to check that I did that is to see that the picture over here matches the picture here. And sure enough, it does. So I did it right.
And the way I want you to find the probabilities, the percentages is look at the percent that is directly under the title. And it's right there, 68.24. So for problem number, I guess, B.
And let's get used to the notation. It's right under the title. The probability that an observation, a cholesterol score, is between 150.3, which is one standard deviation below, and 200. and 31.7, the probability that it's trapped in there is going to be equal to, and I'm looking right here, 68.27%. So that is, how does that compare to the empirical rule? The empirical rule says that's 68%.
That's really, really close to the value. of the empirical rule. And that's why we love the empirical rule. But I'm going to box this because that's my answer. That's what I wanted.
And that's what I'm going to always want you to do is just take it directly off the graph. So now we're asking for below 200. So my advice for all of this is draw your sketch first. Use your brain for, are they asking me for more than, are they asking me for less than, are they asking me for something in between?
Use your brain to get that from the context of the problem and then go to technology and make sure that your picture matches what you know the sketch should look like. So let's do it. So for the next one, so sketch first.
So for the next one, I've got, okay, here's my, here it is. I know that it's 191. What is the... a cholesterol level below 200. So is 200 an observation or is that an area?
It sure isn't an observation. So I'm going to mark it right here. That's my 200. It's just a sketch. And we're interested in below.
So below is going to be going in this direction. Can you see that? A little light below. means that you're going this way, less than.
So estimate the probability of a randomly selected adult has a cholesterol below. So they're actually asking me to find this area. So that's a question mark.
So they've given me a cut point. I know which direction to go in and I know I want to find an area. So I'm going to use that second tab right here because I'm looking for area, looking for area associated with a cut point associated with an observation.
So I'm going to stay right here. I'm in the right place. And the only thing I need to adjust is I want to have a different, I'm not interested.
I'm now interested in things going in this direction. So I'm going to change this and I'm going to pick so that my arrow is going exactly how I want, which is right here. And it says lower tail. And I know that I want to have a threshold of or a cut point of 200. So I'll put that right in here. And I'm going to hope that my picture looks very similar.
And it does. So I now trust the value. And it says 58.75. So the probability that my cholesterol is less than 200, maybe not mine, but a randomly selected adult is equal to 58.75%.
So over half the population has a cholesterol level of 200 or below. And I think they said 200 is norm is borderline. So less than half of adults have.
cholesterol that's deemed in good shape. That's a little disturbing. Okay, the cholesterol level between 200 and 239 is borderline.
So I'm just going to write a note to self because I am actually really surprised about this. Only about... 56% of adults have healthy levels.
It's disturbing because we eat all that fast food. So now here are the borderline people. So again, I want you to draw a little tiny sketch. And I know that 191 is here.
And we're interested in people between 200 and 239. So again, those are observations. So 200, I don't know, maybe it's about right here. And 239, it's just a sketch. So we're interested in this area. And that area is going to be the probability.
So again, we're being asked to look for the area associated with certain given observations. So I'm going to stay in this tab. And so draw your sketch. And now let's play around. We're going to have to set everything stays.
the same in terms of, we still are talking about cholesterol. So we still have a mean of 191. We still have a standard deviation of 40.7, but now we want to trap area between two. So I'm going to change this right here. So my, now I've got a trapped interval. I'm going to ask again.
And my two cut points are 200. and 239. So I'll make sure to put those in 200 and 239. And then I'm going to look at my graph. It's like Christmas. Oh, look at that. It looks just the same. So that makes me happy.
And I'm just going to write exactly what I see under the title. The probability that a randomly selected adult has between a level of 200 and 239, that means their cholesterol is borderline, is 29.34%. So that's telling me, so just barely over half people in the country. adults have good cholesterol and almost 30% have borderline cholesterol. So almost 30% of U.S. adults have borderline cholesterol.
And high cholesterol leads to strokes and heart attack. So you should go home and ask your parents if they know what their cholesterol levels are. I think you can get it checked.
Oh, I think you have to go to a doctor. Okay, so moving on. The CDC has reported that 11% of adults in the U.S. have high cholesterol.
Okay, how does that compare to the analysis of your own? data. So 11% have high cholesterol. So we can approach this in one of two ways.
We can see what score corresponds to a top 11%, or we could take this observation. and see what percent comes after it. So let's look at the area first because that we haven't done that yet. So let's look at percent or area first.
So I'm going to go here, I'm drawing my sketch. And I want to know, given the information they told me about the distribution, what is the threshold value that is associated with this area? This is an area.
This is a probability. So we're no longer going to be using this tab right here. we're not going to be using this. That's if I'm looking to find the area and I have a given threshold, then I use this one. Now I'm going to, I'm given the area and want.
to find the threshold observation. So in this case, I want to find the cholesterol score. So I'm going to use this one.
So I'll click on that one. And so what that's going to allow me to do is plug the areas in and it'll spit out the cholesterols. So let's go back to this.
So top 11%, I need to draw that. So I'm going to do the best I can. I'll just slash like this. And I know that my area is 11%, 11.5% area. So that's what I know.
So what I don't know is what's the score that goes with that? What's that point right here. So that's my question mark.
What's that? So, well, I already, I know that nothing's changed. It's still the same mean. It's still the same mean. It's the same standard deviation.
My picture, this looks like that 11% right here. That's an upper tail. That's the upper 11%. I'm not.
that worried about people have really low cholesterol. So my picture does not look match yet. So I'm going to have to change this. So it's on lower tail. Okay, that looks a little better.
And the area that I want is not 5%. It's 11.5%. And I don't have to put the percent in.
It's already given. And so now does my picture. here match my picture here? Yes, it does. And lo and behold, it's given me a cut point of 239.855.
So what that's telling me is that if you have a cholesterol score of 239.855, you are the lowest value in the top 11.5%. So does that jive with what the CDC said? You're not quite. So I think the CDC would need to, I mean, they must be doing some rounding, but you know what? It's really, really close.
Probably if they chose 11.6%, they get a score that was above their threshold of 240. So. One way to do this is to take the top 11% and see what score goes with it. And it's 239.88, very close to 240. The other way to do it is to, instead of using the area, I will use what's 240? What percentage of people are above 240?
So I'll draw my same sketch. And now I'm going to use the 240 that was given to me. And I'm going to see the area that goes with that question mark on the area. What is that area?
Area connected to the known score. Well, am I going to keep using this tab right here? Which tab am I going to use? Do I need to, am I looking for the area or am I looking for the cut point?
Well, I now have the cut point. So I'm going to go back to finding the probability associated. And I do want an upper tail.
If you have trouble remembering that, just play around until you get the picture you want. So there it's shading the way I want. And I've got. a known value of 240. So I'll change this to 240. Let's hope that it, okay, that looks good.
And the area associated with that is 11.43% of people have above 240. cholesterol score. So was the CDC exaggerating or were they pretty right on? I think they're pretty right on.
It's like almost, it's just a tiny bit less. She went 11.43% as opposed to what they said, which was 11.5%. So I think they did a good job. All right, so but what we did was review, you go to this tab if you want to find the area, and you go to this tab if you want to find the cut point associated with the area.
So to line that up a little bit better, there we go. And we don't really use the explore tab unless I expressly say to you. All right, so we've had enough of cholesterol.
Oh no, we got one more cholesterol question. An adult patient has cholesterol level of 310. That patient asks, is this unusually high? How would you respond?
So if I say do whatever you want, you can go and use these tabs over here. You can use these tabs. But if I specifically say use a z-score to defend your answer, then you have to use a z-score. And we haven't talked about z-scores yet.
So let's do that first. Here is, is this 310, is it an observation or is it an area? This is an observation. Observation. Okay.
So if you recall, z scores, z equals observation minus center over spread. And very specifically, the center. is the center of your distribution, and the spread is the standard deviation of your distribution. So center is going to be in this, so if we want to use z-score, so z equals observation. Minus center over spread.
So I'm going to pop in my observation, which is 310. That was the cholesterol level you got. And then the center of my distribution is 191. And the spread of my distribution is the... 40.7. So if I do that, I will get a Z score of 2.9. I can't remember what I'm writing there.
I think that's 924. I don't have my calculator with me. So forgive me if it's not quite right. but I think that's what it says. It's terrible if you can't read your own writing.
That's close to three standard deviations above. And what we remember about that is if you are, if you are not, here's the center, one is 68, two is 95, three is 99.8, 99. Seven. So almost everybody has scores that are less than that. So I would say this is looking dangerously high. So I'm just going to say, according to your Z score, your cholesterol is higher than almost anyone.
So an adult patient has a cholesterol level of that. The patient's ass is unusually high. What would you respond? So if I'm using Z-scores, I would say you're almost off the charts. Like it's so stop eating those hamburgers, start eating some vegetables and trying to do some other lifestyle changes.
So that's if you want to do Z scores, you could keep using this tool though. And you could say, okay, am I going to be using, am I going to be using this one? am I going to be using this one?
And that two, that 310, I'm giving you an observation. So I'm going to use the find the probability. And so I hit that, I think I'm in it.
And I'm interested in, we're interested in how many people have are higher than this person. So I know it's 191, and I'll just put a slash. I don't know how far away we are. So I'll just put a slash there and say this is 310. So I'm given a cholesterol score and I want to find the area associated with that.
And I want to specifically find how many people, what percent of people have higher scores. So I'll use technology to do that rather than the z-square. So I'm going to come over here and I do, it looks like my tail is checking out. I'm getting an upper tail here, which is good. But I just need to have my threshold be 310. So I'll change that to 310. And no, it doesn't like my ones.
Oh, come on. Hmm, getting really fussy now. Good.
Okay, oh look at that. Just like the z-score is telling us, if you look at this picture over here, it looks like that 310 is not placed right. It should be further out.
definitely almost three standard deviations so that almost no one has so it's much smaller area than i had initially drawn go ahead erase that area and put in the area is over here almost and if i look see that blue tail it is less than That 0.17%, 0.17% is less than 1%. It's way, way less than 1% of people have this score or higher. And it's, I'm just picking 1% because I don't want you, that 0.17% is really, really small. We've already moved the decimal point over.
So that's not 17%. It's already got the percent in here. So it translates to 0.17 out of 100, which is 17. out of 10,000. So 17 people out of 10,000 will have a higher cholesterol level than you.
So you better stop eating those hamburgers. That's what I would say. So we did it two different ways.
Okay. So now let's move on from the fun topic of cholesterol. Let's move on to the fun topic of blood pressure. So blood pressure is another important indicator of health, heart health.
Like cholesterol, blood pressure levels for adults in the United States are normally distributed. However, the individual's blood pressure varies in a normally distributed pattern throughout the day. So you can't just take your cholesterol is pretty stable.
But if you walked up the stairs, if you're nervous, if you just meditated, you're going to be kind of all over the place. So. Uh, and also blood pressure is reported as a systolic pressure, how your heart pumps over di, uh, di stolic pressure. So there's, there's way more factors. That's what they're saying is it's complicated.
Um, so the other funny thing is systolic blood pressure values below 120. are healthy. So I'm going to put, that's a happy thing. I'm going to make that maybe turquoise.
So, um, below 200 and it's how it's mercury, uh, milligrams per HD. Don't know what that is, but it's some kind of measurement are healthy. And then, um, 40, if we're values over 140, are considered high.
So I think I'll put... that as that's worrisome over 140 are high and in between 120 in between the good value and the scary value that's borderline okay so we'll just keep that in in our back pocket the value of the standard deviation it varies um and it depends on what your actual mean So if you have a higher average blood pressure reading, your variation will also is going to be 9.9% of whatever your average is. So just to make things even more complicated, people with higher averages have more variability.
So there's not one picture that we can put all this stuff on. Okay, so... just keep all that in mind that it's complicated.
So a, a patient with a mean systolic blood pressure of 120. So they have an average that is actually almost it's in the, it's on, it's the upper end of good. Um, the stand and they calculated the standard deviation for you to be 11.9. And that little seven up there is not. a, it's a footnote to say where that came from.
So healthcare provider checks this patient's blood pressure. What's the probability of someone who has a mean of 120, which is good ish. What's the probability that they'll actually have a reading that's high, like maybe they walked up the stairs or something.
So what I want you to do is I want you to Try this one and this one. And this we have a different mean and we have a different standard deviation. Okay, so different picture.
So for these, draw a sketch, get your sketch right before you go to the technology and then use technology to calculate the percentage. which is going to be an area. So you're going to be using the find the probability tab. So go ahead and do that, pause that, and then come back and see how you did.
Okay, here we go. Let's see how you did. So I'm going to draw a sketch. I know that the mean for this person from previous history, I know it's 120, and I know the standard deviation is 11.9. And we're wondering what's a probability that if they got a value of 140, so I'll just slash here, 140, what are the chances that that could happen?
So it's going to be that. So we want to find out what kind of scores are going to give, what percentage of scores are going to be higher than 40 for a relatively healthy person. So let's go over here. And I know my center now is 120. And I know my spread is 11.9.
Oh, and it just doesn't like my ones. I got it. Yay. All right. So I've got a slightly different distribution.
And it's giving me an upper tail already. So that means I don't need to change this one, yay, but the threshold value should be 140. So I'll change that to 140. It's going to have a fit while I'm doing it, but everything will be fine later on. And does that picture match, does my picture match up to the computer picture? Yep, it does.
So I'm going to trust the results and there's going to be almost a five percent chance. that a healthy person will have a reading of 140. That's what that's saying. So the probability of someone who actually has a mean of 120 has a result that is greater than 140 is going to be 4.64% of the time. So if I wanted to actually write out what that means, For a person with...
a mean of 120, which is good, blood pressure, we would see 140 or greater 4.64% of the time. So if somebody comes in and they go, I know the nurse takes his blood pressure and she goes, mister, you got high blood pressure. He's like, no, my average is actually 120. You know, it's possible.
All right. So for the next patient, this person has a slightly higher blood pressure. So I'm just going to deliberately shift this a little bit higher.
It's not. they're not stacked beautifully. So that's 130. And because he has, or she has slightly higher average or mean she, or he also has a slightly higher variability standard deviation.
So there's my picture. What is the probability that this person will have a measurement of 140? Well, 140 is a lot closer. So I'm just it's saying I'm going to just say this is 140. I don't know because I haven't got the z score, but I'm 140 is closer to 130 and there's more variation. So it's going to be even closer because the distribution is more spread out.
So if I'm asking for the probability, I'm asking you to find this area right here. that's what I don't know. Area under the curve is the probability. So I'm going to come over to this tab right here. I'm looking at the problem.
I'm finding the area of the probability and I've got to change everything, which is kind of a bummer, but it only takes a moment. So I'm going to change this to 130. So it has a higher center and it has a higher spread 12. 1.9. Good.
And it read that properly for me. And it still is registering an upper tail. So I don't need to change that.
And I also, but it automatically changed my threshold value. So that should be 140. That's the edge of my tail. And just like the picture says, it's closer at what it telling me is the area is 21.91 percent so what that tells me so I'll put a box around that should put a box around this one and put a box around that one and I think I'll write a sentence to um for a patient you with this remembrance borderline, for a patient with a mean of 130, we would see a blood pressure score of 140 or higher almost 22% of the time. So I'm guessing that if I do see a blood pressure score of 140. it's more likely that this patient is borderline than totally healthy.
It's not impossible that they're totally healthy. Almost 5% of the time, it could be a result like that, but it's more likely that they should be working on their blood pressure. All right, a healthcare provider sees a new patient. For the first time, that patient's measurement, that we get a measurement of 140. Should the provider diagnose the person has high blood pressure?
Well, 140 is definitely, if we go back, 140 is the very first, anything over 140 is high, they say. So somebody comes in with 140, is it high or borderline? Well.
It's possible, it's very possible, it's borderline. According to B, part B, it's very possible this person... has borderline, borderline high blood pressure, and it's even possible he or she is in the healthy range for me.
That happens 5% of the time, someone who's healthy will get that kind of a reading. So don't put them on blood pressure yet. Don't put them.
So No meds yet. Some of those blood pressure medications are not nice at all. Instead, take several more readings.
Take more readings throughout the day. Ran out of room. So throughout the day, take more readings just to see. Okay.
Not sure. A different patient has a stolid blood pressure reading of 170. Discuss the likelihood that that actually means that his blood pressure is greater, the true mean is greater than 140. So I think what I would do there, this is... Okay.
So discuss the likelihood that this patient actually has a mean blood pressure greater than 40. So I'll start with, let's assume he has a blood pressure of 140. If, and I'm going to just say it's a he, he had a mean actually equal to 140. Well, if he has a mean of 140, then we already know what the standard deviation is. If I go back, I think I had a picture of that. Oh, no, did we have a picture of 140? Did they ever tell us that?
Oh, it's right there. Equal to 140 and a standard deviation of 13.9. So I'm going by these readings right here.
What could we use to measure how unusual a result of 170 would be? What can we use to see how... unusual 170 is.
If we know the mean and we know the standard deviation, can we somehow rate how far away from the center 170 is going to be the z-score? Let's use z-score. to see how unusual it would be for him to get.
score of 170. He's saying, well, that was weird. I just walked up the stairs weird. That's what it is.
It's I I'm actually perfectly normal. That's what he's saying. So let's do it.
So Z equals observation minus the center of my distribution over the spread of my distribution. And the Z score. I had it right here. I'm just going to write down z equals observation minus center overspread so that you have that formula somewhere. So here it is.
So it's going to be observation is 170. That's what the nurse read off. And the guy's going, oh, no, my average is actually 140. And my standard deviation is actually 13.9. So if the nurse figures that out and measures that, let's see, I'm going to pause this because I don't have a calculator.
Oh my screen sharing is paused. I don't know if my recording is paused. I got my calculator. Let me get the notes back.
So if I do this order of operations, I've got to figure out the numerator first. So that's going to be a 30 over 13.9. And I noticed that quite a few people missed that on the midterm. So you've got to review order of operation, make sure you know how to, when you're going. So it's going to be 2.158.
I'm going to stop there. So my Z score is this. So I'm more than two standard deviations from the norm, from the mean.
So. That's 68, 95. So you're more than 95% of the data is inside of that range. So I think this is unusual. According to the empirical rule, this seems unusual.
This observation is more. than two standard deviations from mean. So very unusual.
So I'm going to tell the guy, the observation is 170. So I don't think that that's actually your average. I think you've got an average that's higher than 140. which means you are definitely outside of, you're outside of the warning range and you're into the over 140 range. So maybe we should talk about medication right now.
That's if we did the Z-score, which I think is the cleanest way to measure if an observation is unusual. Just take the Z-score. But you could do the other approach is to plot the observation where you go, okay, here's my distribution.
It's 140 and it's 13.9 and I have a score of 170. What's the area? What's the probability that I get that score higher? I could do that. I want to know the area which is the probability and so I will just throw in 140, throw in is it 13.9 or is it 12.9? No it's 13.9.
And what are the chances of getting that observation or something more unusual? I'll throw in my observation. of 170 and you would see that result 1.55% of the time. You would see a score or a measurement of 170 or higher, which is almost never. So I'm assuming he's in denial.
So time for some med, time for meds or a serious lifestyle or lifestyle. change. So you should be talking to your parents about this because if you're in your 20s, then it's likely your parents are getting to the point where they are not thinking about it, but they should start thinking about it.
And I'm starting thinking about it right now too. All right. So, but can we, do we worry about blood pressure for children? Do we ever do that? And the answer is yes.
So children's blood pressure varies with age and size. So for every child, you've got to have a distribution that has both their age and their size. So if your kid is eight years old and weighs 95 pounds, you've got to find that special distribution for him to account for the variation. Instead of providing a simple numerical range.
the child's blood pressure falls in the normal distribution of all. So you just look at the whole distribution and they define hypertension. That's high blood pressure is if the kid is in the 95th percentile.
So if only 5% of people, of children, his age and size have more higher blood pressure than... he's got hypertension or she does. So for an eight-year-old boy who is 50 inches tall, the distribution has a mean of 97 and a standard deviation of 9.4. Okay. So I have everything I need to draw a distribution.
What value of blood pressure corresponds to the 95th percentile? So here, Am I giving you, I'm giving you a piece of information and you need to ask yourself, is that piece of information, the cut point, is that his blood pressure measurement or is that the area associated with the measurement? So pause and have a look at that and think.
So this, if I draw this picture and I always want you to draw the picture before you start. Plugging in things to that. So I know my center is 97 and my spread is 9.4. Okay. So I kind of got a picture of my distribution.
And if 95th percentile means that 95% of children have lower scores. So that means I'm going to just grab. I'm just going to do this and it means that this here is 95% because that's what percentile means.
It means lower tail. So we want to find the observation that goes with that percent. So when I come up here.
I'm not going to use this one. I'm going to use this one. This tab will give me the observation if I know the area. So I'm going to pop that in and I'm going to, this is a children's thing, so they have much lower blood pressure. Okay.
And the standard deviation, because they have lower blood pressure, they have lower standard deviation too. So it's 9.4. Make sure it took that right.
And I'm going to look at my picture and I want, I want this, I want this, and I want this. And that's not what I see. I see upper tail there.
So I'm going to switch that around. And now my picture coincides. Oh, and it does it, it's, it, it really, it gave.
automatically had the 95 in there so the point the threshold value is 112.462 so um so we didn't have to do a lot of work for that one it was kind of given to us and the um It's good to put the units in. And where are the units? I thought they were here somewhere. Something about mercury.
Oh, well, find the units. I'm going to put in them. I think it's something over mercury.
So if you want a sentence here, if a child who is eight years old, eight year old boy. who is 50 inches tall, just for him, has a blood pressure score of 112.462 or higher. Um, what value corresponds, um, he is in the 95th percentile. That's the cut point that corresponds to 95% area behind his observation. So high is bad.
Low is also bad. So here, so pause this and answer it yourself and then come back and make the sketch, then go to the district, then go to the technology. and put in your stuff to make the the picture look the same. Okay, very low blood pressure is also dangerous and requires emergency medical treatment.
And we get very excited if the children are below the fifth percentile. So I'm going to draw my distribution. So I assume, are we still dealing with the eight-year-old boys? Yep, same, same boy. maybe so 97 and 50 and 9.4 so and we're down here fifth percentile means that only five percent of children will have lower blood pressure so they're giving us that this is five percent and we don't know, we don't know what this is.
We don't know what the odds are, what the score would be for that. So if I'm going to go back to this tab, because they did not, they gave me the area, but they did not give me the cut point. So this will help me get that.
So I'm going to click on that and I'm, Oh, I'm already there. And Oh, all this is the same but now I still want the lower tail. I don't need to change the direction.
I just need to move it way on down the line. We're still shading in the right direction, but our threshold's way off. And that's because we want an area of 5%. So I'm going to come over here and put in 5. You don't have to.
The percent's already there, and it interpreted S. So try again. Good. And...
Does the picture I drew look like that picture? Yes, it does. And so this ends up being 81.538. So if the eight-year-old boy who's 50 inches tall, so Shorty has a blood pressure measure of 81 point, that's a terrible reversal, 81.538 something milligrams mercury, then only 5%.
of boys have lower measures. So he has a very low, very low blood pressure. So I'm taking the pasta.
All right, we're done. So it was, I don't think there was really a lot of new information here. We just reviewed some scores.
So take a break and do the practice and then do the preview for the next section. All right. Talk to you later. Bye.