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I really don't need a calculator for this. Plus 7 plus 14. So that's five data points, and I'm going to divide by 5. And I get 6. So the population mean for years of experience at my organization is 6. 6 years of experience. Well, that's, I guess, interesting. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And I get 6. So the population mean for years of experience at my organization is 6. 6 years of experience. Well, that's, I guess, interesting. But now I want to ask another question. I want to get some measure of how much spread there is around that mean, or how much do the data points vary around that mean? And obviously, I can give someone all the data points. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Well, that's, I guess, interesting. But now I want to ask another question. I want to get some measure of how much spread there is around that mean, or how much do the data points vary around that mean? And obviously, I can give someone all the data points. But instead, I actually want to come up with a parameter that somehow represents how much all of these things, on average, are varying from this number right here. Or maybe I will call that thing the variance. And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And obviously, I can give someone all the data points. But instead, I actually want to come up with a parameter that somehow represents how much all of these things, on average, are varying from this number right here. Or maybe I will call that thing the variance. And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear. It's a parameter. The population variance I'm going to denote with the Greek letter sigma, lowercase sigma. This is capital sigma. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so what I do, so the variance, and I will do, and this is a population variance that I'm talking about, just to be clear. It's a parameter. The population variance I'm going to denote with the Greek letter sigma, lowercase sigma. This is capital sigma. Lowercase sigma squared. And I'm going to say, well, I'm going to take the distance from each of these points to the mean. And just so I get a positive value, I'm going to square it. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is capital sigma. Lowercase sigma squared. And I'm going to say, well, I'm going to take the distance from each of these points to the mean. And just so I get a positive value, I'm going to square it. And then I'm going to divide by the number of data points that I have. So essentially, I'm going to find the average squared distance. Now, that might sound very complicated, but let's actually work it out. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And just so I get a positive value, I'm going to square it. And then I'm going to divide by the number of data points that I have. So essentially, I'm going to find the average squared distance. Now, that might sound very complicated, but let's actually work it out. So I'll take my first data point. I'll take that data point, and I will subtract our mean from it. So this is going to give me a negative number, but if I square it, it's going to be positive. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Now, that might sound very complicated, but let's actually work it out. So I'll take my first data point. I'll take that data point, and I will subtract our mean from it. So this is going to give me a negative number, but if I square it, it's going to be positive. So it's essentially going to be the squared distance between 1 and my mean. And then to that, I'm going to add the squared distance between 3 and my mean. And to that, I'm going to add the squared distance between 5 and my mean. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is going to give me a negative number, but if I square it, it's going to be positive. So it's essentially going to be the squared distance between 1 and my mean. And then to that, I'm going to add the squared distance between 3 and my mean. And to that, I'm going to add the squared distance between 5 and my mean. And since I'm squaring, it doesn't matter if I do 5 minus 6 or 6 minus 5. When I square it, I'm going to get a positive result regardless. And then to that, I'm going to add the squared distance between 7 and my mean. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And to that, I'm going to add the squared distance between 5 and my mean. And since I'm squaring, it doesn't matter if I do 5 minus 6 or 6 minus 5. When I square it, I'm going to get a positive result regardless. And then to that, I'm going to add the squared distance between 7 and my mean. So 7 minus 6 squared. All of this, this is my population mean that I'm finding the difference between. And then finally, the squared difference between 14 and my mean. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then to that, I'm going to add the squared distance between 7 and my mean. So 7 minus 6 squared. All of this, this is my population mean that I'm finding the difference between. And then finally, the squared difference between 14 and my mean. And then I'm going to find essentially the mean of these squared distances. So I have 5 squared distances right over here. So let me divide by 5. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then finally, the squared difference between 14 and my mean. And then I'm going to find essentially the mean of these squared distances. So I have 5 squared distances right over here. So let me divide by 5. So what will I get when I make this calculation right over here? Well, let's figure this out. This is going to be equal to 1 minus 6 is negative 5. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So let me divide by 5. So what will I get when I make this calculation right over here? Well, let's figure this out. This is going to be equal to 1 minus 6 is negative 5. Negative 5 squared is 25. 3 minus 6 is negative 3. If I square that, I get 9. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is going to be equal to 1 minus 6 is negative 5. Negative 5 squared is 25. 3 minus 6 is negative 3. If I square that, I get 9. 5 minus 6 is negative 1. If I square it, I get positive 1. 7 minus 6 is 1. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
If I square that, I get 9. 5 minus 6 is negative 1. If I square it, I get positive 1. 7 minus 6 is 1. If I square it, I get positive 1. And 14 minus 6 is 8. If I square it, I get 64. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
7 minus 6 is 1. If I square it, I get positive 1. And 14 minus 6 is 8. If I square it, I get 64. And then I'm going to divide all of that by 5. And I don't need to use a calculator, but I tend to make a lot of careless mistakes when I do things while I'm making a video. So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
If I square it, I get 64. And then I'm going to divide all of that by 5. And I don't need to use a calculator, but I tend to make a lot of careless mistakes when I do things while I'm making a video. So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5. So I get 20. So the average squared distance, or the mean squared distance from our population mean is equal to 20. And you might say, wait, these things aren't 20. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So let me get 25 plus 9 plus 1 plus 1 plus 64 divided by 5. So I get 20. So the average squared distance, or the mean squared distance from our population mean is equal to 20. And you might say, wait, these things aren't 20. Remember, it's the squared distance away from my population mean. So I squared each of these things. I liked it because it made it positive. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And you might say, wait, these things aren't 20. Remember, it's the squared distance away from my population mean. So I squared each of these things. I liked it because it made it positive. And we'll see later it has other nice properties about it. Now, the last thing is how can we represent this mathematically? We already saw that we know how to represent a population mean and a sample mean mathematically like this. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I liked it because it made it positive. And we'll see later it has other nice properties about it. Now, the last thing is how can we represent this mathematically? We already saw that we know how to represent a population mean and a sample mean mathematically like this. And hopefully, we don't find it that daunting anymore. But how would we do the exact same thing? How would we denote what we did right over here? | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We already saw that we know how to represent a population mean and a sample mean mathematically like this. And hopefully, we don't find it that daunting anymore. But how would we do the exact same thing? How would we denote what we did right over here? Well, let's just think it through. We're just saying that the population variance, we're taking the sum of each. So we're going to take each item. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
How would we denote what we did right over here? Well, let's just think it through. We're just saying that the population variance, we're taking the sum of each. So we're going to take each item. We'll start with the first item. And we're going to go to the nth item in our population. We're talking about a population here. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So we're going to take each item. We'll start with the first item. And we're going to go to the nth item in our population. We're talking about a population here. And we're not going to just take the item. This would just be the item. But we're going to take the item. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We're talking about a population here. And we're not going to just take the item. This would just be the item. But we're going to take the item. And from that, we're going to subtract the population mean. We're going to subtract this thing. We're going to subtract this thing. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
But we're going to take the item. And from that, we're going to subtract the population mean. We're going to subtract this thing. We're going to subtract this thing. We're going to square it. So the way I've written it right now, this would just be the numerator. I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
We're going to subtract this thing. We're going to square it. So the way I've written it right now, this would just be the numerator. I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it. If I really want to get the way I figured out this variance right over here, I have to divide the whole thing by the number of data points we have. So this might seem very daunting and very intimidating. But all it says is, take each of your data points. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I've just taken the sum of each of these things, the sum of the difference between each data point and the population mean and squared it. If I really want to get the way I figured out this variance right over here, I have to divide the whole thing by the number of data points we have. So this might seem very daunting and very intimidating. But all it says is, take each of your data points. Well, one, it says, figure out your population mean. Figure that out first. And then from each data point in your population, subtract out that population mean, square it, take the sum of all of those things, and then just divide by the number of data points you have. | Variance of a population Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So every day after school, you decide to go to the frozen yogurt store at exactly four o'clock, four o'clock p.m. Now because you like frozen yogurt so much, you are not a big fan of having to wait in line when you get there. You're impatient. You want your frozen yogurt immediately. And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line. So let me make two columns here. Line size is the left column. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of the line. So let me make two columns here. Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in line, exactly eight times. And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
And in your 50 visits, you don't see more than, you never see more than two people in line. I guess this is a very efficient cashier at this frozen yogurt store. So based on this, based on what you have observed, what would be your estimate of the probability of finding no people in line, one people in line, or two people in line, at 4 p.m. on the days after school that you visit the frozen yogurt store? And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right? | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
And you'll say you only visited on weekdays where there are school days. So what's the probability of there being no line, a one person line, or a two person line when you visit at 4 p.m. on a school day? Well, all you can do is estimate the true probability, the true theoretical probability. We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
We don't know what that is, but you've done 50 observations here, right? This is, and notice this adds up to 50. 18 plus eight is 26, 26 plus 24 is 50. So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So you've done 50 observations here, and so you can figure out, well, what are the relative frequencies of having zero people? What is the relative frequency of one person, or the relative frequency of two people in line? And then we can use that as the estimates for the probability. So let's do that. So probability estimate. I'll do it in the next column. So probability, probability estimate. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So let's do that. So probability estimate. I'll do it in the next column. So probability, probability estimate. And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line? | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So probability, probability estimate. And once again, we can do that by looking at the relative frequency. The relative frequency of zero, well, we observe that 24 times out of 50, and so 24 out of 50 is the same thing as 0.48, or you could even say that this is 48%. Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Now, what's the relative frequency of seeing one person in line? Well, you observe that 18 out of the 50 visits, 18 out of the 50 visits, that would be a relative frequency, 18 divided by 50 is 0.36, which is 36% of your visits. And then finally, the relative frequency of seeing a two-person line, that was eight out of the 50 visits, and so that is 0.16, and that is equal to 16% of the visits. And so there's interesting things here. Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
And so there's interesting things here. Remember, these are estimates of the probability. You're doing this by essentially sampling what the line on 50 different days. You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
You don't know, it's not gonna always be exactly this, but it's a good estimate. You did it 50 times. And so based on this, you'd say, well, I'd estimate the probability of having a zero-person line is 48%. I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
I'd estimate that the probability of having a one-person line is 36%. I'd estimate that the probability of having a two-person line is 16%, or 0.16. And it's important to realize that these are legitimate probabilities. Remember, to be a probability, it has to be between zero and one. It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Remember, to be a probability, it has to be between zero and one. It has to be zero and one. And if you look at all of the possible events, it should add up to one, because at least based on your observations, these are the possibilities. Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Obviously, in a real world, there might be some kind of crazy thing where more people go in line, but at least based on the events that you've seen, these three different events, and these are the only three that you've observed, based on your observations, these three should add, because these are the only three things you've observed, they should add up to one, and they do add up to one. Let's see, 36 plus 16 is 52. 52 plus 48, they add up to one. Now, once you do this, you might do something interesting. You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Now, once you do this, you might do something interesting. You might say, okay, you know what? Over the next two years, you plan on visiting 500 times. So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So visiting 500 times. So based on your estimates of the probability of having no line, of a one-person line, or a two-person line, how many times in your next 500 visits would you expect there to be a two-person line, based on your observations so far? Well, it's reasonable to say, well, a good estimate of the number of times you'll see a two-person line when you visit 500 times. Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Well, you say, well, there's gonna be 500 times, and it's a reasonable expectation, based on your estimate of the probability, that 0.16 of the time, you will see a two-person line, or you could say eight out of every 50 times. And so what is this going to be? Let's see, 500 divided by 50 is just 10. So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations. It is completely possible, first of all, that your observations were off, that this is just a random chance that you happened to observe this many, or this few times that there were two people in line. So that could be off, but even if these are very good estimates, it's possible that something, that you see a two-person line 85 out of the 500 times, or 65 out of the 500 times. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
So you would expect, you would expect that 80 out of the 500 times, you would see a two-person line. Now, to be clear, I would be shocked if it's exactly 80 ends up being the case, but this is actually a very good expectation, based on your observations. It is completely possible, first of all, that your observations were off, that this is just a random chance that you happened to observe this many, or this few times that there were two people in line. So that could be off, but even if these are very good estimates, it's possible that something, that you see a two-person line 85 out of the 500 times, or 65 out of the 500 times. All of those things are possible. Whenever you think, and it's always very important to keep in mind, you're estimating the true probability here, which it's very hard to know for sure what the true probability is, but you can make estimates based on sampling the line on different days, by making these observations, by having these experiments, so to speak, each of these observations you can use an experiment, and then you can use those to set an expectation. But none of these things do you know for sure, that they're definitely gonna be exactly 80 out of the next 500 times. | Constructing probability model from observations 7th grade Khan Academy (2).mp3 |
Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
A statistician for a basketball team tracked the number of points that each of the 12 players on the team had in one game, and then made a stem and the leaf plot to show the data. Sometimes it's called a stem plot. How many points did the team score? And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored? | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And when you first look at this plot right over here, it seems a little hard to understand. Under stem you have zero, one, two. Under leaf you have all of these digits here. How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
How does this relate to the number of points each student or each player actually scored? And the way to interpret a stem and leaf plot is the leaf contain, at least the way that this statistician used it, the leaf contains the smallest digit, or the ones digit, in the number of points that each player scored, and the stem contains the tens digits. And usually the leaf will contain the rightmost digit, or the ones digit, and then the stem will contain all of the other digits. And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And what's useful about this is it gives kind of a distribution of where the players were. You see that most of the players scored points that started with a zero, then a few more scored points that started with a one, and then only one score scored points that started with a two, and it was actually 20 points. So let me actually write down all of this data in a way that maybe you're a little bit more used to understanding it. So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So I'm gonna write the zeros in purple. So there's, let's see, one, two, three, four, five, six, seven players had zero as the first digit. So one, two, three, four, five, six, seven. I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
I wrote seven zeros, and then this player also had a zero in his ones digit. This player, I'm gonna try to do all the colors, this player also had a zero in his ones digit. This player right here had a two in his ones digit, so he scored a total of two points. This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
This player, let me do orange, this player had four for his ones digit. This player had seven for his ones digit. Then this player had seven for his ones digit. And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And then, let me see, I'm almost using all the colors, this player had nine for his ones digit. So the way to read this is you had one player with zero points, zero, two, four, seven, nine, and nine. But you can see, I'm just kind of silly saying the zero was the tens digit. You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
You could have even put a blank there, but the zero lets us know that they didn't score, they didn't score anything in the tens place. But these are the actual scores for those seven players. Now let's go to the next row in the stem and leaf plot. So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So over here, all of the digits start with, or all of the points start with one for each of the players, and there's four of them. So one, one, one, and one. And then we have this player over here, it's one, his ones digit, or her ones digit is a one. So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So this player, this represents 11. One in the tens place, one in the ones place. This player over here also got 11. One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
One in the tens place, one in the ones place. This player, let me do orange. This player has three in the ones place. So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So he or she scored 13 points. One in the tens place, three in the ones place, 13 points. And then, I will do this in purple, this player has eight in their ones place. So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So he or she scored 18 points. One in the tens place, eight in the ones place, 18 points. And then finally, you have this player that has two, the tens digit is a two, and then the ones digit is a zero. Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Is a zero, I'll circle that in yellow. It is a zero. So he or she scored 20 points. So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So using, looking at the stem and leaf plot, we're able to extract out all of the number of points that all of the players scored. And once again, what was useful about this is you see how many players scored between zero and nine points, including nine points. How many scored between 10 and 19 points, and then how many scored 20 points or over. And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
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