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No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three? This is the total amount no chocolate. That's this one plus two. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So I took some screen captures from the Khan Academy exercise on correlation coefficient intuition, and they've given us some correlation coefficients, and we need to match them to the various scatter plots. On that exercise, there's a little interface where we can drag these around in a table to match them to the different scatter plots. And the point isn't to figure out how exactly to calculate these. We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that? | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. So I would say scatterplot B is a better fit. A linear model works better for scatterplot B than it works for scatterplot D. So I would give the higher r to scatterplot B, and the lower r, r equals 0.65 to scatterplot D. r is equal to 0.65. And once again, that's because with a linear model, it looks like there's a trend, but there's several data points that really, more data points are way off the line in scatterplot D than in the case of scatterplot B. | Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3 |
It says, Evie read an article that said 6% of teenagers were vegetarians, but she thinks it's higher for students at her school. To test her theory, Evie took a random sample of 25 students at her school, and 20% of them were vegetarians. So just from this first paragraph, some interesting things are being said. It's saying that the true population proportion, if we believe this article, of teenagers that are vegetarian, we could say that is 6%. Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
It's saying that the true population proportion, if we believe this article, of teenagers that are vegetarian, we could say that is 6%. Now, for her school, there is a null hypothesis that the proportion of students at her school that are vegetarian, so this is at her school, that the true proportion, the null would be it's just the same as the proportion of teenagers as a whole. So that would be the null hypothesis. And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
And you can see that she's generating an alternative hypothesis, but she thinks it's higher for students at her large school. So her alternative hypothesis would be the proportion, the true population parameter for her school, school, is greater than 6%. And so to see whether or not you could reject the null hypothesis, you take a sample, and that's exactly what Evie did. She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
She took a random sample of 25 students, and you calculate the sample proportion. And then you figure out what is the probability of getting a sample proportion this high or greater. And if it's lower than a threshold, then you will reject your null hypothesis. And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
And that probability, we call the p-value. The p-value is equal to the probability that your sample proportion, and she's doing this for students at her school, is going to be greater than or equal to 20% if you assumed that your null hypothesis was true. So if you assumed that the true proportion at your school was 6% vegetarians, but you took a sample of 25 students where you got 20%, what is the probability of getting 20% or greater for a sample of 25? Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
Now, there's many ways to approach it, but it looks like she is using a simulation. To see how likely a sample like this was to happen by random chance alone, Evie performed a simulation. She simulated 40 samples of n equals 25 students from a large population, where 6% of the students were vegetarian. She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
She recorded the proportion of vegetarians in each sample. Here are the sample proportions from her 40 samples. So what she's doing here with the simulation, this is an approximation of the sampling distribution of the sample proportions if you were to assume that your null hypothesis is true. And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
And it says below, Evie wants to test her null hypothesis, which is that the true proportion at her school is 6% versus the alternative hypothesis that the true proportion at her school is greater than 6%, where P is the true proportion of students who are vegetarian at her school. And then they ask us, based on these simulated results, what is the approximate P value of the test? And they say the sample result, the sample proportion here was 20%. We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
We saw that right over here. Well, if we assume that this is a reasonably good approximation of our sampling distribution of our sample proportions, there's 40 data points here. And how many of these samples do we get a sample proportion that is greater than or equal to 20%? Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
Well, you could see this is 20% right over here, 20 hundredths. And so you see we have three right over here that meet this constraint. And so that is three out of 40. So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
So if we think this is a reasonably good approximation, we would say that our P value is going to be approximately three out of 40. That if the true population proportion for the school were 6%, if the null hypothesis were true, then approximately three out of every 40 times you would expect to get a sample with 20% or larger being vegetarians. And so three 40ths is what? Let's see. If I multiply both the numerator and the denominator by two and a half, this is approximately equal to, I say two and a half because to go from 40 to 100, and then two and a half times three would be 7.5. So I would say this is approximately 7.5%. And this is actually a multiple choice question. | Estimating a P-value from a simulation AP Statistics Khan Academy.mp3 |
She conducted a poll by calling 100 people whose names were randomly sampled from the phone book. Note that mobile phones and unlisted numbers are not in phone books. The senator's office called those numbers until they got a response from all 100 people chosen. The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here. She didn't ask 100 people to volunteer. Her team went out and got them from the phone book. So this was definitely a case of undercoverage. | Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3 |
The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant. In most studies, in most experiments, the threshold that they think about is the probability of something statistically significant, if the probability of that happening by chance is less than 5%, so this is less than 1%. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
She then created the following scatter plot and trend line. So this is salary in millions of dollars and the winning percentage. And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million. | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%. Yeah, this is the closest statement to what we just said, that if you believe that model, and that's a big if, if you believe this model, then this model says someone making zero dollars will get 39%, and this is frankly why you have to be skeptical of models. | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
In the last few videos, we saw that if we had n points, each of them have x and y coordinates. So let me draw n of those points. So let's call this point 1. It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
It has a coordinate x1, y1. You have the second point over here that has a coordinate x2, y2. And then we keep putting points up here and eventually we get to the nth point over here. So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we have the nth point that has the coordinates xn, yn. What we saw is that there is a line that we can find. We can find a line that minimizes the squared distance. So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So this line right here, I'll call it y is equal to mx plus b. That there is some line that minimizes the squared distance to the point. So let me just review what those squared distances are. Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Sometimes it's called the squared error. So this is the error between the line and point 1. So I'll call that error 1. This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This is the error between the line and point 2. We'll call this error 2. This is the error between the line and point 3. Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Sorry, and point n. So if you wanted the total error, if you want the total squared error, and this is actually how we started off this whole discussion, the total squared error between the points and the line, you literally just take the y value at each point. So for example, you would take y1, that's this value right over here. You would take y1 minus the y value at this point in the line. Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Well, that point in the line is essentially the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus mx1 plus b. This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This right here, that is this y value right over here. That is mx1 plus b. I don't want to get my graph too cluttered, so I'll just delete that there. That is error 1 right over there. That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
That is error 1, and we want the squared errors between each of the points in the line. So that's the first one. Then you do the same thing for the second point. So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we started our discussion this way, y2 minus mx2 plus b squared all the way, I'll do dot, dot, dot to show that there are a bunch of these that we have to do until we get to the nth point, all the way to yn minus mxn plus b squared. Now that we actually know how to find these m's and b's, I showed you the formula, in fact we've proved the formula of how to find these m's and b's. We can find this line, and if we wanted to say, well, you know, how much error is there, we can then calculate it because we now know the m's and the b's. So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? And so let's think about this. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we can calculate it for a certain set of data. Now what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. To do that, we're going to ask ourselves the question, how much, or we could even say what percentage, what percentage of the variation in y is described by the variation in x? And so let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And so let's think about this. How much of the total variation in y, there's obviously variation in y. This y value is over here, this point's y value is over here. There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? So let's think about that. First, let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
There's clearly a bunch of variation in the y, but how much of that is essentially described by the variation in x or described by the line? So let's think about that. First, let's think about what the total variation is. How much of the, we could even say total variation, how much of the total variation in y? So let's just figure out what the total variation in y is. The total variation, and it's really just a tool for measuring, total variation in y, well, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency, and the best central measure we can have of y is the arithmetic mean. So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
How much of the, we could even say total variation, how much of the total variation in y? So let's just figure out what the total variation in y is. The total variation, and it's really just a tool for measuring, total variation in y, well, when we think about variation, and this is even true when we talk about variance, which was the mean variation in y, is we think about the square distance from some central tendency, and the best central measure we can have of y is the arithmetic mean. So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So we could just say the total variation in y is just going to be the sum of the distances of each of the y's, so you get y1, let me do this in another color, you get y1, this y1 over here, this is y1 over here, you get y1 minus the mean of all the y's, minus the mean of all the y's squared, plus y2, plus y2, minus the mean of all of the y squared, plus, and you just keep going all the way to the nth y value, to yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values, find their mean, it'll be some value, maybe it's right over here someplace, maybe that is the mean value of all the y's, and so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if you visualize it, you can imagine a line that's y is equal to the mean of y, which would look just like that, and what we're measuring over here, this error right over here is the square of this distance right over here, between this point vertically and this line. The second one is going to be this distance, is going to be this distance, just right up to the line. The nth one is going to be the distance from there all the way to the line right over there, and then there are these other points in between. This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
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