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Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? | Test taking probability and independent events Precalculus Khan Academy.mp3 |
That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. Or, since these are independent events, you can multiply. You see that there are 12 outcomes because there's 12 possible outcomes. So there's four possible outcomes for problem number one times the three possible outcomes for problem number two, and that's also where you get a 12. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
And I ask them, and there's only two options. They can either have an unfavorable rating or they could have a favorable rating. And let's say after I survey every single member of this population, 40% have an unfavorable rating and 60% have a favorable rating. So if I were to draw the probability distribution, it's going to be a discrete one because there's only two values that any person can take on. They could either have an unfavorable view or they could have a favorable view. And 40% have an unfavorable view. And let me color code this a little bit. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So if I were to draw the probability distribution, it's going to be a discrete one because there's only two values that any person can take on. They could either have an unfavorable view or they could have a favorable view. And 40% have an unfavorable view. And let me color code this a little bit. So this is the 40% right over here, so 0.4. Maybe I'll just write 40% right over there. And then 60% have a favorable view. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
And let me color code this a little bit. So this is the 40% right over here, so 0.4. Maybe I'll just write 40% right over there. And then 60% have a favorable view. Let me just make sure I'm doing this right. 60% have a favorable view. And notice these two numbers add up to 100% because everyone had to pick between these two options. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
And then 60% have a favorable view. Let me just make sure I'm doing this right. 60% have a favorable view. And notice these two numbers add up to 100% because everyone had to pick between these two options. Now, if I were to go and ask you to pick a random member of that population and say, what is the expected favorability rating of that member, what would it be? Or another way to think about it is, what is the mean of this distribution? And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
And notice these two numbers add up to 100% because everyone had to pick between these two options. Now, if I were to go and ask you to pick a random member of that population and say, what is the expected favorability rating of that member, what would it be? Or another way to think about it is, what is the mean of this distribution? And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on. Now, the way I've written it right here, you can't take a probability-weighted sum of u and f. You can't say 40% times u plus 60% times f. You won't get any type of a number. So what we're going to do is define u and f to be some type of value. So let's say that u is 0 and f is 1. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
And for a discrete distribution like this, your mean or your expected value is just going to be the probability-weighted sum of the different values that your distribution can take on. Now, the way I've written it right here, you can't take a probability-weighted sum of u and f. You can't say 40% times u plus 60% times f. You won't get any type of a number. So what we're going to do is define u and f to be some type of value. So let's say that u is 0 and f is 1. And now the notion of taking a probability-weighted sum makes some sense. So the mean of this distribution is going to be 0.4. That's this probability right here. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So let's say that u is 0 and f is 1. And now the notion of taking a probability-weighted sum makes some sense. So the mean of this distribution is going to be 0.4. That's this probability right here. Times 0 plus 0.6 times 1, which is going to be equal to this is just going to be 0. 0.6 times 1 is 0.6. So clearly, no individual can take on the value of 0.6. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
That's this probability right here. Times 0 plus 0.6 times 1, which is going to be equal to this is just going to be 0. 0.6 times 1 is 0.6. So clearly, no individual can take on the value of 0.6. No one can tell you I 60% am favorable and 40% am unfavorable. Everyone has to pick either favorable or unfavorable. So you will never actually find someone who has a 0.6 favorability value. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So clearly, no individual can take on the value of 0.6. No one can tell you I 60% am favorable and 40% am unfavorable. Everyone has to pick either favorable or unfavorable. So you will never actually find someone who has a 0.6 favorability value. It will either be a 1 or a 0. So this is an interesting case where the mean or the expected value is not a value that the distribution can actually take on. It's a value someplace over here that obviously cannot happen, but this is the mean. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So you will never actually find someone who has a 0.6 favorability value. It will either be a 1 or a 0. So this is an interesting case where the mean or the expected value is not a value that the distribution can actually take on. It's a value someplace over here that obviously cannot happen, but this is the mean. This is the expected value. And the reason why that makes sense is if you surveyed 100 people, you multiply 100 times this number, you would expect 60 people to say yes. Or if you summed them all up, 60 would say yes and then 40 would say 0. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
It's a value someplace over here that obviously cannot happen, but this is the mean. This is the expected value. And the reason why that makes sense is if you surveyed 100 people, you multiply 100 times this number, you would expect 60 people to say yes. Or if you summed them all up, 60 would say yes and then 40 would say 0. You sum them all up, you would get 60% saying yes. And that's exactly what our population distribution told us. Now what is the variance? | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Or if you summed them all up, 60 would say yes and then 40 would say 0. You sum them all up, you would get 60% saying yes. And that's exactly what our population distribution told us. Now what is the variance? What is the variance of this population right over here? So the variance, let me write it over here. Let me pick a new color. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Now what is the variance? What is the variance of this population right over here? So the variance, let me write it over here. Let me pick a new color. The variance is just, you could view it as the probability weighted sum of the squared distances from the mean or the expected value of the squared distances from the mean. So what's that going to be? Well, there's two different values that anything can take on. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Let me pick a new color. The variance is just, you could view it as the probability weighted sum of the squared distances from the mean or the expected value of the squared distances from the mean. So what's that going to be? Well, there's two different values that anything can take on. You can either have a 0 or you could either have a 1. The probability that you get a 0 is 0.4. So there's a 0.4 probability that you get a 0. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Well, there's two different values that anything can take on. You can either have a 0 or you could either have a 1. The probability that you get a 0 is 0.4. So there's a 0.4 probability that you get a 0. And if you get a 0, what's the distance from 0 to the mean? The distance from 0 to the mean is 0 minus 0.6. Or I could even say 0.6 minus 0. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So there's a 0.4 probability that you get a 0. And if you get a 0, what's the distance from 0 to the mean? The distance from 0 to the mean is 0 minus 0.6. Or I could even say 0.6 minus 0. Same thing because we're going to square it. 0 minus 0.6 squared. Remember, the variance is the probability or the weighted sum of the squared distances. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Or I could even say 0.6 minus 0. Same thing because we're going to square it. 0 minus 0.6 squared. Remember, the variance is the probability or the weighted sum of the squared distances. So this is the difference between 0 and the mean. And then plus, there's a 0.6 chance that you get a 1. The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Remember, the variance is the probability or the weighted sum of the squared distances. So this is the difference between 0 and the mean. And then plus, there's a 0.6 chance that you get a 1. The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here. Now what is this value going to be? This is going to be 0.4 times 0.6 squared. This is 0.4 times 0. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
The difference between 1 and 0.6, 1 and our mean, 0.6, is that, and then we are also going to square this over here. Now what is this value going to be? This is going to be 0.4 times 0.6 squared. This is 0.4 times 0. Because 0 minus 0.6 is negative 0.6. If you square it, you get positive 0.36. So this value right here, I'm going to color code it, this value right here is times 0.36. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
This is 0.4 times 0. Because 0 minus 0.6 is negative 0.6. If you square it, you get positive 0.36. So this value right here, I'm going to color code it, this value right here is times 0.36. And then this value right here, let me do this in another, so then we're going to have plus 0.6, plus this 0.6 times 1 minus 0.6 squared. Now 1 minus 0.6 is 0.4. 0.4 squared, or 0.4 squared, is 0.16. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So this value right here, I'm going to color code it, this value right here is times 0.36. And then this value right here, let me do this in another, so then we're going to have plus 0.6, plus this 0.6 times 1 minus 0.6 squared. Now 1 minus 0.6 is 0.4. 0.4 squared, or 0.4 squared, is 0.16. So let me do this. So this value right here is going to be 0.16. Let me get my calculator out to actually calculate these values. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
0.4 squared, or 0.4 squared, is 0.16. So let me do this. So this value right here is going to be 0.16. Let me get my calculator out to actually calculate these values. Let me get my calculator out. So this is going to be 0.4 times 0.36 plus 0.6 times 0.16, which is equal to 0.24. So our standard deviation of this distribution is 0.24. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
Let me get my calculator out to actually calculate these values. Let me get my calculator out. So this is going to be 0.4 times 0.36 plus 0.6 times 0.16, which is equal to 0.24. So our standard deviation of this distribution is 0.24. Or if you want to think about the variance of this distribution is 0.24, and the standard deviation of this distribution, which is just the square root of this, is going to be the square root of 0.24. And let's calculate what that is. That is going to be, let's take the square root of 0.24, which is equal to 0.48. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So our standard deviation of this distribution is 0.24. Or if you want to think about the variance of this distribution is 0.24, and the standard deviation of this distribution, which is just the square root of this, is going to be the square root of 0.24. And let's calculate what that is. That is going to be, let's take the square root of 0.24, which is equal to 0.48. Well, I'll just round it up, 0.49. So this is equal to 0.49. So if you were to look at this distribution, the mean of this distribution is 0.6. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
That is going to be, let's take the square root of 0.24, which is equal to 0.48. Well, I'll just round it up, 0.49. So this is equal to 0.49. So if you were to look at this distribution, the mean of this distribution is 0.6. So 0.6 is the mean. And the standard deviation is 0.5. So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So if you were to look at this distribution, the mean of this distribution is 0.6. So 0.6 is the mean. And the standard deviation is 0.5. So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1. So this is one standard deviation above. And then one standard deviation below gets you right about here. And that kind of makes sense. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
So the standard deviation is, so it's actually out here, because if you go add one standard deviation, you're almost getting to 1.1. So this is one standard deviation above. And then one standard deviation below gets you right about here. And that kind of makes sense. It's hard to really have a good intuition for a discrete distribution, because you really can't take on those values. But it makes sense that the distribution is skewed to the right over here. Anyway, I did this example with particular numbers, because I wanted to show you why this distribution is useful, and the next video I'll do these with just general numbers, where this is going to be p, where this is the probability of success, and this is the 1 minus p, which is the probability of failure. | Mean and variance of Bernoulli distribution example Probability and Statistics Khan Academy.mp3 |
What we're going to do in this video is talk about the idea of a residual plot for a given regression and the data that it's trying to explain. So right over here, we have a fairly simple least squares regression. We're trying to fit four points. And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. Maybe some type of nonlinear model, some type of nonlinear curve might better fit the data, or the relationship between the y and the x is nonlinear. Another way you could think about it is when you have a lot of residuals that are pretty far away from the x-axis in the residual plot, you would also say this line isn't such a good fit. If you calculate the r value here, it would only be slightly positive, but it would not be close to one. | Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3 |
The data were skewed to the right with a sample mean of 38.75. She's considering using her data to make a confidence interval to estimate the mean age of faculty members at her university. Which conditions for constructing a t-interval have been met? So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So pause this video and see if you can answer this on your own. Okay, now let's try to answer this together. So there's 700 faculty members over here. She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
She's trying to estimate the population mean, the mean age. She can't talk to all 700, so she takes a sample, a simple random sample of 20. So the n is equal to 20 here. From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
From this 20, she calculates a sample mean of 38.75. Now ideally, she wants to construct a t-interval, a confidence interval using the t-statistic. And so that interval would look something like this. It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
It would be the sample mean, plus or minus the critical value, times the sample standard deviation divided by the square root of n. And we use a t-statistic like this, and a t-table, and a t-distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution, but we can compute the sample standard deviation. Now in order for this to hold true, there's three conditions, just like what we saw when we thought about z-intervals. The first is, is that our sample is random. Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Well, they tell us that here, that she took a simple random sample of 20. And so we know that we are meeting that constraint, and that's actually choice A. The data is a random sample from the population of interest. So we can circle that in. So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So we can circle that in. So the next condition is the normal condition. Now the normal condition when we're doing a t-interval is a little bit more involved, because we do need to assume that the sampling distribution of the sample means is roughly normal. Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Now there's a couple of ways that we can get there. Either our sample size is greater than or equal to 30. The central limit theorem tells us that then our sampling distribution, regardless of what the distribution is in the population, that the sampling distribution actually would then be approximately normal. She didn't meet that constraint right over here. Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
She didn't meet that constraint right over here. Here, her sample size is only 20. So, so far, this isn't looking good. Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Now that's not the only way to meet the normal condition. Another way to meet the normal condition, if we have a smaller sample size, smaller than 30, is one, if the original distribution of ages is normal. So original, distribution, normal. Or even if it's roughly symmetric around the mean. So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Or even if it's roughly symmetric around the mean. So approximately symmetric. But if you look at this, they tell us that it has a right skew. They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
They say the data were skewed to the right with the sample mean of 38.75. So that tells us that the data set that we're getting in our sample is not symmetric. And the original distribution is unlikely to be normal. Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Think about it. It's not going to be, you're likely to have people who are, you could have faculty members who are 30 years older than this, 68 and 3 quarters, but you're very unlikely to have faculty members who are 30 years younger than this. And that's actually what's causing that skew to the right. So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So this one does not meet the normal condition. We can't feel good that our sampling distribution of the sample means is going to be normal. So I'm not gonna fill that one in. Choice C, individual observations can be considered independent. So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Choice C, individual observations can be considered independent. So there's two ways to meet this constraint. One is, is if we sample with replacement. Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Every faculty member we look at after asking them their age, we say, hey, go back into the pool and we might pick them again until we get our sample of 20. It does not look like she did that. It doesn't look like she sampled with replacement. And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70. 70 is 10% of 700. And so this is definitely less than or equal to 10%. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And so even if you're sampling without replacement, the 10% rule says that, look, as long as this is less than 10% or less than or equal to 10% of the population, then we're good. And the 10% of this population is 70. 70 is 10% of 700. And so this is definitely less than or equal to 10%. And so it can be considered independent. And so we can actually meet that constraint as well. So the main issue where our t interval might not be so good is that our sampling distribution, we can't feel so confident that that is going to be normal. | Conditions for valid t intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Let me write that down. Theoretical, theoretical probability. Well, maybe the simplest example, or one of the simplest examples, is if you're flipping a coin. And let's say in theory you're flipping a completely fair coin and you're flipping it in a way that is completely fair. Well, there you know you have two outcomes. The coin will either result, either heads will be on top or head tails will be on top. And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
And let's say in theory you're flipping a completely fair coin and you're flipping it in a way that is completely fair. Well, there you know you have two outcomes. The coin will either result, either heads will be on top or head tails will be on top. And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities. So you have a 1 1⁄2 probability. And once again, if in theory the coin is definitely fair, it's a fair coin and it's flipped in a very fair way, then this is true. You have a 1 1⁄2 probability. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
And so theoretically you say, well look, if I want to figure out the probability of getting a heads, in theory I have two equally likely possibilities and heads is one of those two equally likely possibilities. So you have a 1 1⁄2 probability. And once again, if in theory the coin is definitely fair, it's a fair coin and it's flipped in a very fair way, then this is true. You have a 1 1⁄2 probability. We could also do that with rolling a die. A fair six-sided die is going to have six possible outcomes. One, two, three, four, five, and six. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You have a 1 1⁄2 probability. We could also do that with rolling a die. A fair six-sided die is going to have six possible outcomes. One, two, three, four, five, and six. And if you said, what is the probability of rolling or getting a result that is greater than or equal to three? Well, we have six equally likely possibilities. You see them there. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
One, two, three, four, five, and six. And if you said, what is the probability of rolling or getting a result that is greater than or equal to three? Well, we have six equally likely possibilities. You see them there. And in theory, if they're all equally likely, four of these possibilities meet our constraint of being greater than or equal to three. So we have four out of the six equal of these possibilities meet our constraints. So we have a 2⁄3, 4⁶ is the same thing as 2⁄3 probability of it happening. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You see them there. And in theory, if they're all equally likely, four of these possibilities meet our constraint of being greater than or equal to three. So we have four out of the six equal of these possibilities meet our constraints. So we have a 2⁄3, 4⁶ is the same thing as 2⁄3 probability of it happening. Now these are for simple things like die or flipping a coin. And if you have fancy computers or spreadsheets, you can even say, hey, I'm gonna flip a coin a bunch of times and do all the combinatorics and all of that. But there are things that are even beyond what a computer can find the exact theoretical probability. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
So we have a 2⁄3, 4⁶ is the same thing as 2⁄3 probability of it happening. Now these are for simple things like die or flipping a coin. And if you have fancy computers or spreadsheets, you can even say, hey, I'm gonna flip a coin a bunch of times and do all the combinatorics and all of that. But there are things that are even beyond what a computer can find the exact theoretical probability. Let's say you're playing a game, say football, American football, and you wanted to figure out the probability of scoring a certain number of points. Well, that isn't very simple because that's going to involve what human beings are doing. Minds are very unpredictable, how people will respond to things. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
But there are things that are even beyond what a computer can find the exact theoretical probability. Let's say you're playing a game, say football, American football, and you wanted to figure out the probability of scoring a certain number of points. Well, that isn't very simple because that's going to involve what human beings are doing. Minds are very unpredictable, how people will respond to things. The weather might get involved. There might be, someone might fall sick. The ball might be wet or just how the ball might interact with some player's jersey. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
Minds are very unpredictable, how people will respond to things. The weather might get involved. There might be, someone might fall sick. The ball might be wet or just how the ball might interact with some player's jersey. Who knows what actually might result in the score being one point this way or seven points this way or seven points that way. And so for situations like that, it makes more sense to think more in terms of experimental probability. And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
The ball might be wet or just how the ball might interact with some player's jersey. Who knows what actually might result in the score being one point this way or seven points this way or seven points that way. And so for situations like that, it makes more sense to think more in terms of experimental probability. And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past. So for example, let's say that you had, this is data from your football team and it's a couple of games or many games into the season, and you've been tabulating the number of points. You have a histogram of the number of games that scored between zero and nine points. You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
And experimental probability, we're really just trying to get an estimate of something happening based on data and experience that we've had in the past. So for example, let's say that you had, this is data from your football team and it's a couple of games or many games into the season, and you've been tabulating the number of points. You have a histogram of the number of games that scored between zero and nine points. You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points. You have five games that went from 20 to 29 points. You had three games that went from 30 to 39 points. And then you had two games that go from 40 to 49 points. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You had two games that scored between zero and nine points, four games that scored between 10 and 19 points, or from 10 to 19 points. You have five games that went from 20 to 29 points. You had three games that went from 30 to 39 points. And then you had two games that go from 40 to 49 points. And let's say for your next game, and let's see, you've already had two, let's see how many games you've had so far. This is two, let me write it down. The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
And then you had two games that go from 40 to 49 points. And let's say for your next game, and let's see, you've already had two, let's see how many games you've had so far. This is two, let me write it down. The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11. 11 plus five is 16. So you've had 16 games so far this season. And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
The game so far is two plus four plus five plus three plus two, so this is what, six, plus, this is six plus five is 11. 11 plus five is 16. So you've had 16 games so far this season. And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30. Your points are greater than or equal to 30 for game 17. So once again, this is very hard to find the exact theoretical probability. You don't know exactly, you can't predict the future. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
And you're curious, for your 17th game, so let me write this, game 17, you want to figure out what is the probability of scoring, scoring, scoring, let's say, score, let's say your score, let's say your points, your points are greater than or equal to 30. Your points are greater than or equal to 30 for game 17. So once again, this is very hard to find the exact theoretical probability. You don't know exactly, you can't predict the future. You don't know who's gonna show up sick, how humans are going to interact with each other. Maybe someone screams something in the stand that just phases the quarterback in exactly the right or the wrong way. You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You don't know exactly, you can't predict the future. You don't know who's gonna show up sick, how humans are going to interact with each other. Maybe someone screams something in the stand that just phases the quarterback in exactly the right or the wrong way. You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game. But you can estimate what'll happen based on what you'll see in your past experience. And it depends on the defense of the team you're facing and all that. So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You don't know, this is an incredibly, incredibly complex system, how what might happen over the course of an entire football game. But you can estimate what'll happen based on what you'll see in your past experience. And it depends on the defense of the team you're facing and all that. So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past. Here, the experimental probability, and I would say the probability, the estimate, because I would, you know, you shouldn't walk away saying this, okay, we absolutely know for sure that if we conducted this next game experiment n times, it's definitely gonna turn out the same. Because this might be the toughest defense that you play all year, this might be the easiest defense that you play all year. But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
So it's not going to be, you know, it's not going to be super exact, but you can estimate based on experiments, based on what you've seen in the past. Here, the experimental probability, and I would say the probability, the estimate, because I would, you know, you shouldn't walk away saying this, okay, we absolutely know for sure that if we conducted this next game experiment n times, it's definitely gonna turn out the same. Because this might be the toughest defense that you play all year, this might be the easiest defense that you play all year. But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points. So five out of the 16 situations, you scored more than that. So an estimate of your probability, and you could view this as maybe your experimental probability of scoring more than 30 points based on past experience, is five, five out of the 16 games you've done this in the past. So you'd say it's 5 16ths. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
But if you look at what's happened in the past, out of the 16 games so far, there have been three games, these three plus these two games, where you scored more than, greater than or equal to 30 points. So five out of the 16 situations, you scored more than that. So an estimate of your probability, and you could view this as maybe your experimental probability of scoring more than 30 points based on past experience, is five, five out of the 16 games you've done this in the past. So you'd say it's 5 16ths. Now I wanna really have you take this with a grain of salt. You should not, you know, then go, say okay, I know for sure there's a 5 16ths probability of us winning this game. Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
So you'd say it's 5 16ths. Now I wanna really have you take this with a grain of salt. You should not, you know, then go, say okay, I know for sure there's a 5 16ths probability of us winning this game. Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera. This is really just an estimate. And I actually, I feel even a little bit of reservation is even calling it a probability. I would just say that this has been true of five out of 16 games in the past. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
Because you only have some data points, every team you play is going to be different, it's gonna be different weather conditions, people are gonna be in different moods, et cetera, et cetera. This is really just an estimate. And I actually, I feel even a little bit of reservation is even calling it a probability. I would just say that this has been true of five out of 16 games in the past. So it's an indicator of what might be, what you might say, okay, based on experience, it's more likely than not that we don't score more than 30 points. But it's really just based on experiential data, what's happened in the season. You know, even the makeup of your football team might have changed. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
I would just say that this has been true of five out of 16 games in the past. So it's an indicator of what might be, what you might say, okay, based on experience, it's more likely than not that we don't score more than 30 points. But it's really just based on experiential data, what's happened in the season. You know, even the makeup of your football team might have changed. You might have gotten a different coach. You might have learned to train better. Who knows, one of your team members might have grown by three inches. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
You know, even the makeup of your football team might have changed. You might have gotten a different coach. You might have learned to train better. Who knows, one of your team members might have grown by three inches. All of these things. So all of this has to be taken with a grain of salt. But this is one way of thinking about it, at least having a sense of what may happen. | Experimental probability Statistics and probability 7th grade Khan Academy.mp3 |
I kind of liked it I didn't like it or they might have rated it on a scale of zero to five which would have been numbers But it's numbers that are measuring people's opinions as opposed to here We have numbers that are measuring their actual scores So there's all different types of data, and I don't want to get into all of that But let's just start thinking about different ways to represent this data So this is one way you could view this as a table where you have the name let me and then you have the score So you have your name column, and then you have your score column, and I could construct it as a table so clearly Looks like a table like that. That's one way one very common way of representing of representing data just like that That's actually how most traditional databases record data in the tables like this, but you could also do it in other ways so you could record it as a as a Oftentimes called a bar graph or sometimes a histogram So you could put score on The vertical axis here, and then you could have your names over here, and let's see the scores. Let's see maybe we'll make this a 50 actually let me just Mark them off, so this is 10 20 30 that's too big 10 20 30 40 50 60 70 80 90 so that's and then a hundred so that's a hundred one two three four five That would be 50 right over there, and then you could go person by person so Amy Amy record got a 90 on the exam so the bar will go up to 90 So that is Amy and then you have bill you got in 95 So it's going to be between 90 and 100 so it's going to be right over there Bill got a 95 And so it would look like this Bill So that is bill, and then you have cam who got a hundred on the exam so Make sure I'm hand drawing it So it's not as it's not as precise as if I were to do it on a computer So this right over there that is cams score Effort got the same score as cam so her score is going to be let me do that in the color in a first color That's a phrase score right now there. She also got a hundred so Efra Efra and then finally Farah got an 80 so 60 70 80 so Farah got Got an 80 so this is Farah score right over here, so this is another way of representing the data and Here we see it in visual form, but it has the same information you can look up someone's name And then figure out their score Amy scored a 90 bill scored a 95 cam scored 100 effort Also scored 100 Farah scored an 80 and there's even other ways you can have some of this information In fact sometimes you might not even know their names and so then it would be less information But it might just a list of scores the professor must say here are the five scores that were That people got on the exam and they were list 90 95 95 100 100 100 and 80 now if it was listed and if this was all of the data you got this would be less information than the data That's in this bar graph or this histogram And or the data that's given in this table right over here because here not only do we know the scores But we know who got what score here. We only know the list of scores But there's even other ways and I this is this isn't and this is not an exhaustive video of all of the different ways you Can represent data you could also represent data by looking at the frequency of scores? So the frequency of scores right over here, so instead of writing the people you could write the scores So let's see you could say this is 80 85 90 95 and 100 and then you could record the frequency that people got these scores So how many times do you have a score of an 80? Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many? | Ways to represent data Data and statistics 6th grade Khan Academy.mp3 |
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