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And you see the distribution right over here. But let's actually answer the question that they're asking us to answer. How many points did the team score? So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right? | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So here, we just have to add up all of these numbers right over here. So we're going to add up, I'll start with the largest. So 20 plus 18 plus 13 plus 11 plus 11, 13, 11, 11, plus nine plus seven, plus seven again, plus four, plus two. Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Did I do that right? We have two 11s, then a nine, then two sevens, then a four, then a two, and then these two characters didn't score anything. So let's add up all of these together. Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Let's add them all up. So zero plus eight is eight, plus three is 11, plus one is 12, plus one is 13, plus nine is 22, plus seven is 27, 34, 38, 30, or 40. So that gets us to 40. Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Let me do that one more time. Eight, 11, 11, 12, 13, 22, 29, 29, and then, 29, 36, 40, and 42. Looks like I actually might have messed, let me do it one more time. This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
This is the hardest part, adding these up. So let me try that one last time. I'm just going to state where my sum is. So zero, eight, add three, 11, 12, 13, 22, 29, 36, 40, 42. So it's a good thing that I double-checked that, I made a mistake the first time. Four plus two is six, seven, eight, nine, 10. So we get to 102 points. | Stem-and-leaf plots Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And we got that to be 30. That's our total sum of squares. And we asked ourselves, how much of that variation is due to variation within each of these groups versus variation between the groups themselves. So for the variation within the groups, we had our sum of squares within. And there we got 6. And then the balance of this 30, there's no units here, the balance of this variation came from variation between the groups. And we actually calculated it and we got 24. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So for the variation within the groups, we had our sum of squares within. And there we got 6. And then the balance of this 30, there's no units here, the balance of this variation came from variation between the groups. And we actually calculated it and we got 24. What I want to do in this video is actually use this type of information, essentially these statistics we've calculated, to do some inferential statistics, to come to some type of conclusion, or maybe not to come to some type of conclusion. And what I want to do is just to put some context around these groups. We've just been dealing with them in the abstract right now. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And we actually calculated it and we got 24. What I want to do in this video is actually use this type of information, essentially these statistics we've calculated, to do some inferential statistics, to come to some type of conclusion, or maybe not to come to some type of conclusion. And what I want to do is just to put some context around these groups. We've just been dealing with them in the abstract right now. But you can imagine that these are kind of the results of some type of experiment. Let's say that I gave 3 different types of pills, or 3 different types of food, to people taking a test. And these are the scores on the test. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
We've just been dealing with them in the abstract right now. But you can imagine that these are kind of the results of some type of experiment. Let's say that I gave 3 different types of pills, or 3 different types of food, to people taking a test. And these are the scores on the test. So this is food 1, this is food 1, this is food 2, this is food 2, and then this right over here is food 3. And I want to figure out if the type of food that people take going into the test, does it really affect their scores? Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And these are the scores on the test. So this is food 1, this is food 1, this is food 2, this is food 2, and then this right over here is food 3. And I want to figure out if the type of food that people take going into the test, does it really affect their scores? Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1. But is that difference purely random, random chance? Or can I be pretty confident that it's due to actual differences in the population means of all of the people who would ever take food 3 versus food 2 versus food 1? What I want to do is say, so my question here is, are the true population means the same? | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
Are the differences in these scores, if you look at these means, it looks like they perform best in group 3, then in group 2, or then in group 1. But is that difference purely random, random chance? Or can I be pretty confident that it's due to actual differences in the population means of all of the people who would ever take food 3 versus food 2 versus food 1? What I want to do is say, so my question here is, are the true population means the same? So if the true population means, this is a sample mean just based on 3 samples, but if I knew the true population means, so my question is, is the mean of the population of people taking food 1 equal to the mean of food 2? Obviously I'll never be able to give that food to every human being that could ever live and then make them all take an exam, but we're trying to get a sense of, there is some true mean there, it's just not really measurable. So my question is, this equal to this equal to the mean 3, the true population mean 3. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
What I want to do is say, so my question here is, are the true population means the same? So if the true population means, this is a sample mean just based on 3 samples, but if I knew the true population means, so my question is, is the mean of the population of people taking food 1 equal to the mean of food 2? Obviously I'll never be able to give that food to every human being that could ever live and then make them all take an exam, but we're trying to get a sense of, there is some true mean there, it's just not really measurable. So my question is, this equal to this equal to the mean 3, the true population mean 3. And my question is, are these equal? It's because if they're not equal, then that means that the food, that the food actually, the different foods that you get, actually do have some type of impact on how people perform on a test. So let's do a little bit of a hypothesis test here. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So my question is, this equal to this equal to the mean 3, the true population mean 3. And my question is, are these equal? It's because if they're not equal, then that means that the food, that the food actually, the different foods that you get, actually do have some type of impact on how people perform on a test. So let's do a little bit of a hypothesis test here. So let's say that my null hypothesis, let's say that my null hypothesis is that the means are the same, or food doesn't make a difference. Food doesn't make a difference. The food doesn't make a difference, but my alternate hypothesis is that it does. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So let's do a little bit of a hypothesis test here. So let's say that my null hypothesis, let's say that my null hypothesis is that the means are the same, or food doesn't make a difference. Food doesn't make a difference. The food doesn't make a difference, but my alternate hypothesis is that it does. And a way of thinking about this a little quantitatively is that if it doesn't make a difference, the true population means of the groups will be the same. So that means the true population mean of the group that took food 1 will be the same as the group that took food 2, which will be the same as the group that took food 3. If our alternate hypothesis is correct, then these means will not all be the same. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
The food doesn't make a difference, but my alternate hypothesis is that it does. And a way of thinking about this a little quantitatively is that if it doesn't make a difference, the true population means of the groups will be the same. So that means the true population mean of the group that took food 1 will be the same as the group that took food 2, which will be the same as the group that took food 3. If our alternate hypothesis is correct, then these means will not all be the same. So how can we test this hypothesis? So what we're going to do, we're going to assume the null hypothesis. This is what we always do in our hypothesis testing. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
If our alternate hypothesis is correct, then these means will not all be the same. So how can we test this hypothesis? So what we're going to do, we're going to assume the null hypothesis. This is what we always do in our hypothesis testing. We're going to assume our null hypothesis, and then essentially figure out what are the chances of getting a certain statistic this extreme. And I haven't even defined what that statistic are. So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
This is what we always do in our hypothesis testing. We're going to assume our null hypothesis, and then essentially figure out what are the chances of getting a certain statistic this extreme. And I haven't even defined what that statistic are. So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic. So our F statistic, which has an F distribution, and we won't go in real deep into the details of the F distribution, but you can already start to think of it as a ratio of 2 chi-squared distributions that may or may not have different degrees of freedom. Our F statistic is going to be the ratio of our sum of squares between the samples. So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So we're going to define, we're going to assume our null hypothesis, and then we're going to come up with a statistic called the F statistic. So our F statistic, which has an F distribution, and we won't go in real deep into the details of the F distribution, but you can already start to think of it as a ratio of 2 chi-squared distributions that may or may not have different degrees of freedom. Our F statistic is going to be the ratio of our sum of squares between the samples. So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within. So that's what I had done up here. The sum of squares within in blue, divided by the sum of squares within, divided by the degrees of freedom of the sum of squares within. And that was M times N minus 1. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So it's going to be our sum of squares between, divided by our degrees of freedom between, and sometimes this is called the mean squares between MSB, either way, it's going to be that, divided by the sum of squares within. So that's what I had done up here. The sum of squares within in blue, divided by the sum of squares within, divided by the degrees of freedom of the sum of squares within. And that was M times N minus 1. Now let's just think about what this is doing right here. If this number, if the numerator is much larger than the denominator, then what that tells us is that the variation in this data is due mostly to the differences between the actual means, and is due less to the actual variation within the means. That's if this numerator is much bigger than this denominator over here. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And that was M times N minus 1. Now let's just think about what this is doing right here. If this number, if the numerator is much larger than the denominator, then what that tells us is that the variation in this data is due mostly to the differences between the actual means, and is due less to the actual variation within the means. That's if this numerator is much bigger than this denominator over here. So that would make us believe, that should make us believe, that there is a difference in the true population mean. So if this number is really big, it should tell us that there's a lower probability that our null hypothesis is correct. If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
That's if this numerator is much bigger than this denominator over here. So that would make us believe, that should make us believe, that there is a difference in the true population mean. So if this number is really big, it should tell us that there's a lower probability that our null hypothesis is correct. If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples. So that means that our variation within each of these samples is a bigger percentage of the total variation versus the variation between the samples. So that would make us believe that, hey, any difference that we actually see in the means is probably just random. And that would make it maybe a little harder to reject our null hypothesis. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
If this number is really small, let's say that this is larger, let's say that our denominator is larger, that means that our variation within each sample makes up more of the total variation than our variation between the samples. So that means that our variation within each of these samples is a bigger percentage of the total variation versus the variation between the samples. So that would make us believe that, hey, any difference that we actually see in the means is probably just random. And that would make it maybe a little harder to reject our null hypothesis. So let's just actually calculate it for this. So in this case, our sum of squares between, we calculated over here was 24, and we had 2 degrees of freedom, and our sum of squares within was 6, and we had how many degrees of freedom? Our degrees of freedom there were also 6, 6 degrees of freedom right over there. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And that would make it maybe a little harder to reject our null hypothesis. So let's just actually calculate it for this. So in this case, our sum of squares between, we calculated over here was 24, and we had 2 degrees of freedom, and our sum of squares within was 6, and we had how many degrees of freedom? Our degrees of freedom there were also 6, 6 degrees of freedom right over there. So this is going to be 24 divided by 2, which is 12 divided by 1. So our F statistic that we've calculated is going to be equal to 12. And this stands for Fisher, who was the biologist and statistician who came up with this. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
Our degrees of freedom there were also 6, 6 degrees of freedom right over there. So this is going to be 24 divided by 2, which is 12 divided by 1. So our F statistic that we've calculated is going to be equal to 12. And this stands for Fisher, who was the biologist and statistician who came up with this. So our F statistic is going to be 12. And what we're going to see is this is a pretty high number. Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And this stands for Fisher, who was the biologist and statistician who came up with this. So our F statistic is going to be 12. And what we're going to see is this is a pretty high number. Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level. And so let's say the significance level that we care about for our hypothesis test is 10%, is 0.10, which means that if we assume, if assuming the null hypothesis, there is less than a 10% chance of getting the result that we got, of getting this F statistic, then we will reject the null hypothesis. So what we want to do is figure out a critical F statistic value that getting that extreme of a value or greater is 10%, and if this is bigger than our critical F statistic value, then we're going to reject the null hypothesis. If it's less, we can't reject the null hypothesis. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
Now one thing I forgot to mention, any hypothesis test, we need to have some type of significance level. And so let's say the significance level that we care about for our hypothesis test is 10%, is 0.10, which means that if we assume, if assuming the null hypothesis, there is less than a 10% chance of getting the result that we got, of getting this F statistic, then we will reject the null hypothesis. So what we want to do is figure out a critical F statistic value that getting that extreme of a value or greater is 10%, and if this is bigger than our critical F statistic value, then we're going to reject the null hypothesis. If it's less, we can't reject the null hypothesis. And so I'm not going to go a lot into the guts of the F statistic, but you can already appreciate that each of these sum of squares has a chi-square distribution. This has a chi-square distribution, and this has a different chi-square distribution. This is a chi-square distribution with 2 degrees of freedom. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
If it's less, we can't reject the null hypothesis. And so I'm not going to go a lot into the guts of the F statistic, but you can already appreciate that each of these sum of squares has a chi-square distribution. This has a chi-square distribution, and this has a different chi-square distribution. This is a chi-square distribution with 2 degrees of freedom. It's a chi-square distribution with, and we haven't normalized it and all of that, but roughly a chi-square distribution with 6 degrees of freedom. So the F statistic is actually, the ratio or the F distribution is the ratio of 2 chi-square distributions. And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
This is a chi-square distribution with 2 degrees of freedom. It's a chi-square distribution with, and we haven't normalized it and all of that, but roughly a chi-square distribution with 6 degrees of freedom. So the F statistic is actually, the ratio or the F distribution is the ratio of 2 chi-square distributions. And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into. But this is what an F distribution looks like. And obviously it's going to look different depending on the degrees of freedom of the numerator and the denominator. There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
And I got this, this is a screenshot from a professor's course at UCLA, hope they don't mind, I actually needed, I had to find an F table for us to look into. But this is what an F distribution looks like. And obviously it's going to look different depending on the degrees of freedom of the numerator and the denominator. There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom. But with that said, let's calculate the critical F statistic. The critical F statistic for alpha is equal to 0.10, and you're actually going to see different F tables for each different alpha, where our numerator degree of freedom is 2 and our denominator degree of freedom is 6. So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
There's kind of 2 degrees of freedom to think about, the numerator's degree of freedom and the denominator's degree of freedom. But with that said, let's calculate the critical F statistic. The critical F statistic for alpha is equal to 0.10, and you're actually going to see different F tables for each different alpha, where our numerator degree of freedom is 2 and our denominator degree of freedom is 6. So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6. So our critical F value is 3.46. So our critical F value is 3.46. So this right over here, this value right here is 3.46. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
So this table that I got, this whole table is for an alpha, our significance level of 10% or 0.10, and our numerator's degree of freedom was 2 and our denominator's degree of freedom is 6. So our critical F value is 3.46. So our critical F value is 3.46. So this right over here, this value right here is 3.46. The value that we got based on our data is much larger than that, way above it. It's going to have a very, very small p-value. The probability of getting something this extreme just by chance, assuming the null hypothesis, is very low. | ANOVA 3 Hypothesis test with F-statistic Probability and Statistics Khan Academy.mp3 |
We already know a little bit about random variables. What we're going to see in this video is that random variables come in two varieties. You have discrete random variables and you have continuous random variables. Continuous. And discrete random variables, these are essentially random variables that can take on distinct or separate values. And we'll give examples of that in a second. That comes straight from the meaning of the word discrete in the English language. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Continuous. And discrete random variables, these are essentially random variables that can take on distinct or separate values. And we'll give examples of that in a second. That comes straight from the meaning of the word discrete in the English language. Distinct or separate values. While continuous, and I guess there's another definition for the word discrete in the English language, which would be polite or not obnoxious or kind of subtle, that is not what we're talking about. We are not talking about random variables that are polite. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
That comes straight from the meaning of the word discrete in the English language. Distinct or separate values. While continuous, and I guess there's another definition for the word discrete in the English language, which would be polite or not obnoxious or kind of subtle, that is not what we're talking about. We are not talking about random variables that are polite. We're talking about ones that can take on distinct values. And continuous random variables, they can take on any value in a range. And that range could even be infinite. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
We are not talking about random variables that are polite. We're talking about ones that can take on distinct values. And continuous random variables, they can take on any value in a range. And that range could even be infinite. So any value in an interval. So with those two definitions out of the way, let's look at some actual random variable definitions. And I want to think together about whether you would classify them as discrete or continuous random variables. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
And that range could even be infinite. So any value in an interval. So with those two definitions out of the way, let's look at some actual random variable definitions. And I want to think together about whether you would classify them as discrete or continuous random variables. So let's say that I have a random variable capital X. And it is equal to, well this is one that we covered in the last video. It's one if my fair coin is heads. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
And I want to think together about whether you would classify them as discrete or continuous random variables. So let's say that I have a random variable capital X. And it is equal to, well this is one that we covered in the last video. It's one if my fair coin is heads. It's zero if my fair coin is tails. So is this a discrete or a continuous random variable? Well this random variable right over here can take on distinct values. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
It's one if my fair coin is heads. It's zero if my fair coin is tails. So is this a discrete or a continuous random variable? Well this random variable right over here can take on distinct values. It can take on either a one or it could take on a zero. Another way to think about it is you can count the number of different values it can take on. This is the first value it can take on. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Well this random variable right over here can take on distinct values. It can take on either a one or it could take on a zero. Another way to think about it is you can count the number of different values it can take on. This is the first value it can take on. This is the second value that it can take on. So this is clearly a discrete random variable. Let's think about another one. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
This is the first value it can take on. This is the second value that it can take on. So this is clearly a discrete random variable. Let's think about another one. Let's define random variable Y as equal to the mass of a random animal selected at the New Orleans Zoo. New Orleans Zoo where I grew up. The Ottoman Zoo. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Let's think about another one. Let's define random variable Y as equal to the mass of a random animal selected at the New Orleans Zoo. New Orleans Zoo where I grew up. The Ottoman Zoo. At the New Orleans Zoo. The Ottoman Zoo. Y is the mass of a random animal selected at the New Orleans Zoo. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
The Ottoman Zoo. At the New Orleans Zoo. The Ottoman Zoo. Y is the mass of a random animal selected at the New Orleans Zoo. Is this a discrete random variable or a continuous random variable? Well the exact mass, and I should probably put that qualifier here. I'll even add it here just to make it really clear. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Y is the mass of a random animal selected at the New Orleans Zoo. Is this a discrete random variable or a continuous random variable? Well the exact mass, and I should probably put that qualifier here. I'll even add it here just to make it really clear. The exact mass of a random animal or a random object in our universe, it can take on any of a whole set of values. I mean who knows exactly the exact number of electrons that are a part of that object right at that moment. Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
I'll even add it here just to make it really clear. The exact mass of a random animal or a random object in our universe, it can take on any of a whole set of values. I mean who knows exactly the exact number of electrons that are a part of that object right at that moment. Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment. So that mass, for example at the zoo, it might take on a value anywhere between, well maybe close to zero. Close to zero. There's no animal that has zero mass. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Who knows the neutrons, the protons, the exact number of molecules in that object or part of that animal exactly that moment. So that mass, for example at the zoo, it might take on a value anywhere between, well maybe close to zero. Close to zero. There's no animal that has zero mass. But it could be close to zero if we're thinking about an ant or we're thinking about a dust mite or maybe if you consider even a bacterium an animal. I believe bacterium is a singular of bacteria. And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
There's no animal that has zero mass. But it could be close to zero if we're thinking about an ant or we're thinking about a dust mite or maybe if you consider even a bacterium an animal. I believe bacterium is a singular of bacteria. And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind. I don't know what it would be in kilograms but it would be fairly large. So maybe you can get up all the way to 3,000 kilograms or probably larger. Say 5,000 kilograms. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
And it could go all the way, maybe the most massive animal in the zoo is the elephant of some kind. I don't know what it would be in kilograms but it would be fairly large. So maybe you can get up all the way to 3,000 kilograms or probably larger. Say 5,000 kilograms. I don't know what the mass of a very heavy elephant or a very massive elephant I should say actually is. Maybe something fun for you to look at. But any animal could have a mass anywhere in between here. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Say 5,000 kilograms. I don't know what the mass of a very heavy elephant or a very massive elephant I should say actually is. Maybe something fun for you to look at. But any animal could have a mass anywhere in between here. It does not take on discrete values. You could have an animal that is exactly maybe 123.75921 kilograms. And even there that actually might not be the exact mass. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
But any animal could have a mass anywhere in between here. It does not take on discrete values. You could have an animal that is exactly maybe 123.75921 kilograms. And even there that actually might not be the exact mass. You might have to get even more precise. 10732. I think you get the picture. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
And even there that actually might not be the exact mass. You might have to get even more precise. 10732. I think you get the picture. Even though this is the way I've defined it now, a finite interval, you can take on any value in between here. There are not discrete values. So this one here is clearly a continuous random variable. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
I think you get the picture. Even though this is the way I've defined it now, a finite interval, you can take on any value in between here. There are not discrete values. So this one here is clearly a continuous random variable. Let's think about another one. Let's say that random variable Y, instead of it being this, let's say it's the year that a random student in the class was born. Random student in a class was born. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
So this one here is clearly a continuous random variable. Let's think about another one. Let's say that random variable Y, instead of it being this, let's say it's the year that a random student in the class was born. Random student in a class was born. Is this a discrete or a continuous random variable? Well that year, you literally can define it as a specific discrete year. It could be 1992 or it could be 1985 or it could be 2000 and 2001. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Random student in a class was born. Is this a discrete or a continuous random variable? Well that year, you literally can define it as a specific discrete year. It could be 1992 or it could be 1985 or it could be 2000 and 2001. There are discrete values that this random variable can actually take on. It won't be able to take on any value between 2000 and 2001. It will either be 2000 or it will be 2001 or 2002. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
It could be 1992 or it could be 1985 or it could be 2000 and 2001. There are discrete values that this random variable can actually take on. It won't be able to take on any value between 2000 and 2001. It will either be 2000 or it will be 2001 or 2002. Once again, you can count the values it can take on. Most of the times that you're dealing with, as in the case right here, a discrete random variable. Let me make it clear. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
It will either be 2000 or it will be 2001 or 2002. Once again, you can count the values it can take on. Most of the times that you're dealing with, as in the case right here, a discrete random variable. Let me make it clear. This one over here is also a discrete random variable. Most of the time that you're dealing with a discrete random variable, you're probably going to be dealing with a finite number of values. But it does not have to be a finite number of values. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Let me make it clear. This one over here is also a discrete random variable. Most of the time that you're dealing with a discrete random variable, you're probably going to be dealing with a finite number of values. But it does not have to be a finite number of values. You can actually have an infinite potential number of values that it can take on as long as the values are countable. As long as you can literally say, okay, this is the first value it can take on, the second, the third, and you might be counting forever. But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
But it does not have to be a finite number of values. You can actually have an infinite potential number of values that it can take on as long as the values are countable. As long as you can literally say, okay, this is the first value it can take on, the second, the third, and you might be counting forever. But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable. Notice, in this scenario with the zoo, you could not list all of the possible masses. You could not even count them. You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
But as long as you can literally list, and it could be an infinite list, but if you could list the values that it could take on, then you're dealing with a discrete random variable. Notice, in this scenario with the zoo, you could not list all of the possible masses. You could not even count them. You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on. You might say, okay, maybe you could take on 0.01 and maybe 0.02. But wait, you just skipped an infinite number of values that it could take on because it could have taken on 0.011, 0.012. Even between those, there's an infinite number of values it could take on. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
You might attempt to, and it's a fun exercise to try at least once, to try to list all of the values this might take on. You might say, okay, maybe you could take on 0.01 and maybe 0.02. But wait, you just skipped an infinite number of values that it could take on because it could have taken on 0.011, 0.012. Even between those, there's an infinite number of values it could take on. There's no way for you to count the number of values that a continuous random variable can take on. There's no way for you to list them. With a discrete random variable, you can count the values. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Even between those, there's an infinite number of values it could take on. There's no way for you to count the number of values that a continuous random variable can take on. There's no way for you to list them. With a discrete random variable, you can count the values. You can list the values. Let's do another example. Let's let random variable Z be the number of ants born tomorrow in the universe. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
With a discrete random variable, you can count the values. You can list the values. Let's do another example. Let's let random variable Z be the number of ants born tomorrow in the universe. Now, you're probably arguing that there aren't ants on other planets, or maybe they're ant-like creatures, but they're not going to be ants as we define them, but how do we know? The number of ants born in the universe, maybe some ants have figured out interstellar travel of some kind. The number of ants born tomorrow in the universe, that's my random variable Z. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Let's let random variable Z be the number of ants born tomorrow in the universe. Now, you're probably arguing that there aren't ants on other planets, or maybe they're ant-like creatures, but they're not going to be ants as we define them, but how do we know? The number of ants born in the universe, maybe some ants have figured out interstellar travel of some kind. The number of ants born tomorrow in the universe, that's my random variable Z. Is this a discrete random variable or a continuous random variable? Once again, we can count the number of values this could take on. This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
The number of ants born tomorrow in the universe, that's my random variable Z. Is this a discrete random variable or a continuous random variable? Once again, we can count the number of values this could take on. This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1. We can actually count the values. Those values are discrete. Once again, this right over here is a discrete random variable. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
This could be 1, it could be 2, it could be 3, it could be 4, it could be 5 quadrillion ants, it could be 5 quadrillion and 1. We can actually count the values. Those values are discrete. Once again, this right over here is a discrete random variable. This is fun, so let's keep doing more of these. Let's say that I have random variable X. We're not using this definition anymore. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Once again, this right over here is a discrete random variable. This is fun, so let's keep doing more of these. Let's say that I have random variable X. We're not using this definition anymore. Now, I'm going to define random variable X to be the winning time. Let me write it this way, the exact winning time for the men's 100 meter dash at the 2016 Olympics. The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
We're not using this definition anymore. Now, I'm going to define random variable X to be the winning time. Let me write it this way, the exact winning time for the men's 100 meter dash at the 2016 Olympics. The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be. Actually, he's aging a little bit. Remember the exact winning time for the men's 100 meter dash at the 2016 Olympics. Not the one that you necessarily see on the clock. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
The exact time that it took for the winner, who's probably going to be Usain Bolt, but it might not be. Actually, he's aging a little bit. Remember the exact winning time for the men's 100 meter dash at the 2016 Olympics. Not the one that you necessarily see on the clock. The exact, the precise time that you would see at the men's 100 meter dash. Is this a discrete or continuous random variable? The way I've defined it, and this one's a little bit tricky, because you might say it's countable. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Not the one that you necessarily see on the clock. The exact, the precise time that you would see at the men's 100 meter dash. Is this a discrete or continuous random variable? The way I've defined it, and this one's a little bit tricky, because you might say it's countable. You might say it could be 9.56 seconds, or 9.57 seconds, or 9.58 seconds. You might be tempted to believe that, because when you watch the 100 meter dash at the Olympics, they measure it to the nearest hundredths. They round to the nearest hundredths. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
The way I've defined it, and this one's a little bit tricky, because you might say it's countable. You might say it could be 9.56 seconds, or 9.57 seconds, or 9.58 seconds. You might be tempted to believe that, because when you watch the 100 meter dash at the Olympics, they measure it to the nearest hundredths. They round to the nearest hundredths. That's how precise their timing is. I'm talking about the exact winning time, the exact number of seconds it takes for that person from the starting gun to cross the finish line. There, it can take on any value. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
They round to the nearest hundredths. That's how precise their timing is. I'm talking about the exact winning time, the exact number of seconds it takes for that person from the starting gun to cross the finish line. There, it can take on any value. It can take on any value between, well, I guess they're limited by the speed of light. It could take on any value you could imagine. It might be anywhere between 5 seconds and maybe 12 seconds. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
There, it can take on any value. It can take on any value between, well, I guess they're limited by the speed of light. It could take on any value you could imagine. It might be anywhere between 5 seconds and maybe 12 seconds. It could be anywhere in between there. It might not be 9.57. That might be what the clock says, but in reality, the exact winning time could be 9.571. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
It might be anywhere between 5 seconds and maybe 12 seconds. It could be anywhere in between there. It might not be 9.57. That might be what the clock says, but in reality, the exact winning time could be 9.571. Or it could be 9.572359. I think you see what I'm saying. The exact precise time could be any value in an interval. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
That might be what the clock says, but in reality, the exact winning time could be 9.571. Or it could be 9.572359. I think you see what I'm saying. The exact precise time could be any value in an interval. This right over here is a continuous random variable. Now, what would be the case, instead of saying the exact winning time, if instead I defined x to be the winning time of the men's 100-meter dash at the 2016 Olympics, rounded to the nearest hundredth? Is this a discrete or a continuous random variable? | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
The exact precise time could be any value in an interval. This right over here is a continuous random variable. Now, what would be the case, instead of saying the exact winning time, if instead I defined x to be the winning time of the men's 100-meter dash at the 2016 Olympics, rounded to the nearest hundredth? Is this a discrete or a continuous random variable? Let me delete this. I've changed the random variable now. Is this going to be a discrete or a continuous random variable? | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Is this a discrete or a continuous random variable? Let me delete this. I've changed the random variable now. Is this going to be a discrete or a continuous random variable? Now we can actually count the actual values that this random variable can take on. It might be 9.56. It could be 9.57. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
Is this going to be a discrete or a continuous random variable? Now we can actually count the actual values that this random variable can take on. It might be 9.56. It could be 9.57. It could be 9.58. We can actually list them. In this case, when we round it to the nearest hundredth, we can actually list the values. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
It could be 9.57. It could be 9.58. We can actually list them. In this case, when we round it to the nearest hundredth, we can actually list the values. We are now dealing with a discrete random variable. Anyway, I'll let you go there. Hopefully this gives you a sense of the distinction between discrete and continuous random variables. | Discrete and continuous random variables Probability and Statistics Khan Academy.mp3 |
And then in the other data set, I have a one, I'm gonna do this on the right side of the screen, a one, a one, a six, and a four. Now the first thing I wanna think about is, well how do I, is there a number that can give me a measure of center of each of these data sets? And one of the ways that we know how to do that is by finding the mean. So let's figure out the mean of each of these data sets. So this first data set, the mean, well we just need to sum up all of the numbers. So it's gonna be two plus two plus four plus four, and then we're gonna divide by the number of numbers that we have. So we have one, two, three, four numbers, so that's that four right over there. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So let's figure out the mean of each of these data sets. So this first data set, the mean, well we just need to sum up all of the numbers. So it's gonna be two plus two plus four plus four, and then we're gonna divide by the number of numbers that we have. So we have one, two, three, four numbers, so that's that four right over there. And this is going to be two plus two is four, plus four is eight, plus four is 12, so it's gonna be 12 over four, which is equal to three. So actually let's just, let's see if we can visualize this a little bit on a number line. So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So we have one, two, three, four numbers, so that's that four right over there. And this is going to be two plus two is four, plus four is eight, plus four is 12, so it's gonna be 12 over four, which is equal to three. So actually let's just, let's see if we can visualize this a little bit on a number line. So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values. So if this is zero, one, two, three, four, and five. And so we have two twos, and so why don't I just do, so for each of these twos, actually I'll just do it in yellow, so I have one two, and then I'll have another two, I'm just gonna do a dot plot here, and then I have two fours. So one four and another four, right over there. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So, and actually I'll do kind of a, I'll do a little bit of a dot plot here, so we can see all of the values. So if this is zero, one, two, three, four, and five. And so we have two twos, and so why don't I just do, so for each of these twos, actually I'll just do it in yellow, so I have one two, and then I'll have another two, I'm just gonna do a dot plot here, and then I have two fours. So one four and another four, right over there. And we calculated that the mean is three. The mean is three. A measure of central tendency, it is three. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So one four and another four, right over there. And we calculated that the mean is three. The mean is three. A measure of central tendency, it is three. So I'll just put three right over here, I'll just mark it with that dotted line. That's where the mean is. All right, well we've visualized that a little bit, and that does look like it's the center. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
A measure of central tendency, it is three. So I'll just put three right over here, I'll just mark it with that dotted line. That's where the mean is. All right, well we've visualized that a little bit, and that does look like it's the center. It's a pretty, it makes sense. So now let's look at this other data set right over here. So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
All right, well we've visualized that a little bit, and that does look like it's the center. It's a pretty, it makes sense. So now let's look at this other data set right over here. So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points. And this is two plus six is eight plus four is 12. 12 divided by four, this is also three. So this also has the same mean. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So the mean, the mean over here is going to be equal to one plus one plus six plus four, all of that over, we still have four data points. And this is two plus six is eight plus four is 12. 12 divided by four, this is also three. So this also has the same mean. We have different numbers, but we have the same mean. But there's something about this data set that feels a little bit different about this. And let's visualize it to see if we can see a difference. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So this also has the same mean. We have different numbers, but we have the same mean. But there's something about this data set that feels a little bit different about this. And let's visualize it to see if we can see a difference. Let's see if we can visualize it. So now I have to go all the way up to six. So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And let's visualize it to see if we can see a difference. Let's see if we can visualize it. So now I have to go all the way up to six. So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. So we have a one, we have a one, we have another one, we have a six, and then we have a four. And we calculated that the mean is three. So we calculated that the mean is three. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So let's say this is zero, one, two, three, four, five, six, and I'll go one more, seven. So we have a one, we have a one, we have another one, we have a six, and then we have a four. And we calculated that the mean is three. So we calculated that the mean is three. So the mean is three. So when we measure it by the mean, the central point or measure of that central point, which we use as the mean, well, it looks the same, but the data sets look different. And how do they look different? | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So we calculated that the mean is three. So the mean is three. So when we measure it by the mean, the central point or measure of that central point, which we use as the mean, well, it looks the same, but the data sets look different. And how do they look different? Well, we've talked about notions of variability or variation and it looks like this data set is more spread out. It looks like the data points are on average further away from the mean than these data points are. And so that's an interesting question that we ask ourselves in statistics. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And how do they look different? Well, we've talked about notions of variability or variation and it looks like this data set is more spread out. It looks like the data points are on average further away from the mean than these data points are. And so that's an interesting question that we ask ourselves in statistics. We just don't want a measure of center like the mean. We might also want a measure of variability. And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean? | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And so that's an interesting question that we ask ourselves in statistics. We just don't want a measure of center like the mean. We might also want a measure of variability. And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean? And that might sound a little complicated, but we're gonna figure out what that means in a second, and not to overuse the word mean. So we wanna figure out on average how far each of these data points from the mean. And what we're about to calculate, this is called mean absolute deviation. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And one of the more straightforward ways to think about variability is, well, on average, how far are each of the data points from the mean? And that might sound a little complicated, but we're gonna figure out what that means in a second, and not to overuse the word mean. So we wanna figure out on average how far each of these data points from the mean. And what we're about to calculate, this is called mean absolute deviation. Absolute deviation. Mean absolute deviation, or if you just use the acronym MAD, MAD, for mean absolute deviation. And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And what we're about to calculate, this is called mean absolute deviation. Absolute deviation. Mean absolute deviation, or if you just use the acronym MAD, MAD, for mean absolute deviation. And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean. Doesn't matter if they're less or more, they're one away from the mean. And then we find the mean of all of the deviations. So what does that mean? | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
And all we're talking about, we're gonna figure out how much do each of these points, their distance, so absolute deviation, how much do they deviate from the mean, but the absolute of it, so each of these points at two, they are one away from the mean. Doesn't matter if they're less or more, they're one away from the mean. And then we find the mean of all of the deviations. So what does that mean? I'm using the word mean, well, using it a little bit too much. So let's figure out the mean absolute deviation of this first data set. So we've been able to figure out what the mean is. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So what does that mean? I'm using the word mean, well, using it a little bit too much. So let's figure out the mean absolute deviation of this first data set. So we've been able to figure out what the mean is. The mean is three. So we take each of the data points, and we figure out what's its absolute deviation from the mean? So we take the first two. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So we've been able to figure out what the mean is. The mean is three. So we take each of the data points, and we figure out what's its absolute deviation from the mean? So we take the first two. So we say two minus the mean. Two minus the mean, and we take the absolute value. So that's its absolute deviation. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So we take the first two. So we say two minus the mean. Two minus the mean, and we take the absolute value. So that's its absolute deviation. Then we have another two. So we find that's absolute deviation from three. Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
So that's its absolute deviation. Then we have another two. So we find that's absolute deviation from three. Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation. How far is it from three? It's fairly easy to calculate in this case. Then we have a four and another four. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
Remember, if we're just taking two minus three and taking the absolute value, that's just saying it's absolute deviation. How far is it from three? It's fairly easy to calculate in this case. Then we have a four and another four. So let me write that. So then we have the mean, or we have the absolute deviation of four from three, from the mean. And then plus, we have another four. | Mean absolute deviation Data and statistics 6th grade Khan Academy.mp3 |
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