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We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant. | Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3 |
Let's say that we are trying to understand a relationship in a classroom of 200 students between the amount of time studied and the percent correct. And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that. We would figure out the percentage, 60 to 79, that would be 30 out of 86. 30 out of 86, whatever percentage that is, and so on and so forth, in order to get that entire distribution. | Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3 |
So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters based on this data here. So it spits out two typical measures of central tendency. The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range? | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency. And that's why you'll see when people talk about salaries, they'll often talk about median because you could have some skewed salaries, especially on the upside. When you talk about things like home prices, you'll see median often measured more typically than mean because home prices in a neighborhood or in a city, a lot of the houses might be in the $200,000, $300,000 range but maybe there's one ginormous mansion that is $100 million. And if you calculated mean, that would skew and give a false impression of the average or the central tendency of prices in that city. | Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3 |
Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get? | Random number list to run experiment Probability AP Statistics Khan Academy.mp3 |
First-year students at a certain large university are required to live on campus in one of the 24 available residence halls. After their first year, students have the option to live away from campus, but many choose to continue living in the residence halls. Estella oversees 12 of these residence halls. Her department surveyed a large, simple, random sample of first-year students who live in those 12 residence halls about their overall satisfaction with campus living. Estella can safely generalize the results of the survey to which population. So pause this video and see if you can figure it out. Alright, so let's do this together. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
Her department surveyed a large, simple, random sample of first-year students who live in those 12 residence halls about their overall satisfaction with campus living. Estella can safely generalize the results of the survey to which population. So pause this video and see if you can figure it out. Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see. Choice A is only those students who were surveyed. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see. Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls. All students, first-year or not, but only those who live in these 12 residence halls. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls. All students, first-year or not, but only those who live in these 12 residence halls. Well the issue here is that a second-year student or third-year student might just have a different perspective, even if they live in that same building, and we did a large, random sample of first-year students. We didn't do a large, random sample of all people in those 12 residence halls, so we'll rule that out. All first-year students at the entire university, but not students beyond their first year. | Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3 |
For a year, James records whether each day is sunny, cloudy, rainy, or snowy, as well as whether this train arrives on time or is delayed. His results are displayed in the table below. Alright, this is interesting. These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days. And so based on this data, because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed, I would make the statement that these are not independent. So for these days, are the events delayed and snowy independent? | Conditional probability and independence Probability AP Statistics Khan Academy.mp3 |
That should be choices. Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. | Test taking probability and independent events Precalculus Khan Academy.mp3 |
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