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He just really needs 15 pieces of paper. And he then counts the number of windows in the room and chooses the volunteer with that number. So the question here is, is the number of windows in the room, is it random, and is it evenly distributed? So maybe you could make a case that depending on what building it's in and someone's house, it's somewhat random on how many windows that house happens to have, the house that's happening to host the birthday party. But it's not going to be evenly distributed. Most, I don't know. There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window. | Picking fairly Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
So maybe you could make a case that depending on what building it's in and someone's house, it's somewhat random on how many windows that house happens to have, the house that's happening to host the birthday party. But it's not going to be evenly distributed. Most, I don't know. There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window. And definitely more likely that they have 10 windows than, let's say, 30 windows, or even maybe 15 windows even. And so it's not going to be evenly distributed. Every house, I guess, has a somewhat different number of windows. | Picking fairly Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
There's probably some, if you were to sit and plot all of the houses that had a birthday party, you'd probably see that it's more likely that they have 10 windows than one window. And definitely more likely that they have 10 windows than, let's say, 30 windows, or even maybe 15 windows even. And so it's not going to be evenly distributed. Every house, I guess, has a somewhat different number of windows. And the house that is happening to host the party seems to be somewhat random. But it's not going to be evenly distributed here. And I would say it's not a really good random number generator. | Picking fairly Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
Every house, I guess, has a somewhat different number of windows. And the house that is happening to host the party seems to be somewhat random. But it's not going to be evenly distributed here. And I would say it's not a really good random number generator. Because it's not evenly generated. And so I would say method three is not fair. The number of windows is not a really good random number generator. | Picking fairly Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And I would say it's not a really good random number generator. Because it's not evenly generated. And so I would say method three is not fair. The number of windows is not a really good random number generator. A good random number generator, we would want, say, a number 1 through 75, where any of these have an equally likely chance of getting picked. It's somewhat random, the number of windows that building has. But they're not all equally likely. | Picking fairly Probability and combinatorics Probability and Statistics Khan Academy.mp3 |
And let me just show you how to figure out a histogram for some data, and I think you're going to get the point pretty easily. So I have some data here, and I want to represent it with a histogram. And what we're going to see is how frequent are each of these numbers. And in order to figure that out, let me just write the numbers down, and let me just categorize them in their respective buckets. So I have a 1 here. So I have 1 1 right there. I have a 4. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And in order to figure that out, let me just write the numbers down, and let me just categorize them in their respective buckets. So I have a 1 here. So I have 1 1 right there. I have a 4. Let me put, I want to leave space for the 2, the 3, and put a 4 there. I have a 2. That 2, I have a 1. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
I have a 4. Let me put, I want to leave space for the 2, the 3, and put a 4 there. I have a 2. That 2, I have a 1. Let me put that 1 on a stack right above that 1. I have a 0. Let me put the 0 to the left of the 1. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
That 2, I have a 1. Let me put that 1 on a stack right above that 1. I have a 0. Let me put the 0 to the left of the 1. I want to put them in order. I have a 2, another 2. Let me stack that above my first 2. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Let me put the 0 to the left of the 1. I want to put them in order. I have a 2, another 2. Let me stack that above my first 2. I have another 1. Let me stack that above my other 2 1's. I have another 0. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Let me stack that above my first 2. I have another 1. Let me stack that above my other 2 1's. I have another 0. Let me stack it there. I have another 1, another 1 right there. Then I have another 2, another 1. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
I have another 0. Let me stack it there. I have another 1, another 1 right there. Then I have another 2, another 1. I have 2 more 0's. I have 2 more 2's. I have a 3. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Then I have another 2, another 1. I have 2 more 0's. I have 2 more 2's. I have a 3. I have 2 more 1's, another 3. And then I have a 6. So no 5's, and then I have a 6. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
I have a 3. I have 2 more 1's, another 3. And then I have a 6. So no 5's, and then I have a 6. And that space right there was unnecessary. But right here, I've just rewritten these numbers, and I've essentially categorized them. Now what I want to do is calculate how many of each of these numbers we have. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So no 5's, and then I have a 6. And that space right there was unnecessary. But right here, I've just rewritten these numbers, and I've essentially categorized them. Now what I want to do is calculate how many of each of these numbers we have. So let me go down here. So I want to look at the frequency of each of these numbers. So I have 1, 2, 3, 4 0's. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Now what I want to do is calculate how many of each of these numbers we have. So let me go down here. So I want to look at the frequency of each of these numbers. So I have 1, 2, 3, 4 0's. I have 1, 2, 3, 4, 5, 6, 7 1's. I have 1, 2, 3, 4, 5 2's. I have 1, 2 3's. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So I have 1, 2, 3, 4 0's. I have 1, 2, 3, 4, 5, 6, 7 1's. I have 1, 2, 3, 4, 5 2's. I have 1, 2 3's. I have 1, 4, and 1, 6. So we could write it this way. We could write the number, and then we can have the frequency. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
I have 1, 2 3's. I have 1, 4, and 1, 6. So we could write it this way. We could write the number, and then we can have the frequency. And so I have the numbers 0, 1, 2, 3, 4. We could even throw 5 in there. Although 5 has a frequency of 0. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
We could write the number, and then we can have the frequency. And so I have the numbers 0, 1, 2, 3, 4. We could even throw 5 in there. Although 5 has a frequency of 0. And we have a 6. So 0 showed up 4 times in this data set. 1 showed up 7 times in this data set. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Although 5 has a frequency of 0. And we have a 6. So 0 showed up 4 times in this data set. 1 showed up 7 times in this data set. 2 showed up 5 times. 3 showed up 2 times. 4 showed up 1 time. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
1 showed up 7 times in this data set. 2 showed up 5 times. 3 showed up 2 times. 4 showed up 1 time. 5 didn't show up. And 6 showed up 1 time. All I did is I counted this data set. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
4 showed up 1 time. 5 didn't show up. And 6 showed up 1 time. All I did is I counted this data set. I mean, I did this first, but you could say, how many times do I see a 0? Or I see it 1, 2, 3, 4 times. How many times do I see a 1? | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
All I did is I counted this data set. I mean, I did this first, but you could say, how many times do I see a 0? Or I see it 1, 2, 3, 4 times. How many times do I see a 1? 1, 2, 3, 4, 5, 6, 7 times. That's what we mean by frequency. Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
How many times do I see a 1? 1, 2, 3, 4, 5, 6, 7 times. That's what we mean by frequency. Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers. It's going to look a lot like this original thing that I drew. So let me draw some axes here. So the different buckets here are the numbers. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Now a histogram is really just a plot, a kind of a bar graph, plotting the frequency of each of these numbers. It's going to look a lot like this original thing that I drew. So let me draw some axes here. So the different buckets here are the numbers. And that worked out because we're dealing with very clean integers that tend to repeat. If you're dealing with things that are more, that aren't just, you know, the exact number doesn't repeat, oftentimes people will put the numbers into buckets or ranges. But here they fit into nice little buckets. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So the different buckets here are the numbers. And that worked out because we're dealing with very clean integers that tend to repeat. If you're dealing with things that are more, that aren't just, you know, the exact number doesn't repeat, oftentimes people will put the numbers into buckets or ranges. But here they fit into nice little buckets. So you have the numbers 0, 1, 2, 3, 4, 5, and 6. This is the actual numbers. And then on the vertical axis, we're going to plot the frequency. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
But here they fit into nice little buckets. So you have the numbers 0, 1, 2, 3, 4, 5, and 6. This is the actual numbers. And then on the vertical axis, we're going to plot the frequency. So 1, 2, 3, 4, 5, 6, 7. So that's 7, 6, 5, 4, 3, 2, 1. So 0 shows up four times. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And then on the vertical axis, we're going to plot the frequency. So 1, 2, 3, 4, 5, 6, 7. So that's 7, 6, 5, 4, 3, 2, 1. So 0 shows up four times. So we'll draw a little bar graph here. 0 shows up four times. Draw it just like that. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So 0 shows up four times. So we'll draw a little bar graph here. 0 shows up four times. Draw it just like that. 0 shows up four times. That is that information right there. 1 shows up seven times. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Draw it just like that. 0 shows up four times. That is that information right there. 1 shows up seven times. So I'll do a little bar graph. 1 shows up seven times, just like that. 1 shows up. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
1 shows up seven times. So I'll do a little bar graph. 1 shows up seven times, just like that. 1 shows up. I want to make it a little bit straighter than that. 1 shows up seven times. 2, I'll do it in a different color. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
1 shows up. I want to make it a little bit straighter than that. 1 shows up seven times. 2, I'll do it in a different color. 2 shows up five times. So a bar graph, go all the way up to 5. 2 shows up five times. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
2, I'll do it in a different color. 2 shows up five times. So a bar graph, go all the way up to 5. 2 shows up five times. 3 shows up two times. We have 1, 3, 2, 3's. 4 shows up one time here. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
2 shows up five times. 3 shows up two times. We have 1, 3, 2, 3's. 4 shows up one time here. 5 doesn't show up at all. So it doesn't even get any height there. And then finally, 6 shows up one time. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
4 shows up one time here. 5 doesn't show up at all. So it doesn't even get any height there. And then finally, 6 shows up one time. So I'll do 6 showing up one time. What I just plotted here, this is a histogram. This right here is a histogram. | Histograms Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Represent the following data using a box and whiskers plot. Once again, exclude the median when computing the quartiles. And they gave us a bunch of data points. And it says if it helps, you might drag the numbers around, which I will do, because that will be useful. And they say the order isn't checked, and that's because I'm doing this on Khan Academy exercises up here in the top right where you can't see there's actually a check answer. So I encourage you to use the exercises yourself. But let's just use this as an example. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
And it says if it helps, you might drag the numbers around, which I will do, because that will be useful. And they say the order isn't checked, and that's because I'm doing this on Khan Academy exercises up here in the top right where you can't see there's actually a check answer. So I encourage you to use the exercises yourself. But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest. So let's see, there is a 1 here. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest. So let's see, there is a 1 here. And we've got some 2's here. And some 3's. I have one 4. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So let's see, there is a 1 here. And we've got some 2's here. And some 3's. I have one 4. Then 5's. I have a 6. I have a 7. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
I have one 4. Then 5's. I have a 6. I have a 7. I have a couple of 8's. And I have a 10. So there you go. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
I have a 7. I have a couple of 8's. And I have a 10. So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1. So my lowest number is 1. My largest number is 10. So the whiskers help me visualize the range. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
My lowest number is 1. So my lowest number is 1. My largest number is 10. So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median. Of course I'm going to exclude the median. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median. Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides. The 7 has 3 to the left, remember, of the top half, and 3 to the right. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides. The 7 has 3 to the left, remember, of the top half, and 3 to the right. And so the 7 is, I guess you could say, the right side of our box. And we're done. We've constructed our box and whiskers plot, which helps us visualize the entire range, but also you could say the middle, roughly the middle half of our numbers. | Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3 |
All right, now let's do this together. A group of doctors was interested in comparing the effectiveness of placebo pills and real pills in treating migraines. And placebo pills are pills that look just like the regular pill, and from a patient's point of view, they don't know which one they're getting. And the reason why we do this in studies is because there's something called the placebo effect. Oftentimes, just by taking something that you think is good for you, some medicine that you think might help you, it actually does help you. And we can think about why the placebo effect happens, but this is well-documented. And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And the reason why we do this in studies is because there's something called the placebo effect. Oftentimes, just by taking something that you think is good for you, some medicine that you think might help you, it actually does help you. And we can think about why the placebo effect happens, but this is well-documented. And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect? And so that's why they are putting them in these two, that's why they're comparing the real pills to the placebo pills. So they randomly assigned a group of 300 patients suffering from migraines into two groups. So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And so when people are trying to test medicine, they wanna say, well, does this have an effect above and beyond the placebo effect? And so that's why they are putting them in these two, that's why they're comparing the real pills to the placebo pills. So they randomly assigned a group of 300 patients suffering from migraines into two groups. So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group. One group was given a real pill, oh, we already read that. Both groups were told which kind of pill they got. That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
So they have their 300, they have their 300 patients right over here, and they're randomly putting them into the placebo group, placebo group, or the real, or the real pill group. One group was given a real pill, oh, we already read that. Both groups were told which kind of pill they got. That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect. If you're just told you're given a sugar pill, well, then the impact of, well, anyway. This is, that right over there is definitely bad practice. So let's keep going here. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
That is sketchy, because the whole point about a placebo is that you think you got the real thing, or you think you might have gotten the real thing, and if you're told you got a placebo, that tends to undermine the placebo effect. If you're just told you're given a sugar pill, well, then the impact of, well, anyway. This is, that right over there is definitely bad practice. So let's keep going here. Before taking the pills, and a day afterwards, the patients were asked to fill out questionnaires regarding their condition. Then the doctors analyzed the overall changes in questionnaires for each group and compared them. All right, so first of all, what type of study is this? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
So let's keep going here. Before taking the pills, and a day afterwards, the patients were asked to fill out questionnaires regarding their condition. Then the doctors analyzed the overall changes in questionnaires for each group and compared them. All right, so first of all, what type of study is this? Well, we're taking our groups, we are randomly putting them into two different groups. You could call the placebo group, maybe the control group, and then you're putting the real pill as your actual experimental group. And so this is a classic experiment. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
All right, so first of all, what type of study is this? Well, we're taking our groups, we are randomly putting them into two different groups. You could call the placebo group, maybe the control group, and then you're putting the real pill as your actual experimental group. And so this is a classic experiment. This is a classic experiment. You're trying to establish a causal relationship. You want to see whether this real pill actually makes migraines better. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And so this is a classic experiment. This is a classic experiment. You're trying to establish a causal relationship. You want to see whether this real pill actually makes migraines better. Migraines better. So does it actually do it? And does it actually do it better than a placebo? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
You want to see whether this real pill actually makes migraines better. Migraines better. So does it actually do it? And does it actually do it better than a placebo? And you're randomly putting the people in both groups to try to distribute any confounding variables that there might be there. So this is clearly an experiment. Now the other options, you might say, well, is this maybe an observational study? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And does it actually do it better than a placebo? And you're randomly putting the people in both groups to try to distribute any confounding variables that there might be there. So this is clearly an experiment. Now the other options, you might say, well, is this maybe an observational study? And remember, an observational study, observational, that's more of where you look at a population, you look at a group, you ask them a bunch of questions often, or you make a bunch of observations, and you see if there's correlations between two variables. So variable one and variable two, and you're able to make some type of correlational statement. And you're not trying to get at causality. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
Now the other options, you might say, well, is this maybe an observational study? And remember, an observational study, observational, that's more of where you look at a population, you look at a group, you ask them a bunch of questions often, or you make a bunch of observations, and you see if there's correlations between two variables. So variable one and variable two, and you're able to make some type of correlational statement. And you're not trying to get at causality. And in a sample study, a sample study, this is just you trying to estimate a parameter for the entire population. So a sample study might have been of the entire population, what percentage gets migraines? And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And you're not trying to get at causality. And in a sample study, a sample study, this is just you trying to estimate a parameter for the entire population. So a sample study might have been of the entire population, what percentage gets migraines? And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines? And then you say, okay, that percentage of our sample that get migraines, that's a good estimate for the parameter of what percentage of my entire population actually gets migraines. So this was clearly an experiment. Now the next question is, was this experiment conducted well? | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
And you can't talk to the entire population, maybe the entire population is millions of people, so you take a sample of, say, 100 people, and you ask them, do you get migraines? And then you say, okay, that percentage of our sample that get migraines, that's a good estimate for the parameter of what percentage of my entire population actually gets migraines. So this was clearly an experiment. Now the next question is, was this experiment conducted well? Well, even when I read it, I was bothered by both groups were told which kind of pill they got. That completely defeats the purpose of a placebo. The placebo effect is, hey, I think I'm taking something good for me, and it's been documented that when you think you're taking a pill that helps you, it oftentimes does help you. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
Now the next question is, was this experiment conducted well? Well, even when I read it, I was bothered by both groups were told which kind of pill they got. That completely defeats the purpose of a placebo. The placebo effect is, hey, I think I'm taking something good for me, and it's been documented that when you think you're taking a pill that helps you, it oftentimes does help you. And so if someone's coming up with a new medicine, it better perform, or it better be more effective than just that placebo. So I don't like the fact, and this is a very bad study, to tell both groups what kind of pill they got. You actually should tell neither group which type of pill they got. | Worked example identifying experiment Study design AP Statistics Khan Academy.mp3 |
So let's just think about this a little bit. Let's say I have negative 10, 0, 10, 20, and 30. Let's say that's one data set right there. And let's say the other data set is 8, 9, 10, 11, and 12. Now, let's calculate the arithmetic mean for both of these data sets. So let's calculate the mean. And when you go further on in statistics, you're going to understand the difference between a population and a sample. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And let's say the other data set is 8, 9, 10, 11, and 12. Now, let's calculate the arithmetic mean for both of these data sets. So let's calculate the mean. And when you go further on in statistics, you're going to understand the difference between a population and a sample. We're assuming that this is the entire population of our data. This is the entire population of our data. So we're going to be dealing with the population mean. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And when you go further on in statistics, you're going to understand the difference between a population and a sample. We're assuming that this is the entire population of our data. This is the entire population of our data. So we're going to be dealing with the population mean. We're going to be dealing with, as you see, the population measures of dispersion. I know these are all fancy words. In the future, you're not going to have all of the data. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So we're going to be dealing with the population mean. We're going to be dealing with, as you see, the population measures of dispersion. I know these are all fancy words. In the future, you're not going to have all of the data. You're just going to have some samples of it. You're going to try to estimate things for the entire population. So I don't want you to worry too much about that just now. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
In the future, you're not going to have all of the data. You're just going to have some samples of it. You're going to try to estimate things for the entire population. So I don't want you to worry too much about that just now. But if you are going to go further in statistics, I just want to make that clarification. Now, the population mean, or the arithmetic mean, of this data set right here, it is negative 10 plus 0 plus 10 plus 20 plus 30 over, we have 5 data points, over 5. And what is this equal to? | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So I don't want you to worry too much about that just now. But if you are going to go further in statistics, I just want to make that clarification. Now, the population mean, or the arithmetic mean, of this data set right here, it is negative 10 plus 0 plus 10 plus 20 plus 30 over, we have 5 data points, over 5. And what is this equal to? That negative 10 cancels out with that 10. 20 plus 30 is 50. Divided by 5, it's equal to 10. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And what is this equal to? That negative 10 cancels out with that 10. 20 plus 30 is 50. Divided by 5, it's equal to 10. Now, what's the mean of this data set? 8 plus 9 plus 10 plus 11 plus 12. All of that over 5. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Divided by 5, it's equal to 10. Now, what's the mean of this data set? 8 plus 9 plus 10 plus 11 plus 12. All of that over 5. And the way we could think about it, 8 plus 12 is 20. 9 plus 11 is another 20, so that's 40. And then we have a 50 there, add another 10. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
All of that over 5. And the way we could think about it, 8 plus 12 is 20. 9 plus 11 is another 20, so that's 40. And then we have a 50 there, add another 10. So this is, once again, going to be 50 over 5. So this has the exact same population means. Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And then we have a 50 there, add another 10. So this is, once again, going to be 50 over 5. So this has the exact same population means. Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean. When you average all these numbers and divide by 5, or when you take the sum of these numbers, divide by 5, you get 10, sum of these numbers, divide by 5, you get 10 as well, but clearly these sets of numbers are different. If you just looked at this number, you'd say, oh, maybe these sets are very similar to each other. But when you look at these two data sets, one thing might pop out at you. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Or if you don't want to worry about the word population or sample and all of that, both of these data sets have the exact same arithmetic mean. When you average all these numbers and divide by 5, or when you take the sum of these numbers, divide by 5, you get 10, sum of these numbers, divide by 5, you get 10 as well, but clearly these sets of numbers are different. If you just looked at this number, you'd say, oh, maybe these sets are very similar to each other. But when you look at these two data sets, one thing might pop out at you. All of these numbers are very close to 10. I mean, the furthest number here is 2 away from 10. 12 is only 2 away from 10. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
But when you look at these two data sets, one thing might pop out at you. All of these numbers are very close to 10. I mean, the furthest number here is 2 away from 10. 12 is only 2 away from 10. Here, these numbers are further away from 10. Even the closer ones are still 10 away, and then these guys are 20 away from 10. So this right here, this data set right here, is more dispersed. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
12 is only 2 away from 10. Here, these numbers are further away from 10. Even the closer ones are still 10 away, and then these guys are 20 away from 10. So this right here, this data set right here, is more dispersed. These guys are further away from our mean than these guys are from this mean. So let's think about different ways we can measure dispersion, or how far away we are from the center on average. Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So this right here, this data set right here, is more dispersed. These guys are further away from our mean than these guys are from this mean. So let's think about different ways we can measure dispersion, or how far away we are from the center on average. Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number. And literally, take the largest number, which is 30 in our example, and from that you subtract the smallest number. So 30 minus negative 10, which is equal to 40, which tells us that the difference between the largest and the smallest number is 40. So we have a range of 40 for this data set. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Now, one way, this is kind of the most simple way, is the range, and you won't see it used too often, but it's kind of a very simple way of understanding how far is the spread between the largest and the smallest number. And literally, take the largest number, which is 30 in our example, and from that you subtract the smallest number. So 30 minus negative 10, which is equal to 40, which tells us that the difference between the largest and the smallest number is 40. So we have a range of 40 for this data set. Here, the range is the largest number, 12, minus the smallest number, which is 8, which is equal to 4. So here, range is actually a pretty good measure of dispersion. We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So we have a range of 40 for this data set. Here, the range is the largest number, 12, minus the smallest number, which is 8, which is equal to 4. So here, range is actually a pretty good measure of dispersion. We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range. So that tells me this is a more dispersed set. But range is always not going to tell you the whole picture. You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
We say, OK, both of these guys have a mean of 10, but when I look at the range, this guy has a much larger range. So that tells me this is a more dispersed set. But range is always not going to tell you the whole picture. You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie. Now, the one that you'll see used most often is called the variance. Actually, you're going to see the standard deviation in this video. That's probably what's used most often, but it has a very close relationship to the variance. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
You might have two data sets with the exact same range, where still, based on how things are bunched up, it could still have very different distributions of where the numbers lie. Now, the one that you'll see used most often is called the variance. Actually, you're going to see the standard deviation in this video. That's probably what's used most often, but it has a very close relationship to the variance. So the symbol for the variance, and we're going to deal with the population variance. Once again, we're assuming that this is all of the data for our whole population, that we're not just sampling, taking a subset of the data. So the variance, its symbol is literally this sigma, this Greek letter squared. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
That's probably what's used most often, but it has a very close relationship to the variance. So the symbol for the variance, and we're going to deal with the population variance. Once again, we're assuming that this is all of the data for our whole population, that we're not just sampling, taking a subset of the data. So the variance, its symbol is literally this sigma, this Greek letter squared. That is the symbol for variance. And we'll see that the sigma letter actually is the symbol for standard deviation. And that is for a reason. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So the variance, its symbol is literally this sigma, this Greek letter squared. That is the symbol for variance. And we'll see that the sigma letter actually is the symbol for standard deviation. And that is for a reason. But anyway, the definition of variance is you literally take each of these data points, find the difference between those data points and your mean, square them, and then take the average of those squares. I know that sounds very complicated, but when I actually calculate it, you're going to see it's not too bad. So remember, the mean here is 10. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
And that is for a reason. But anyway, the definition of variance is you literally take each of these data points, find the difference between those data points and your mean, square them, and then take the average of those squares. I know that sounds very complicated, but when I actually calculate it, you're going to see it's not too bad. So remember, the mean here is 10. So I take the first data point. I say, let me do it over here. Let me scroll down a little bit. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So remember, the mean here is 10. So I take the first data point. I say, let me do it over here. Let me scroll down a little bit. So I take the first data point, negative 10. From that, I'm going to subtract our mean, and I'm going to square that. So I just found the difference from that first data point to the mean and squared it. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Let me scroll down a little bit. So I take the first data point, negative 10. From that, I'm going to subtract our mean, and I'm going to square that. So I just found the difference from that first data point to the mean and squared it. And that's essentially to make it positive. Plus the second data point, 0 minus 10 minus the mean. This is the mean. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So I just found the difference from that first data point to the mean and squared it. And that's essentially to make it positive. Plus the second data point, 0 minus 10 minus the mean. This is the mean. This is that 10 right there. Squared. Plus 10 minus 10 squared. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
This is the mean. This is that 10 right there. Squared. Plus 10 minus 10 squared. That's the middle 10 right there. Plus 20 minus 10. That's the 20. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Plus 10 minus 10 squared. That's the middle 10 right there. Plus 20 minus 10. That's the 20. Squared plus 30 minus 10. Squared. So this is the squared differences between each number and the mean. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
That's the 20. Squared plus 30 minus 10. Squared. So this is the squared differences between each number and the mean. This is the mean. This is the mean right there. That is the mean. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So this is the squared differences between each number and the mean. This is the mean. This is the mean right there. That is the mean. I'm finding the difference between every data point and the mean, squaring them, summing them up, and then dividing by that number of data points. I'm taking the average of these numbers, of the squared distances. So when you say it kind of verbally, it sounds very complicated. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
That is the mean. I'm finding the difference between every data point and the mean, squaring them, summing them up, and then dividing by that number of data points. I'm taking the average of these numbers, of the squared distances. So when you say it kind of verbally, it sounds very complicated. But you're just taking each number, what's the difference between that, the mean, square it, take the average of those. So I have 1, 2, 3, 4, 5. Divide by 5. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
So when you say it kind of verbally, it sounds very complicated. But you're just taking each number, what's the difference between that, the mean, square it, take the average of those. So I have 1, 2, 3, 4, 5. Divide by 5. So what is this going to be equal to? Negative 10 minus 10 is negative 20. Negative 20 squared is 400. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Divide by 5. So what is this going to be equal to? Negative 10 minus 10 is negative 20. Negative 20 squared is 400. 0 minus 10 is negative 10 squared is 100. So plus 100. 10 minus 10 squared, that's just 0 squared, which is 0. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Negative 20 squared is 400. 0 minus 10 is negative 10 squared is 100. So plus 100. 10 minus 10 squared, that's just 0 squared, which is 0. Plus 20 minus 10 is 10 squared is 100. Plus 30 minus 10, which is 20, squared is 400. All of that over 5. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
10 minus 10 squared, that's just 0 squared, which is 0. Plus 20 minus 10 is 10 squared is 100. Plus 30 minus 10, which is 20, squared is 400. All of that over 5. And what do we have here? 400 plus 100 is 500. Plus another 500 is 1,000. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
All of that over 5. And what do we have here? 400 plus 100 is 500. Plus another 500 is 1,000. It's equal to 1,000 over 5, which is equal to 200. So in this situation, our variance is going to be 200. That's our measure of dispersion there. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
Plus another 500 is 1,000. It's equal to 1,000 over 5, which is equal to 200. So in this situation, our variance is going to be 200. That's our measure of dispersion there. And let's compare it to this data set over here. Let's compare it to the variance of this less dispersed data set. So let me scroll over a little bit. | Range, variance and standard deviation as measures of dispersion Khan Academy.mp3 |
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