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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 |
What is the smallest sample size required to obtain the desired margin of error? So let's just remind ourselves what the confidence interval will look like and what part of it is the margin of error and then we can think about what is her sample size that she would need. So she wants to estimate the true population proportion that favor a tax increase. She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
She doesn't know what this is so she is going to take a sample size of size n and in fact this question is all about what n does she need in order to have the desired margin of error. Well whatever sample she takes there she's going to calculate a sample proportion and then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value and this critical value is based on the confidence level. We'll talk about that in a second. What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
What z-star, what critical value would correspond to a 95% confidence level times and then you would have times the standard error of her statistic and so in this case it would be the square root. It would be the standard error of her sample proportion which is the sample proportion times one minus the sample proportion, all of that over her sample size. Now she wants the margin of error to be no more than 2%. So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So the margin of error is this part right over here. So this part right over there she wants to be no more than 2%. Has to be less than or equal to 2%. That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out? | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
That green color is kind of too shocking. It's unpleasant. All right, less than or equal to 2% right over here. So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So how do we figure that out? Well the first thing, let's just make sure we incorporate the 95% confidence level. So we could look at a z-table. Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
Remember, 95% confidence level, that means if we have a normal distribution here, if we have a normal distribution here, 95% confidence level means the number of standard deviations we need to go above and beyond this in order to capture 95% of the area right over here. So this would be 2.5% that is unshaded at the top right over there, and then this would be 2.5% right over here. And we could look up in a z-table, and if you were to look up in a z-table, you would not look up 95%. You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
You would look up the percentage that would leave 2.5% unshaded at the top. So you would actually look up 97.5%. But it's good to know in general that at a 95% confidence level, you're looking at a critical value of 1.96. And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And that's just something good to know. We could of course look it up on a z-table. So this is 1.96. And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And so this is going to be 1.96 right over here. But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take? Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
Well, remember, we want this stuff right over here that I'm now circling or squaring in this less, less bright color, this blue color. We want this thing to be less than or equal to 2%. This is our margin of error. And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And so what we could do is we could pick a sample proportion. We don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe. So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So the p hat, the maximum p hat. And so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
It's actually going to be p hat is 0.5. And I wanna emphasize, we don't know. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
She didn't even take the random sample and calculate the sample proportion. But we wanna figure out what n to take. And so to be safe, she says, okay, well, what sample proportion would maximize my margin of error? And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And so let me just assume that, and then let me calculate n. So let me set up an inequality here. We want 1.96, that's our critical value, times the square root of, we're just going to assume 0.5 for our sample proportion, although, of course, we don't know what it is yet until we actually take the sample. So that's our sample proportion. That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
That's one minus our sample proportion. All of that over n needs to be less than or equal to 2%. We don't want our margin of error to be any larger than 2%. And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And let me just write this as a decimal, 0.02. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So this could be rewritten as, we could divide both sides by 1.96, 1.96, one over 1.96. And so this would be equal to, on the left-hand side, we'd have the square root of all of this. But that's the same thing as the square root of 0.5 times 0.5, so that'd just be 0.5 over the square root of n. Needs to be less than or equal to, actually, let me write it this way. This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
This is the same thing as two over 100. So two over 100 times one over 1.96 needs to be less than or equal to two over 196. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
This is fancier algebra than we typically do in statistics, or at least in introductory statistics class. All right, so let's see. We could take the reciprocal of both sides. We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
We could say the square root of n over 0.5 and 1.96 over two. And let's see, what's 196 divided by two? That is going to be 98. So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So this would be 98. And so if we take the reciprocal of both sides, then you're gonna swap the inequality, so it's gonna be greater than or equal to. Let's see, I can multiply both sides of this by 0.5. So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared? | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
So 0.5, that's what my, I said 0.5, but my fingers wrote down 0.4. Let's see, 0.5. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And what's 49 squared? Well, you know 50 squared is 2,500, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be d. But if you wanna multiply it out, we can. 49 times 49, nine times nine is 81. Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
Nine times four is 36, plus eight is 44. Four times nine, 36. Four times four is 16 plus three, we have 19. And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
And then you add all of that together, and you indeed do get, so that's 10. And so this is a 14. You do indeed get 2,401. So that's the minimum sample size that Della should take if she genuinely wanted her margin of error to be no more than 2%. Now, it might turn out that her margin of error, when she actually takes the sample of size 2,401, if her sample proportion is less than 0.5, or greater than 0.5, well, then she's going to be in a situation where her margin of error might be less than this, but she just wanted it to be no more than that. Another important thing to appreciate is, it just, the math all worked out very nicely just now, where I got our n to be actually a whole number, but if I got 2,401.5, then you would have to round up to the nearest whole number because you can't have a, your sample size is always going to be a whole number value. So I will leave you there. | Determining sample size based on confidence and margin of error AP Statistics Khan Academy.mp3 |
If the significance level was lowered to 1 hundredth, which of the following would be true? So pause this video and see if you can answer it on your own. Okay, now let's do this together. And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
And let's see, they're talking about how the probability of a type two error or the power, and or the power would change. So before I even look at the choices, let's think about this. We have talked about in previous videos that if we increase our level of significance, that will increase our power. And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
And power is the probability of not making a type two error. So that would decrease the probability of making a type two error. But in this question, we're going the other way. We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
We're decreasing the level of significance, which would lower the probability of making a type one error, but this would decrease the power. It actually would increase, it actually would increase the probability of making a type two, a type two error. And so which of these choices are consistent with that? Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
Well, choice A says that both the type two error and the power would decrease. Well, those don't, these two things don't move together. If one increases, the other decreases. So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
So we rule that one out. Choice B also has these two things moving together, which can't be true. If one increases, the other decreases. Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
Choice C, the probability of a type two error would increase. That's consistent with what we have here. And the power of the test would decrease. Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
Yep, that's consistent with what we have here. So that looks good. And choice D is the opposite of that. The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
The probability of a type two error would decrease. So this is, they're talking about this scenario over here, and that would have happened if they increased our significance level, not decreased it. So we could rule that one out as well. Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
Let's do another example. Asha owns a car wash and is trying to decide whether or not to purchase a vending machine so that customers can buy coffee while they wait. She'll get the machine if she's convinced that more than 30% of her customers would buy coffee. She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
She plans on taking a random sample of N customers and asking them whether or not they would buy coffee from the machine. And she'll then do a significance test using alpha equals 0.05 to see if the sample proportion who say yes is significantly greater than 30%. Which situation below would result in the highest power for her test? So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
So again, pause this video and try to answer it. Well, before I even look at the choices, we could think about what her hypotheses would be. Her null hypothesis is, you could kind of view it as a status quo, no news here. And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
And that would be that the true population proportion of people who want to buy coffee is 30%. And that her alternative hypothesis is that no, the true population proportion, the true population parameter there, is greater than, is greater than 30%. And so if we're talking about what would result in the highest power for her test. So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
So a high power, a high power means the lowest probability of making a type two error. And in other videos, we've talked about it. It looks like she's dealing with the sample size and what is the true proportion of customers that would buy coffee. And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
And the sample size is under her control. The true proportion isn't. Don't wanna make it seem like somehow you can change the true proportion in order to get a higher power. You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
You can change the sample size. But the general principle is, the higher the sample size, the higher the power. So you want a highest possible sample size. And you're going to have a higher power if the true proportion is further from your hypothesis, your null hypothesis proportion. And so we want the highest possible N, and that looks like an N of 200, which is there and there. And we want a true proportion of customers that would actually buy coffee as far away as possible from our null hypothesis, which once again would not be under Asha's control. But you can clearly see that 50% is further from 30 than 32 is, so this one, choice D, is the one that looks good. | Examples thinking about power in significance tests AP Statistics Khan Academy.mp3 |
A group of four friends likes to bowl together, and each friend keeps track of his all-time highest score in a single game. Their high scores are all between 180 and 220, except for Adam, whose high score is 250. Adam then bowls a great game and has a new high score of 290. How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
We have four friends, and they're each going to have, they each keep track of their all-time high score. So we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends. This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
This is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
So before Adam bowls this super awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
Well, remember, the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're going to take the average of whatever this question mark is in 220. That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
That's going to be the median. Now, over here, after Adam has scored a new high score, how would we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers, whatever this friend's high score was, and it hasn't changed. And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
And so we're going to have the same median. It's going to be 220 plus question mark divided by two. It's going to be halfway between question mark and 220. So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
So our median won't change. So median no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
Well, the mean, you take the sum of all these numbers and then you divide by four. And then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
Well, the first three numbers are the same, but in the second list, you have a higher number, 290. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four because their sum is going to be larger. And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
And so the mean is going to go up. The mean will increase. So median no change and mean increase. All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
No, median doesn't change. The mean will increase, yep, and the median will stay the same. Yep, that's exactly what we're talking about. And if you want to make it a little bit more tangible, you could replace question mark with some number. You could replace it, maybe this question mark is 200. And if you try it out with 200 just to make things tangible, you're going to see that that is indeed going to be the case. The median would be halfway between these two numbers, and I just arbitrarily picked 200. | Impact on median and mean when increasing highest value 6th grade Khan Academy.mp3 |
An ecologist surveys the age of about 100 trees in a local forest. He uses a box and whisker plot to map his data, shown below. What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years. And you can even see it. | Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So ideally, you would like to know the mean height of men in the United States. Let me write this down. Mean height, height of men, men in the United States. So how would you do that? When I talk about the mean, when I talk about the mean, I'm talking about the arithmetic mean. If I were to talk about some other types of means, and there are other types of means, like the geometric mean, I would specify. But when people just say mean, they're usually talking about the arithmetic mean. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So how would you do that? When I talk about the mean, when I talk about the mean, I'm talking about the arithmetic mean. If I were to talk about some other types of means, and there are other types of means, like the geometric mean, I would specify. But when people just say mean, they're usually talking about the arithmetic mean. So how would you go about finding the mean height of men in the United States? Well, the obvious one is, is well, you go and ask every, or measure every man in the United States, take their height, add them all together, and then divide by the number of men there are in the United States. But the question you need to ask yourself is whether that is practical. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
But when people just say mean, they're usually talking about the arithmetic mean. So how would you go about finding the mean height of men in the United States? Well, the obvious one is, is well, you go and ask every, or measure every man in the United States, take their height, add them all together, and then divide by the number of men there are in the United States. But the question you need to ask yourself is whether that is practical. Because you have on the order, let's see, there's about 300 million people in the United States, roughly half of them will be men, or at least they'll be male. And so you will have 150 million, roughly 150 million men in the United States. So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
But the question you need to ask yourself is whether that is practical. Because you have on the order, let's see, there's about 300 million people in the United States, roughly half of them will be men, or at least they'll be male. And so you will have 150 million, roughly 150 million men in the United States. So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men. And even if you did try to do that, by the time you're done, many of them might have passed away, the new men will have been born, and so your data will go stale immediately. So it is seemingly impossible, or almost impossible to get the exact height of every man in the United States in a snapshot of time. And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So if you wanted the true mean height of all of the men in the United States, you would have to somehow survey, or not even, you would have to be able to go and measure all 150 million men. And even if you did try to do that, by the time you're done, many of them might have passed away, the new men will have been born, and so your data will go stale immediately. So it is seemingly impossible, or almost impossible to get the exact height of every man in the United States in a snapshot of time. And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample. I don't wanna just go sample 100 people who happen to play basketball, or play basketball for their college. I don't wanna go sample 100 people who are volleyball players. I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
And so instead, what you do is say, well look, okay, I can't get every man, but maybe I can take a sample, I could take a sample of the men in the United States, and I'm going to make an effort, I'm gonna make an effort that it's a random sample. I don't wanna just go sample 100 people who happen to play basketball, or play basketball for their college. I don't wanna go sample 100 people who are volleyball players. I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height. So you take a sample, and from that sample, you can calculate a mean of at least the sample, and you'll hope that that is indicative of, especially if this was a reasonably random sample, you'll hope that that was indicative of the mean of the entire population. And what you're going to see in much of statistics, in much of statistics, it is all about, it is all about using information, using things that we can calculate about a sample, to infer things about a population, because we can't directly measure the entire population. So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
I wanna randomly sample, just maybe the first person who comes out of a mall in a random town, or in several towns, or something like that, something that should not be based in any way, or skewed in any way, by height. So you take a sample, and from that sample, you can calculate a mean of at least the sample, and you'll hope that that is indicative of, especially if this was a reasonably random sample, you'll hope that that was indicative of the mean of the entire population. And what you're going to see in much of statistics, in much of statistics, it is all about, it is all about using information, using things that we can calculate about a sample, to infer things about a population, because we can't directly measure the entire population. So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be. But let's just say you're a little bit lazy, and you just sample five men. And so you get their five heights. Let's say one is 6.2 feet. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So for example, let's say, and I wouldn't, if you're actually trying to do this, I would recommend doing at least 100 data points, or 1,000, and later on we'll talk about how you can think about whether you've measured enough, or how confident you can be. But let's just say you're a little bit lazy, and you just sample five men. And so you get their five heights. Let's say one is 6.2 feet. Let's say one is 5.5 feet. 5.5 feet would be five foot six inches. One would be, let's say one ends up being 5.75, seven five feet. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
Let's say one is 6.2 feet. Let's say one is 5.5 feet. 5.5 feet would be five foot six inches. One would be, let's say one ends up being 5.75, seven five feet. Another one is 6.3 feet. Another is 5.9 feet. Now, if these are the ones that you happen to sample, what would you get for the mean of this sample? | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
One would be, let's say one ends up being 5.75, seven five feet. Another one is 6.3 feet. Another is 5.9 feet. Now, if these are the ones that you happen to sample, what would you get for the mean of this sample? Well, let's get our calculator out, and we get 6.2 plus 5.5 plus 5.75 plus 6.3 plus 5.9. The sum is 29.65, and then we wanna divide by the number of data points we have. So we have five data points. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
Now, if these are the ones that you happen to sample, what would you get for the mean of this sample? Well, let's get our calculator out, and we get 6.2 plus 5.5 plus 5.75 plus 6.3 plus 5.9. The sum is 29.65, and then we wanna divide by the number of data points we have. So we have five data points. So let's divide 29.65 divided by five, and we get 5.93 feet. So here our sample mean, and I'm going to denote it with an X with a bar over it, is, and I already forgot the number, 5.93 feet. 5.93 feet. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So we have five data points. So let's divide 29.65 divided by five, and we get 5.93 feet. So here our sample mean, and I'm going to denote it with an X with a bar over it, is, and I already forgot the number, 5.93 feet. 5.93 feet. This is our sample mean, or if we wanna make it clear, sample arithmetic mean. And when we're taking this calculation based on a sample, and then somehow we're trying to estimate it for the entire population, we call this right over here, we call it a statistic. We call it a statistic. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
5.93 feet. This is our sample mean, or if we wanna make it clear, sample arithmetic mean. And when we're taking this calculation based on a sample, and then somehow we're trying to estimate it for the entire population, we call this right over here, we call it a statistic. We call it a statistic. Now, you might be saying, well, what notation do we use if somehow we are able to measure it for the population? Well, let's say we can't even measure it for the population, but we at least wanna denote what the population mean is. Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
We call it a statistic. Now, you might be saying, well, what notation do we use if somehow we are able to measure it for the population? Well, let's say we can't even measure it for the population, but we at least wanna denote what the population mean is. Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu. So the population mean is usually denoted by the Greek letter mu. And so in a lot of statistics, it's calculating a sample mean in an attempt to estimate this thing that you might not know, the population mean. And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
Well, if you wanna do that, the population mean is usually denoted by the Greek letter mu. So the population mean is usually denoted by the Greek letter mu. And so in a lot of statistics, it's calculating a sample mean in an attempt to estimate this thing that you might not know, the population mean. And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters. This is called parameters. So what you're going to find in much of statistics, it's all about calculating statistics for a sample, finding these sample statistics in order to estimate parameters for an entire population. Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
And these calculations on the entire population, sometimes you might be able to do it, oftentimes you will not be able to do it, these are called parameters. This is called parameters. So what you're going to find in much of statistics, it's all about calculating statistics for a sample, finding these sample statistics in order to estimate parameters for an entire population. Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult. But hopefully, after the next few minutes, you'll appreciate that it's really just doing exactly what we did here, adding up the numbers and dividing by the number of numbers you had. If you had to do the population mean, it's the exact same thing, it's just many, many more numbers in this context, you would have to add up 150 million numbers and divide by 150 million. So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers? | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
Now, the last thing I wanna do is introduce you to some of the notation that you might see in a statistics textbook that looks very mathy and very difficult. But hopefully, after the next few minutes, you'll appreciate that it's really just doing exactly what we did here, adding up the numbers and dividing by the number of numbers you had. If you had to do the population mean, it's the exact same thing, it's just many, many more numbers in this context, you would have to add up 150 million numbers and divide by 150 million. So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers? Well, let's first think about the sample mean, the sample, well, the sample mean, because that's where we actually did the calculation. So a mathematician might call each of these data points, they'll call it, let's say they'll call this first one right over here, they'll call this x sub one, they'll call this one x sub two, they'll call this one x sub three, they'll call this one, when I say sub, I'm literally saying subscript one, subscript two, subscript three. They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
So how do mathematicians talk about an operation like that, adding up a bunch of numbers and then dividing by the number of numbers? Well, let's first think about the sample mean, the sample, well, the sample mean, because that's where we actually did the calculation. So a mathematician might call each of these data points, they'll call it, let's say they'll call this first one right over here, they'll call this x sub one, they'll call this one x sub two, they'll call this one x sub three, they'll call this one, when I say sub, I'm literally saying subscript one, subscript two, subscript three. They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five. So this is literally saying, this is literally saying, this is equal to x sub one plus x sub two plus x sub three, all the way, all the way to the nth one. Once again, in this case, we only had five. Now, are we done? | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
They could call this x subscript four, they could call this x subscript five, and so if you had n of these, you would just keep going, x subscript six, x subscript seven, all the way to x subscript n. And so to take the sum of all of these, they would denote it as the sum, let me write it right over here, so they will say that the sample mean is equal to the sum, the sum of all of my x sub i's, my x sub i's, so the way you can conceptualize it, these i's will change, so the i's are going to, in this case, the i started at one, the i's are going to start at one until the size of our actual sample, so all the way until n. In this case, n was equal to five. So this is literally saying, this is literally saying, this is equal to x sub one plus x sub two plus x sub three, all the way, all the way to the nth one. Once again, in this case, we only had five. Now, are we done? Is this what the sample mean is? Well, no, we aren't done. We don't just add up all of the data points. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
Now, are we done? Is this what the sample mean is? Well, no, we aren't done. We don't just add up all of the data points. We then have to divide by the number of data points there are. So we then have to divide, we then have to divide by the number of data points that there actually are. So this might look like very fancy notation, but it's really just saying, add up your data points and divide by the number of data points you have. | Inferring population mean from sample mean Probability and Statistics Khan Academy.mp3 |
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