More Standard Deviation
成绩单
More on standard deviation. This lesson will cover three more advanced topics concerning standard deviation.
所以，马上我会说，如果你只是担心获得标准偏差的基础，你在最后一个视频中学到的一切，
that will really help you with standard deviation. These are relatively specialized rare questions, and
如果你真的在数学中做得好，并且在测试最艰难的问题中，你只会看到这些。
所以，如果你只是担心标准偏差的基础知识，我就不会担心这个，你可以跳过这个视频。 These are very advanced topics. Topic number one concerns how the standard deviation would 当我们将新成员添加到列表时，更改，使列表更长。出于几个原因，这是一个棘手的问题。
假设我们有一个带有20个成员的集合，并且该组的平均值为50和标准偏差为5。 所以我们在这里被称为摘要统计数据，我们知道整体均值，整体标准偏差，我们没有单独的数据列表。 假设我们将包括两个数字 将成员总数达到22。
Suppose we include 80 and 80 as the 21st and 22nd members of the list. Of course, this would change the mean, and if we wanted to, we could calculate the new mean. The new mean would be slightly more then 50. But all the deviations change, so it is impossible to calculate the new standard deviation.
So first of all, this is a subtle distinction if we have the summary data just the old mean the old standard deviation and 然后我们添加这两个新点。我们绝对无法计算新的标准差。 现在,如果我们的列表,原始列表20 values, and then we added the 21st and 22nd values, if we had all the values, 然后我们可以计算标准偏差。
但即便如此，这是一个计算，测试不会让你这样做。因此，无论如何，你不会担心这个计算。 All we can say, is that if we include new members that are far away from the mean of the set, the standard deviation of the new set will be larger. 所以，这很清楚。80的两个值远离所有其他数字。
They're really big outliers, so adding really big outliers, that's gonna increase the standard deviation. That you need to know. We can say a little bit more if we include a pair of 不改变平均值的数字。如果我们包括围绕平均值同样间隔的数字， 所以一个是k个单位在平均值之上，一个是k个单位低于平均值，然后我们不会改变平均值。
So the deviations from the mean for all the numbers on the list will stay the same and 我们可以绘制一些关于标准偏差的结论。再次，让我们从同一个集合，20个数字，平均值开始， 标准偏差5.假设我们包括40和60作为我们的新成员，所以首先要注意的是40岁 60, one is 10 above the mean, one is 10 below the mean, they're equally spaced around the mean.
这样的意思，他们不会改变平均值。所以新套装的平均值也是50。 好吧，马上那是好的，这意味着没有其他偏差变化。现在，让我们考虑一下这个，40和60 在标准偏差仅为5时，距离平均值有10个距离。所以这些从平均值中进一步， in fact each one is two standard deviations away from the mean, each one is further from the mean than the standard deviation.
So adding bigger deviations than the standard deviation will increase the standard deviation of the list. Now, we're not gonna have to actually calculate the new value of the standard deviation. It's just enough to realize that if we add 40 and 60 to this set, we're going to increase the standard deviation, because we've added numbers that are further from the mean than the standard deviation.
好的。重启。 We start out with our set of 20. Now we're gonna include 45 and 55. 再一次，添加两个对称间隔的数字，五个上面，五个，下面的五个，所以这不是改变平均值， 我们将坚持同样的平均值，平均值为50，现在我们添加了两个数字，该数字完全在任一侧的标准偏差。
One is one standard deviation above the mean and the other is one standard deviation below the mean. 每个人的偏差与标准偏差的尺寸完全相等，因此它们根本不会改变标准偏差。 This new set will have exactly the same standard deviation as the old set so the, the old standard deviation and the new standard deviation both equal five.
因此，这是您唯一需要知道新标准偏差的时间，知道数字值，因为它保持不变， 它与旧的数值相同。好的，重置回到20的20。 现在假设我们包括两个更接近均值的数字，比如47和53.好吧，现在我们添加两个数字， 再次，对称间隔，一个是三个下方的平均值，三个上方的平均值，所以这不会改变平均值，平均值将保持不变。
因此，请注意，这些分离的平均值小于标准偏差。 They are closer to the mean then the value of the standard deviation. And so that means they would decrease the standard deviation. 现在我们不需要能够计算新的标准偏差，但我们需要识别出标准偏差会降低 我们添加了这两个数字。
现在，我们可能会变得好奇，我们可以包括哪一对数字，这将大部分降低标准偏差？ 嗯，我们必须包括与平均值的最小距离对。 The smallest possible distance of the mean, of course, is 0. You can't have a distance smaller than 0.
如果我们包括的集合的两个新成员是50和50，则这2个成员有距离，每个成员距离平均值为0，所以 它们最多降低了标准偏差。在所有可能的新成员对中，我们可以在一组中包含， including two new members equal to the mean of the set is the pair that would decrease the standard deviation the most.
The test likes to ask about that idea. So here we go, you can see, we've added those two points right at the mean. 他们会有0的偏差，因此如果您想到偏差列表，请在该列表上放入两个，这会降低标准偏差。 再一次，我们不需要，需要能够计算这种新的标准偏差，但我们只需要认识到它已经减少了最多。
第二个主题涉及标准偏差作为测量单位。 我的意思是什么？在非常大的套装中，例如，国家/地区或 每个人都坐着的人，某种巨大的一套，我们可能想指定个人对人口的位置。
如果我们被告知某一套的平均值是50，有人的分数是60，那究竟是什么意思？ Yes, that score of 60 is above the mean, is this just something kind of mildly above the mean, or is this a really really impressive score far above the mean? Well, think about it this way, if the mean is 50 and the standard deviation is 20, then 60 is above the mean, it's half a standard deviation above the mean.
但是，据推测，许多数字将高于那个。一个设置有数字，它不会是不寻常的 与平均值甚至进一步的标准偏差远远。所以我们肯定希望有一些标准偏差的数字 above the mean, maybe even one and a half standard deviations above the mean. So we'd expect scores of 70 and 80, so if 60 yeah, it's above the mean but it's not among the highest scores.
And so it's not, it's kind of on the good side of average rather then being a super impressive score in this particular set. Notice what happens if we change the standard deviation though. By contrast, if the mean is 50, 标准偏差为2，然后60非常，远远高于平均值。因为它是标准偏差，告诉我们某些人有多有意义 separation from the mean is, mathematicians generally use the standard deviation of any set as a unit of measurement within that set.
如果这个个体是10个单位，则平均值和标准偏差为2，那么该个体是5个标准偏差在平均值之上。 现在很难强调，在平均值之上的5个标准偏差是多么强调。 In terms of musical abilities, that would be the best musician in the world. In terms of athletic abilities, that would be the best athlete in the world.
这将只是一个人离开图表，就像每百年的人才一样。 That's what, that's how impressive this score would be if it were five standard deviations above the mean. Here's a practice problem, pause the video and then we'll talk about this.
在一定的测试中，分数具有300的平均值和25的标准偏差。 If John scored three standard deviations above the mean, what was John's score? Well, the mean is 300, and he scored 3 25s above 300. Well, that would be 300 plus 75, which is 375. So, it turns out, a question like this is, actually involves a very, very simple calculation.
You're just doing, you just have to be not intimidated by the question itself. The final topic, very advanced, concerns the actual calculation of standard deviation. The test will not, repeat, not ask you to calculate the standard deviation of the list from scratch, but on a very advanced question, it could ask 关于一些细节，一些概念与此计算的细节有关。
Here are the steps in the calculation. 所以，首先，我们从数字列表开始，我们必须找到均值。正如我们所说，我们减去了平均值和 这会创建第二个列表，偏差列表。然后我们会采取那个偏差清单，有些是积极的， 有些是否定的，我们将被广场列出。
所以这将是一个平方偏差的列表。第三个列表，该列表将所有正数为基础，因为我们已经平衡了它们。 现在我们将花费这个第三个清单。平均平均偏差，该数字实际上称为方差。 Once we have the variance, we're gonna take the square root and this is the standard deviation.
好的，所以现在我们会做一个样本计算。我们将从一个非常简单的列表开始，只是1到9的整数。 因此，这是一个很好的对称均匀间隔列表，当然是中间的数字，5，这都是这个列表的均值和中位数。 So we have the mean, so it's easy to figure out the deviations. So you get the list of deviations, I'm just gonna take this list number one and 从列表中的每个数字中减去五个。
So, 5 of course has a deviation of zero. The numbers less than five have negative deviations. The numbers bigger than five have positive deviations. That's the second list. Now to get the third list, we just square everything. So those are the squares, notice 0 squared is 0, 其他一切都是积极的，所以现在我们拥有平方偏差的名单。
现在，我们平均第三个列表。第三个列表的平均值是一个称为方差的东西。 因此，差异，该列表的平均值发生在于超过3.这是方差，这是第一个列表的方差， the first list has a variance of 20 over 3. To find the standard deviation, we take the square route of the variance.
So we could write it in radical form, we could simplify that radical if we wanted to, typically the standard deviation is just written as a decimal. 所以，我们将以十进制形式写入。所以请注意，我们在这里发现了2.582，这是标准偏差 列表从1到9，而且还因为我们在最后一个视频中学到了什么，这应该是任何九个连续整数的标准偏差。
So, any non-consecutive integers at all would have a standard deviation of not, of 2.582. That is the entire calculation for standard. Deviation in, in all its gory detail. The test will not ask you to repeat that entire procedure. Conceivably on the very hardest quant problems, 它可以呈现该程序的某些部分，它可能会询问一些细节。
细节。一件事要注意， 顺便提一下，因为我们正在平衡，更大的数字，具有更大偏差的数字对标准偏差产生更大的贡献。 贡献更大。平方放大效果的效果， 输入具有更大偏差的数字。
That's an important thing to notice. You may wonder why the standard deviation is defined in this particular way. This is related to its principal use. We could find the standard deviation of any list or 设置，但在某些方面，标准偏差旨在伴随着正常分布，我们将在下一个视频中讨论。
Here's a practice problem. Pause the video and then we'll talk about this. 好的。所以让我们谈谈这个问题。一个营地30个女孩的高度平均为130厘米和 a standard deviation of four centimeters. Suppose four more girls had joined the camp, so there'll be 34 all together.
这四个额外女孩的哪一套高度将大部分增加营地所有女孩的标准偏差？ Well certainly if we wanted to least increase the standard, if we wanted to decrease the standard deviation, we'd be adding girls. Who all had values of 130. They'd all be equal to the mean.
Well, we don't have any like that but notice with C, two of them are equal to the mean and the other two are very close to the mean, so 它们中的所有四个都更接近平均值然后是标准偏差。所以这会降低平均值。 So that's certainly not gonna increase. If we add the ones in B, notice all four of those.
我们的偏差是负三个，负面的一个，一个和三个。所以那些将偏差再次较少 比偏差，所以这是ALC，也会降低标准偏差。所以这不是正确的。 Very interesting if we look at A, there we have deviations of negative four, negative four, four and four.
所有这四个数字是一个标准deviation away from the mean. So, in fact, adding those four girls will keep the standard deviation at exactly four, because we're just adding more standard deviations, more deviations of the same value. So, that's gonna stay the same, and so that's not an increase. So the only one that's left that's D and if we look at D, what we're adding there are four outliers.
Each one of them is two and a half standard deviations from the mean and that's far away from the mean, and so 这会改变整个营地的平均值，它会扰乱所有偏差。并且网络结果是你会有更大的偏差和 更大的标准偏差。所以D是答案。
In summary, we discussed the effect of the stan, 在集合中包括一对新数字的标准偏差。并注意我们可以最明智地讨论这一点 添加两个不改变均值的数字。我们使用标准偏差作为一个单位讨论 表明个人在大人物中的位置，谈论在平均值上方或低于或低于平均值的标准偏差。
我们讨论了技术目的，详细说明了标准偏差的确切计算。
阅读完整成绩单
所以，如果你只是担心标准偏差的基础知识，我就不会担心这个，你可以跳过这个视频。 These are very advanced topics. Topic number one concerns how the standard deviation would 当我们将新成员添加到列表时，更改，使列表更长。出于几个原因，这是一个棘手的问题。
假设我们有一个带有20个成员的集合，并且该组的平均值为50和标准偏差为5。 所以我们在这里被称为摘要统计数据，我们知道整体均值，整体标准偏差，我们没有单独的数据列表。 假设我们将包括两个数字 将成员总数达到22。
Suppose we include 80 and 80 as the 21st and 22nd members of the list. Of course, this would change the mean, and if we wanted to, we could calculate the new mean. The new mean would be slightly more then 50. But all the deviations change, so it is impossible to calculate the new standard deviation.
So first of all, this is a subtle distinction if we have the summary data just the old mean the old standard deviation and 然后我们添加这两个新点。我们绝对无法计算新的标准差。 现在,如果我们的列表,原始列表20 values, and then we added the 21st and 22nd values, if we had all the values, 然后我们可以计算标准偏差。
但即便如此，这是一个计算，测试不会让你这样做。因此，无论如何，你不会担心这个计算。 All we can say, is that if we include new members that are far away from the mean of the set, the standard deviation of the new set will be larger. 所以，这很清楚。80的两个值远离所有其他数字。
They're really big outliers, so adding really big outliers, that's gonna increase the standard deviation. That you need to know. We can say a little bit more if we include a pair of 不改变平均值的数字。如果我们包括围绕平均值同样间隔的数字， 所以一个是k个单位在平均值之上，一个是k个单位低于平均值，然后我们不会改变平均值。
So the deviations from the mean for all the numbers on the list will stay the same and 我们可以绘制一些关于标准偏差的结论。再次，让我们从同一个集合，20个数字，平均值开始， 标准偏差5.假设我们包括40和60作为我们的新成员，所以首先要注意的是40岁 60, one is 10 above the mean, one is 10 below the mean, they're equally spaced around the mean.
这样的意思，他们不会改变平均值。所以新套装的平均值也是50。 好吧，马上那是好的，这意味着没有其他偏差变化。现在，让我们考虑一下这个，40和60 在标准偏差仅为5时，距离平均值有10个距离。所以这些从平均值中进一步， in fact each one is two standard deviations away from the mean, each one is further from the mean than the standard deviation.
So adding bigger deviations than the standard deviation will increase the standard deviation of the list. Now, we're not gonna have to actually calculate the new value of the standard deviation. It's just enough to realize that if we add 40 and 60 to this set, we're going to increase the standard deviation, because we've added numbers that are further from the mean than the standard deviation.
好的。重启。 We start out with our set of 20. Now we're gonna include 45 and 55. 再一次，添加两个对称间隔的数字，五个上面，五个，下面的五个，所以这不是改变平均值， 我们将坚持同样的平均值，平均值为50，现在我们添加了两个数字，该数字完全在任一侧的标准偏差。
One is one standard deviation above the mean and the other is one standard deviation below the mean. 每个人的偏差与标准偏差的尺寸完全相等，因此它们根本不会改变标准偏差。 This new set will have exactly the same standard deviation as the old set so the, the old standard deviation and the new standard deviation both equal five.
因此，这是您唯一需要知道新标准偏差的时间，知道数字值，因为它保持不变， 它与旧的数值相同。好的，重置回到20的20。 现在假设我们包括两个更接近均值的数字，比如47和53.好吧，现在我们添加两个数字， 再次，对称间隔，一个是三个下方的平均值，三个上方的平均值，所以这不会改变平均值，平均值将保持不变。
因此，请注意，这些分离的平均值小于标准偏差。 They are closer to the mean then the value of the standard deviation. And so that means they would decrease the standard deviation. 现在我们不需要能够计算新的标准偏差，但我们需要识别出标准偏差会降低 我们添加了这两个数字。
现在，我们可能会变得好奇，我们可以包括哪一对数字，这将大部分降低标准偏差？ 嗯，我们必须包括与平均值的最小距离对。 The smallest possible distance of the mean, of course, is 0. You can't have a distance smaller than 0.
如果我们包括的集合的两个新成员是50和50，则这2个成员有距离，每个成员距离平均值为0，所以 它们最多降低了标准偏差。在所有可能的新成员对中，我们可以在一组中包含， including two new members equal to the mean of the set is the pair that would decrease the standard deviation the most.
The test likes to ask about that idea. So here we go, you can see, we've added those two points right at the mean. 他们会有0的偏差，因此如果您想到偏差列表，请在该列表上放入两个，这会降低标准偏差。 再一次，我们不需要，需要能够计算这种新的标准偏差，但我们只需要认识到它已经减少了最多。
第二个主题涉及标准偏差作为测量单位。 我的意思是什么？在非常大的套装中，例如，国家/地区或 每个人都坐着的人，某种巨大的一套，我们可能想指定个人对人口的位置。
如果我们被告知某一套的平均值是50，有人的分数是60，那究竟是什么意思？ Yes, that score of 60 is above the mean, is this just something kind of mildly above the mean, or is this a really really impressive score far above the mean? Well, think about it this way, if the mean is 50 and the standard deviation is 20, then 60 is above the mean, it's half a standard deviation above the mean.
但是，据推测，许多数字将高于那个。一个设置有数字，它不会是不寻常的 与平均值甚至进一步的标准偏差远远。所以我们肯定希望有一些标准偏差的数字 above the mean, maybe even one and a half standard deviations above the mean. So we'd expect scores of 70 and 80, so if 60 yeah, it's above the mean but it's not among the highest scores.
And so it's not, it's kind of on the good side of average rather then being a super impressive score in this particular set. Notice what happens if we change the standard deviation though. By contrast, if the mean is 50, 标准偏差为2，然后60非常，远远高于平均值。因为它是标准偏差，告诉我们某些人有多有意义 separation from the mean is, mathematicians generally use the standard deviation of any set as a unit of measurement within that set.
如果这个个体是10个单位，则平均值和标准偏差为2，那么该个体是5个标准偏差在平均值之上。 现在很难强调，在平均值之上的5个标准偏差是多么强调。 In terms of musical abilities, that would be the best musician in the world. In terms of athletic abilities, that would be the best athlete in the world.
这将只是一个人离开图表，就像每百年的人才一样。 That's what, that's how impressive this score would be if it were five standard deviations above the mean. Here's a practice problem, pause the video and then we'll talk about this.
在一定的测试中，分数具有300的平均值和25的标准偏差。 If John scored three standard deviations above the mean, what was John's score? Well, the mean is 300, and he scored 3 25s above 300. Well, that would be 300 plus 75, which is 375. So, it turns out, a question like this is, actually involves a very, very simple calculation.
You're just doing, you just have to be not intimidated by the question itself. The final topic, very advanced, concerns the actual calculation of standard deviation. The test will not, repeat, not ask you to calculate the standard deviation of the list from scratch, but on a very advanced question, it could ask 关于一些细节，一些概念与此计算的细节有关。
Here are the steps in the calculation. 所以，首先，我们从数字列表开始，我们必须找到均值。正如我们所说，我们减去了平均值和 这会创建第二个列表，偏差列表。然后我们会采取那个偏差清单，有些是积极的， 有些是否定的，我们将被广场列出。
所以这将是一个平方偏差的列表。第三个列表，该列表将所有正数为基础，因为我们已经平衡了它们。 现在我们将花费这个第三个清单。平均平均偏差，该数字实际上称为方差。 Once we have the variance, we're gonna take the square root and this is the standard deviation.
好的，所以现在我们会做一个样本计算。我们将从一个非常简单的列表开始，只是1到9的整数。 因此，这是一个很好的对称均匀间隔列表，当然是中间的数字，5，这都是这个列表的均值和中位数。 So we have the mean, so it's easy to figure out the deviations. So you get the list of deviations, I'm just gonna take this list number one and 从列表中的每个数字中减去五个。
So, 5 of course has a deviation of zero. The numbers less than five have negative deviations. The numbers bigger than five have positive deviations. That's the second list. Now to get the third list, we just square everything. So those are the squares, notice 0 squared is 0, 其他一切都是积极的，所以现在我们拥有平方偏差的名单。
现在，我们平均第三个列表。第三个列表的平均值是一个称为方差的东西。 因此，差异，该列表的平均值发生在于超过3.这是方差，这是第一个列表的方差， the first list has a variance of 20 over 3. To find the standard deviation, we take the square route of the variance.
So we could write it in radical form, we could simplify that radical if we wanted to, typically the standard deviation is just written as a decimal. 所以，我们将以十进制形式写入。所以请注意，我们在这里发现了2.582，这是标准偏差 列表从1到9，而且还因为我们在最后一个视频中学到了什么，这应该是任何九个连续整数的标准偏差。
So, any non-consecutive integers at all would have a standard deviation of not, of 2.582. That is the entire calculation for standard. Deviation in, in all its gory detail. The test will not ask you to repeat that entire procedure. Conceivably on the very hardest quant problems, 它可以呈现该程序的某些部分，它可能会询问一些细节。
细节。一件事要注意， 顺便提一下，因为我们正在平衡，更大的数字，具有更大偏差的数字对标准偏差产生更大的贡献。 贡献更大。平方放大效果的效果， 输入具有更大偏差的数字。
That's an important thing to notice. You may wonder why the standard deviation is defined in this particular way. This is related to its principal use. We could find the standard deviation of any list or 设置，但在某些方面，标准偏差旨在伴随着正常分布，我们将在下一个视频中讨论。
Here's a practice problem. Pause the video and then we'll talk about this. 好的。所以让我们谈谈这个问题。一个营地30个女孩的高度平均为130厘米和 a standard deviation of four centimeters. Suppose four more girls had joined the camp, so there'll be 34 all together.
这四个额外女孩的哪一套高度将大部分增加营地所有女孩的标准偏差？ Well certainly if we wanted to least increase the standard, if we wanted to decrease the standard deviation, we'd be adding girls. Who all had values of 130. They'd all be equal to the mean.
Well, we don't have any like that but notice with C, two of them are equal to the mean and the other two are very close to the mean, so 它们中的所有四个都更接近平均值然后是标准偏差。所以这会降低平均值。 So that's certainly not gonna increase. If we add the ones in B, notice all four of those.
我们的偏差是负三个，负面的一个，一个和三个。所以那些将偏差再次较少 比偏差，所以这是ALC，也会降低标准偏差。所以这不是正确的。 Very interesting if we look at A, there we have deviations of negative four, negative four, four and four.
所有这四个数字是一个标准deviation away from the mean. So, in fact, adding those four girls will keep the standard deviation at exactly four, because we're just adding more standard deviations, more deviations of the same value. So, that's gonna stay the same, and so that's not an increase. So the only one that's left that's D and if we look at D, what we're adding there are four outliers.
Each one of them is two and a half standard deviations from the mean and that's far away from the mean, and so 这会改变整个营地的平均值，它会扰乱所有偏差。并且网络结果是你会有更大的偏差和 更大的标准偏差。所以D是答案。
In summary, we discussed the effect of the stan, 在集合中包括一对新数字的标准偏差。并注意我们可以最明智地讨论这一点 添加两个不改变均值的数字。我们使用标准偏差作为一个单位讨论 表明个人在大人物中的位置，谈论在平均值上方或低于或低于平均值的标准偏差。
我们讨论了技术目的，详细说明了标准偏差的确切计算。
