Sampling Distributions
AP Statistics Topic 5 7:32 English narration · English + 中文 subtitles burned in
Chapters
Transcript
Two thousand students go to this school, and some of them cycle here.
这所学校有两千名学生，其中有些人骑车上学。
We want to know what share.
我们想知道这个比例是多少。
Nobody can ask all two thousand, so we ask a random hundred.
没有人能问遍两千个人，所以我们随机问一百个人。
Forty six of them cycle.
其中四十六人骑车。
Your friend asks a different random hundred, and gets forty one.
你的朋友另外随机问一百个人，得到四十一。
A third sample gives fifty two.
第三个样本给出五十二。
Same school, same question, three different answers.
同一所学校，同一个问题，三个不同的答案。
Nobody made a mistake.
没有人算错。
The number we work out simply changes from one sample to the next.
我们算出来的这个数，只是会随着样本的不同而改变。
So how much does a number like that move around?
那么这样一个数会上下浮动多少呢？
That is what this unit answers.
这正是本单元要回答的问题。
We will meet the words parameter and statistic, then the Central Limit Theorem, then bias and variability, and finally the two sampling distributions the exam really asks for: one for a proportion, and one for a mean.
我们会认识"参数"和"统计量"这两个词，然后是中心极限定理，接着是偏倚与变异性， 最后是考试真正会考的两个抽样分布：一个关于比例，一个关于均值。
Two words first, and the exam is strict about them.
先说两个词，考试对它们要求很严格。
A parameter describes the whole population, and it is fixed — it never changes.
参数描述整个总体，它是固定的，永远不变。
A statistic is worked out from one sample, and it does change, every time you draw a new sample.
统计量是从一个样本算出来的，每抽一个新样本，它就会变。
The letters help: p for parameter, p for population; s for statistic, s for sample.
有个记忆窍门：参数对应总体，统计量对应样本。
Now collect that statistic from every possible sample of one size.
现在把这个统计量从所有可能的、同样大小的样本里都算一遍。
Together those values form the sampling distribution, centered on the true parameter.
这些值合在一起，就构成了抽样分布，以真实参数为中心。
Here it is happening.
我们来看它是怎么形成的。
The population at the top is badly skewed, nothing like a bell.
上面的总体严重偏斜，一点也不像钟形。
We take a sample, work out its mean, and drop that one number into the picture below.
我们抽一个样本，算出它的均值，把这一个数落到下面的图里。
Then another sample, and another, hundreds of them.
然后再抽一个样本，再抽一个，抽上几百个。
The means pile up into a shape of their own, and that shape is the sampling distribution.
这些均值堆积成了它们自己的形状，而这个形状就是抽样分布。
Look at the shape they made. It is a bell.
看看它们堆出来的形状：一个钟形。
That is the normal model, and it is what turns a sampling distribution into an answer, because under a normal curve an area is a probability.
这就是正态模型， 正是它把一个抽样分布变成一个可以计算的答案， 因为在正态曲线下面，面积就是概率。
About sixty eight percent of the values sit within one standard deviation of the centre, and about ninety five percent within two.
大约百分之六十八的值落在离中心一个标准差以内，大约百分之九十五落在两个标准差以内。
But why should the sample mean be normal when the population is not?
可是，总体不是正态的，为什么样本均值会是正态的呢？
That is the Central Limit Theorem.
这就是中心极限定理。
If the sample is large enough, the sampling distribution of the sample mean is roughly normal, whatever shape the population has.
只要样本足够大，样本均值的抽样分布就近似正态，无论总体是什么形状。
Look at the picture.
看这张图。
The population on the left is strongly right skewed.
左边的总体严重右偏。
On the right, at a sample of two the curve is still skewed, but by thirty it lies almost exactly on the dashed normal curve.
右边，样本量只有二时曲线仍然偏斜， 但到了三十，它几乎完全贴在虚线的正态曲线上。
Look again at those curves.
再看一次那些曲线。
As the sample grows they do not only straighten, they also pull in.
随着样本变大，它们不只是变得更对称，也在向内收紧。
The spread of a sampling distribution has its own name: the standard error.
抽样分布的离散程度有它自己的名字：标准误。
For a mean it is the population standard deviation divided by the square root of the sample size.
对均值来说，它等于总体标准差除以样本量的平方根。
The sample size sits under a square root, and that changes everything.
样本量在平方根下面，这一点改变了一切。
With nine in a sample the standard error is a third of the population value.
样本量是九时，标准误是总体标准差的三分之一。
Take four times as much data and it is only a sixth.
把数据增加到四倍，它才变成六分之一。
Four times the data buys you half the spread.
四倍的数据只换来一半的离散程度。
So halving your error costs four times the sample.
所以要把误差减半，样本量得扩大到四倍。
Two very different faults can spoil an estimate, and the exam keeps them apart.
有两种完全不同的毛病会毁掉一个估计，考试要求把它们分清楚。
The dashed line in each picture is the truth.
每张小图里的虚线代表真值。
Along the top row the curves are centred on it: the estimator is unbiased, right on average.
上面一行的曲线以它为中心：估计量是无偏的，平均而言是对的。
Along the bottom row they sit to one side. That is bias, and extra data never removes it.
下面一行的曲线偏在一边，这就是偏倚，再多的数据也去不掉它。
Left to right is variability, the width.
从左到右变化的是变异性，也就是曲线的宽度。
A bigger sample narrows the curve, but a biased estimator is consistently wrong, and extra data cannot drag it back onto the truth.
更大的样本能让曲线变窄，但有偏的估计量是一贯错误的，再多数据也拉不回真值。
Now the two distributions the exam really asks for.
现在来看考试真正会考的两个分布。
First, a sample proportion.
第一个是样本比例。
Its centre is the population proportion, so the sample proportion is unbiased.
它的中心就是总体比例，所以样本比例是无偏的。
Its spread is the square root of the proportion, times one minus the proportion, divided by the sample size.
它的离散程度等于总体比例乘以一减总体比例、再除以样本量，最后开平方根。
And the shape is normal only when the Large Counts condition holds: both of those counts must be at least ten.
而它的形状只有在大计数条件成立时才是正态的：那两个计数都必须至少是十。
One last check. The sample must be at most ten percent of the population, so the draws stay close to independent.
最后还有一项检查：样本最多占总体的百分之十，这样各次抽取才接近相互独立。
Let's do one properly.
我们完整地做一道。
Forty percent of the voters in a large town support a plan.
某个大城镇里有百分之四十的选民支持一项方案。
You take a random sample of one hundred voters.
你随机抽取一百名选民。
What is the chance that more than half your sample supports it?
样本中支持者超过一半的概率是多少？
Start with the conditions.
先从条件开始。
One hundred times zero point four is forty; one hundred times zero point six is sixty.
一百乘以零点四等于四十；一百乘以零点六等于六十。
Both are at least ten, so a normal model is safe.
两个都不小于十，所以用正态模型是安全的。
Next, the centre and the spread.
接着是中心和离散程度。
The centre is zero point four, and the spread is zero point zero four nine.
中心是零点四，离散程度是零点零四九。
Now standardise.
现在做标准化。
One half sits a little over two standard errors above the centre.
一半比中心高出略多于两个标准误。
So the probability is about zero point zero two.
所以概率大约是零点零二。
In context, only about two samples in a hundred would show a majority.
结合情境来说，一百个样本里大约只有两个会显示出多数支持。
Second, a sample mean.
第二个是样本均值。
Its centre is the population mean, so this one is unbiased too.
它的中心就是总体均值，所以这一个也是无偏的。
Its spread is the population standard deviation divided by the square root of the sample size — the standard error again.
它的离散程度等于总体标准差除以样本量的平方根，也就是刚才说的标准误。
The shape rule comes in two branches.
形状的规则分成两支。
If the population is already normal, the sample mean is normal for any sample size.
如果总体本身就是正态的，样本均值对任何样本量都是正态的。
If it is not, lean on the Central Limit Theorem and check that the sample is at least thirty.
如果不是，就依靠中心极限定理，并检查样本量至少是三十。
Either way, keep the sample under ten percent of the population.
无论哪一种，都要让样本保持在总体的百分之十以内。
One more, with means.
再做一道，这次是关于均值的。
A population has a mean of seventy and a standard deviation of twelve.
某个总体的均值是七十，标准差是十二。
You take a random sample of thirty six.
你随机抽取三十六个个体。
What is the chance the sample mean comes out above seventy three?
样本均值高于七十三的概率是多少？
Shape first.
先看形状。
Thirty six is at least thirty, so the Central Limit Theorem gives a normal model.
三十六不小于三十，所以中心极限定理给出一个正态模型。
Centre and spread next.
接着是中心和离散程度。
The centre is seventy, and the standard error is twelve divided by the square root of thirty six, which is two.
中心是七十，标准误是十二除以三十六的平方根，等于二。
Now standardise.
现在做标准化。
Seventy three is three above seventy, and three divided by two is one point five.
七十三比七十高三，三除以二等于一点五。
So the probability is about zero point zero six seven.
所以概率大约是零点零六七。
In context, about seven samples in a hundred land that high — uncommon, but not shocking.
结合情境来说， 一百个样本里大约有七个会落得这么高，这并不常见，但也不算离奇。
The exam also compares two groups.
考试还会比较两个组。
When you subtract one independent statistic from another, the centres subtract, as you would expect.
当你用一个独立的统计量减去另一个时， 中心是相减的，这和你预料的一样。
But the spreads do not subtract.
但离散程度不能相减。
You add the two variances — the squares of the two standard deviations — and only then take the square root.
你要把两个方差相加，也就是两个标准差的平方相加，然后才开平方根。
Adding is right for a difference just as much as for a sum.
对差和对和都一样，相加的都是方差。
Subtracting standard deviations is always wrong.
把标准差相减，永远是错的。
Three marks students lose every year.
学生每年都会丢的三个分。
First, the spread of a sample mean is the population standard deviation divided by the square root of the sample size, never the standard deviation alone.
第一，样本均值的离散程度是总体标准差除以样本量的平方根， 绝不是总体标准差本身。
A mean varies far less than a single value.
均值的波动远小于单个观测值。
Second, when a question asks you to check a condition, show the numbers.
第二，题目要你检验条件时，要把数字写出来。
The rule with no arithmetic earns nothing.
只写规则、不算数，一分也拿不到。
Third, add variances, never standard deviations.
第三，相加的是方差，绝不是标准差。
And always answer in context, naming the population and the units.
最后，答案一定要结合情境，说出总体，带上单位。