From 46314731456916100f36c817739a277c918132e6 Mon Sep 17 00:00:00 2001 From: wizardforcel <562826179@qq.com> Date: Sat, 27 Jan 2018 22:46:33 +0800 Subject: [PATCH] ch12pic --- 12.md | 52 ++++++++++++++++++++++++++++++++++++++++++++++++++ img/12-1.png | Bin 0 -> 10386 bytes img/12-10.png | Bin 0 -> 17413 bytes img/12-11.png | Bin 0 -> 17606 bytes img/12-12.png | Bin 0 -> 17371 bytes img/12-13.png | Bin 0 -> 17242 bytes img/12-14.png | Bin 0 -> 17443 bytes img/12-15.png | Bin 0 -> 11313 bytes img/12-16.png | Bin 0 -> 11951 bytes img/12-17.png | Bin 0 -> 11243 bytes img/12-18.png | Bin 0 -> 11019 bytes img/12-19.png | Bin 0 -> 15594 bytes img/12-2.png | Bin 0 -> 10630 bytes img/12-20.png | Bin 0 -> 27044 bytes img/12-21.png | Bin 0 -> 11667 bytes img/12-22.png | Bin 0 -> 13942 bytes img/12-25.png | Bin 0 -> 29906 bytes img/12-26.png | Bin 0 -> 22432 bytes img/12-27.png | Bin 0 -> 14547 bytes img/12-3.png | Bin 0 -> 10660 bytes img/12-4.png | Bin 0 -> 22255 bytes img/12-5.png | Bin 0 -> 20554 bytes img/12-6.png | Bin 0 -> 10254 bytes img/12-7.png | Bin 0 -> 10918 bytes img/12-8.png | Bin 0 -> 13687 bytes img/12-9.png | Bin 0 -> 13117 bytes 26 files changed, 52 insertions(+) create mode 100644 img/12-1.png create mode 100644 img/12-10.png create mode 100644 img/12-11.png create mode 100644 img/12-12.png create mode 100644 img/12-13.png create mode 100644 img/12-14.png create mode 100644 img/12-15.png create mode 100644 img/12-16.png create mode 100644 img/12-17.png create mode 100644 img/12-18.png create mode 100644 img/12-19.png create mode 100644 img/12-2.png create mode 100644 img/12-20.png create mode 100644 img/12-21.png create mode 100644 img/12-22.png create mode 100644 img/12-25.png create mode 100644 img/12-26.png create mode 100644 img/12-27.png create mode 100644 img/12-3.png create mode 100644 img/12-4.png create mode 100644 img/12-5.png create mode 100644 img/12-6.png create mode 100644 img/12-7.png create mode 100644 img/12-8.png create mode 100644 img/12-9.png diff --git a/12.md b/12.md index d4834ca..a7e6c78 100644 --- a/12.md +++ b/12.md @@ -100,6 +100,7 @@ np.mean(same_distribution) 均值是分布直方图的物理属性。这里是`not_symmetric`的分布直方图,或者等价的`same_distribution`的分布直方图。 +![](img/12-1.png) 想象一下,直方图是由纸板组成的图形,它附着在一条线上,线沿着横轴延伸。并且,将这些条形想象为附加在值 2, 3 和 9 上的权重。假设你尝试在线上的某个点平衡这个图形。如果该点接近 2,图形就向右倾斜。如果该点接近 9,则图形就向左倾斜。之间的某个地方是这个数字取得平衡的点。这个点是 4.25,就是均值。 @@ -109,6 +110,8 @@ np.mean(same_distribution) 因为均值是一个平衡点,有时在直方图的底部显示为一个支点或三角形。 +![](img/12-2.png) + ### 均值和中位数 如果一个学生的考试成绩低于平均水平,这是否意味着该学生在该考试中处于后一半? @@ -121,6 +124,8 @@ np.mean(same_distribution) symmetric = make_array(2, 3, 3, 4) ``` +![](img/12-3.png) + ```py np.mean(symmetric) 3.0 @@ -132,6 +137,7 @@ percentile(50, symmetric) 如果分布不对称呢? 我们来比较`symmetric`和`not_symmetric`。 +![](img/12-4.png) 蓝色直方图表示原始的`symmetric`分布。 `not_symmetric `的金色从左端起始,和蓝色一样,但是最右边的条形到了数值 9。棕色部分是两个直方图重叠的位置。 @@ -157,6 +163,8 @@ sf2015 = Table.read_table('san_francisco_2015.csv').where('Salaries', are.above( sf2015.select('Total Compensation').hist(bins = np.arange(10000, 700000, 25000)) ``` +![](img/12-5.png) + 这个直方图向右偏斜;它的右侧有个尾巴。 平均值拉向了尾巴的方向。 所以我们预计平均薪酬会比中位数大,事实确实如此。 @@ -330,6 +338,8 @@ nba13 nba13.select('Height').hist(bins=np.arange(68, 88, 1)) ``` +![](img/12-6.png) + NBA 球员身材高大并不奇怪! 他们的平均身高只有 79 英寸(6'7"),比美国男子的平均身高高出 10 英寸。 ```py @@ -412,6 +422,8 @@ nba13.sort('Height').show(3) nba13.select('Age in 2013').hist(bins=np.arange(15, 45, 1)) ``` +![](img/12-7.png) + ```py ages = nba13.column('Age in 2013') mean_age = np.mean(ages) @@ -574,6 +586,8 @@ united.hist('Delay (Standard Units)', bins=np.arange(-5, 15.5, 0.5)) plots.xticks(np.arange(-6, 17, 3)); ``` +![](img/12-8.png) + ## 标准差和正态曲线 我们知道均值是直方图的平衡点。 标准差与平均值不同,通常不容易通过查看直方图来识别。 @@ -598,6 +612,8 @@ positions = np.arange(-3, 3.1, 1)*sd_height + mean_height plots.xticks(positions); ``` +![](img/12-9.png) + 上面单元格中的最后两行代码更改了横轴的标签。 现在,对于`z=0, ±1, ±2, ±3`,标签对应于“标签上下`z`个标准差”。 由于分布的形状,“中心”具有明确的含义,在 64 处清晰可见。 ### 如何定位钟形曲线上的 SD @@ -618,6 +634,8 @@ plots.xticks(positions); ![](https://www.zhihu.com/equation?tex=%5Cphi%28z%29%20%3D%20%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B2%20%5Cpi%7D%7D%7D%20e%5E%7B-%5Cfrac%7B1%7D%7B2%7Dz%5E2%7D%2C%20%7E%7E%20-%5Cinfty%20%3C%20z%20%3C%20%5Cinfty) +![](img/12-10.png) + 与往常一样,当你检查新的直方图时,首先查看横轴。在标准正态曲线的横轴上,这些值是标准单位。 这里是曲线的一些属性。有些是通过观察显而易见的,有些则需要大量的数学才能建立起来。 @@ -646,6 +664,8 @@ from scipy import stats 让我们使用这个函数来求出标准正态曲线下,`z=1`左侧的面积。 +![](img/12-11.png) + 阴影区域的数值可以通过调用`stats.norm.cdf`来求出。 ```py @@ -657,6 +677,8 @@ stats.norm.cdf(1) `z = 1`右侧的面积大概是`100% - 84% = 16%`。 +![](img/12-12.png) + ```py 1 - stats.norm.cdf(1) 0.15865525393145707 @@ -664,6 +686,8 @@ stats.norm.cdf(1) `z = -1`和`z = 1`之间的面积可以用几种不同的方式来计算。 它是下面的曲线下方的金色区域。 +![](img/12-13.png) + 例如,我们可以将面积计算为“`100% -`两个相等的尾巴”,结果大致是`100% - 2X16% = 68%`。 或者我们可以注意到,`z = 1`和`z = -1`之间的区域等于`z = 1`左边的所有区域,减去`z = -1`左边的所有区域。 @@ -675,6 +699,8 @@ stats.norm.cdf(1) - stats.norm.cdf(-1) 通过类似的计算,我们看到`-2`和`2`之间的区域大约是 95%。 +![](img/12-14.png) + ```py stats.norm.cdf(2) - stats.norm.cdf(-2) 0.95449973610364158 @@ -762,6 +788,8 @@ red red.select('Winnings: Red').hist(bins=np.arange(-1.5, 1.6, 1)) ``` +![](img/12-15.png) + 现在假设你多次对红色下注。 你的净收益将是来自上述分布的,多个带放回随机抽样的总和。 这将需要一些数学,来列出净收益的所有可能值,以及所有的记录。 我们不会那样做;相反,我们将通过模拟来逼近概率分布,就像我们在这个过程中一直做的那样。 @@ -786,6 +814,8 @@ results = Table().with_column( results.hist(bins=np.arange(-80, 50, 6)) ``` +![](img/12-16.png) + 这是一个大致钟形的直方图,即使我们正在绘制的分布并不是钟形。 中心。分布集中在`-$20`附近。 要知道为什么,请注意,你的奖金在 18/38 左右的下注中为 1 美元,剩下的 20/38 则为负一美元。 所以每个一美元赌注的平均奖金大概是 -5.26 美分: @@ -830,6 +860,8 @@ united = Table.read_table('united_summer2015.csv') united.select('Delay').hist(bins=np.arange(-20, 300, 10)) ``` +![](img/12-17.png) + 平均延误约为 16.6 分钟,SD 约为 39.5 分钟。 注意 SD 与平均值相比有多大。 但是右侧的较大偏差会产生影响,尽管它们在数据中占很小的比例。 ```py @@ -869,6 +901,8 @@ results = Table().with_column( results.hist(bins=np.arange(10, 25, 0.5)) ``` +![](img/12-18.png) + 即使我们从非常偏斜的分布抽样,我们再次看到了大致的钟形。 正如我们所期望的那样,这个钟形的中心在 16 到 17 之间。 ### 中心极限定律 @@ -920,6 +954,8 @@ results = Table().with_column('Sample Proportion: 200', props) results.hist(bins=np.arange(0.65, 0.85, 0.01)) ``` +![](img/12-19.png) + 正如你所期望的那样,中央极限定理预测了,正态曲线再次集中于 0.75 左右。 如果我们增加样本量,这个分布如何变化? 让我们再次运行代码,样本量为 800 ,并将模拟结果收集在同一个表中,我们在里面收集了样本量为 200 的模拟结果。我们使重复次数与之前相同,以便两列具有相同的长度。 @@ -938,6 +974,8 @@ results = results.with_column('Sample Proportion: 800', props2) results.hist(bins=np.arange(0.65, 0.85, 0.01)) ``` +![](img/12-20.png) + 两个分布都大致是正态,但一个比另一个更窄。 样本量为 800 的比例,比样本量为 200 的比例更紧密地聚集在 0.75 左右。增加样本量可以减少样本比例的可变性。 这应该不会令人惊讶。 我们多次产生了这样的直觉,更大的样本量通常会降低统计量的可变性。 然而,在样本均值的案例中,我们可以量化样本量和可变性之间的关系。 @@ -960,6 +998,8 @@ pop_mean 16.658155515370705 ``` +![](img/12-21.png) + 现在我们来随机抽样,来查看样本均值的概率分布。 像往常一样,我们将使用模拟来得到这种分布的经验近似。 我们将定义一个函数`simulate_sample_mean`来实现,因为我们将在稍后改变样本量。 参数是表的名称,包含变量的列标签,样本量和模拟次数。 @@ -1002,6 +1042,8 @@ Population SD: 39.4801998516 SD of sample means: 3.90507237968 ``` +![](img/12-22.png) + ```py simulate_sample_mean(delay, 'Delay', 400, 10000) plots.xlim(5, 35) @@ -1013,6 +1055,8 @@ Population SD: 39.4801998516 SD of sample means: 1.98326299651 ``` +![](img/12-23.png) + ```py simulate_sample_mean(delay, 'Delay', 625, 10000) plots.xlim(5, 35) @@ -1024,6 +1068,8 @@ Population SD: 39.4801998516 SD of sample means: 1.60089096006 ``` +![](img/12-24.png) + 你可以在实践中看到中心极限定律 - 样本均值的直方图是大致正态的,即使延误本身的直方图与正态分布相差甚远。 你还可以看到,样本均值的三个直方图中的每一个中心都非常接近总体均值。 在每种情况下,“样本均值的均值”非常接近 16.66 分钟,是总体均值。 每个直方图上方的打印输出都提供了这两个值。 像预期一样,样本均值是对总体均值的无偏估计。 @@ -1094,6 +1140,8 @@ sd_comparison sd_comparison.plot('Sample Size n') ``` +![](img/12-25.png) + 那里确实有两条曲线。但他们彼此如此接近,看起来好像只有一个。 我们看到了一个普遍结果的实例。 请记住,上面的图表基于每个样本量的 10,000 个重复。 但是每个样本量有超过 10,000 个样本。 样本均值的概率分布基于大小固定的所有可能样本的均值。 @@ -1165,6 +1213,8 @@ sd_comparison.plot('Sample Size n') 那么我们卡住了吗? 不,因为我们可以限制人口的标准差。 这里是两个这样的分布的直方图,一个是相等比例的 1 和 0 ,另一个是 90% 的 1 和 10% 的 0。 哪一个标准差更大? +![](img/12-26.png) + 请记住,总体中的可能值只有 0 和 1。 蓝色直方图(50% 的 1 和 50% 的 0)比金色延展度更大。 它的均值是 0.5。 距离均值的偏差,一半等于 0.5,另一半等于 -0.5,所以标准差是 0.5。 @@ -1209,6 +1259,8 @@ zero_one_sds zero_one_sds.scatter("Population Proportion of 1's") ``` +![](img/12-27.png) + 总结:01 总体的标准差最大为 0.5。 当 50% 的总体为 1 而另外 50% 为 0 时,这就是标准差的值。 ### 样本量 diff --git a/img/12-1.png b/img/12-1.png new file mode 100644 index 0000000000000000000000000000000000000000..3fa92438b147fcde56420ec87d29c6327618b66c GIT binary patch literal 10386 zcmdUVby!vJ_UAznP!KqRh;&Fu2?j_@rywCEDWM`E-604brKCYX1Qb-HyFpS?Nk7 z9g@sBADz{d*w0EXMm1OBKAE5V#2YBp@#BTi#?gdjH7lt=ezRzShxgOGn2qHQ;jr9s zliz3j>x|E!(cyk&-)Vg@n3irGXKCpVu3W-QOzI*rv}p8Iy0%aZhCIw%PFnhPb3QW@ zQ)6Ti_WF$*6H`;BJ7@|DixL%CS-fxc_1y)JR2!O_CR@UpuYY7=V$x;dghHhs$vOMNTaSLBY#;d3gkcgi;a`IKi^a{5B)(rVp8z>~H1Aj;>8Lqv_~i zodoSk!k2E{x`pz6`)TUKMqE)*5gVhescAgj5`I1X3WSY$(Wae6DTaZS6;DD! z;>Y3^5-JT1vgOJR71W)F51H&89MDu$uiv~Oot~b)^HH5qOw6!zFyd2ocA!>fv}33L z{pP%zwn=6i@ye{O?(T~G^r*wda9zs5oRSX?LX%Gy2->f5Yqu<%yNomNeF&W&Gx|M} zjn8&WkFIi5D^> z;j4Lx{Y-0Hc3SVk``(3YNh$~hM~09Eq{=kGf15(4Ly&Wmc~}fae(nDvb;T5K(|F+D~Eq?xo$A^15)t-XoPKy>LR*kPO%B!fz z!c8Hc?%lhGO6$giPz(H>ydfGA`vZ+i@$8&OWisLo$vu+<9rA?C! zku*1-=?_(qmgf0VFoN^*tFAv(!&m*X{KowKeRl#sRFR&wK(u<3o><8$)S@$7g5BspFv$+=1j6tMcgn@xvF*EIZi5qhIQ~zHa~3u!`#9#f!Om zS&*ItL_~%ENb`Z&5lC+0Htvg`Z%~DFL?o9oZ+Xh5bu<2^*536#-G24f-)T43^|{|t zcylj5#sFS2km&M5ftaYM7e9X7`|#ld!_AxI$H&JItGe+ze;gE43_MF-Rn^vEIn>*P zCab($Z#@`4Gqk?9S0jH9BKcth$5-*ApVsh$9zoi*B1hcWs2I*^$ADnp#c3>oZd+$t z8=;}0p|dF+9o_2ow(QCA5f19rt5+=@9Wsv}vr9@!zKM)H9jzF!yXz{*M1~G0j5xxL z7AM24Ti!~Q2{PtS2!H$boPV8fON+eKSf%bdH!);7PX=d&^>A?!W$%LHj|Io;9~haK zSV+CjnQ3WJ5Yq}dexG?sLNBYO6>}v8gTZ{bQjT;@yVdbJC7wq}`WHlCC;4G#c?`EZ!+=i5HVoYWUKKMGf zu<%OEYyZ8&qYUA)NDSuFQa&L8!JRvIP`*L2xSK1#YZ~vy!#xs{=I`c-i+jHWNVu<` zkEP(zCy*dr?NPdxqLHDZQ~GSeoQ(_@C#QP<%Lg3fnSUb8clPFz>pMEmQv{t!5_0-B zm+Uh(sb%QnTs9*uEp1lelBH;9ctf z8LM`S8~OS9tJ}l2X(=fa5sO7d!O$#-Q`>l0SkA0eZV;y{$NMGNl%E|N8w+ihk(R!4 zx!b(NdYHdG{sz9An_FnR6#ScDH`^xyKL9oYISbexdxCCrmp1N__l1Rh&(&a8Xrvs6 zK7RNx-c0Y)0QnybFLW+LH962Q$97Eux|o$TeE~JU6{T;`rPxasFB;g3z6*i6>Xe1A zrKhhu8^bjb(y^mtsv;9DH1M9p^}_*3YkM`8-`3h%jurA_V7PUwNa{I~0<3?fz_l`E zQe2%ZEd*T!Sztli*dYp2h#H$p;vEwc#@4pBJi94bg?d04g%qh~?8wN-`|ihY-n|Qj zmT%J!x7Uck;zp~b$(qb{CaK?rc6;UW_txbE+v?{D)T+w>B6H09aihX3mA!vlO--5M zj@GI&4DkN^5WlNe7b4o+H|L*EPn*#d0H9|;YHqLm{{0ZvjEj%Y%)#-c(Lm?=_3Na7$?rrxnE3fA zCG8!QDk{AcI zzP=9K&liieii(Q5lcmMX!$Zs(sn7#~{$Q2b+F4(Jy1Kf0b$vZzv%|J}&qDd_&9hV? 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