{ "metadata": { "name": "", "signature": "sha256:073dffa71cee5b9202311aa8d6c3060e9986ff4a2469e7363f21581431f58479" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Recall the \"normal\" probability distribution:\n", "$$\\qquad\\qquad{\\rm gaussian}(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}{\\rm e}^{\\textstyle-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def gaussian(x,mu=0.,sigma=1.):\n", " return exp(-(x-mu)**2/(2.*sigma**2))/(sigma*sqrt(2.*pi))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": [ "x=arange(-5,5.1,.1)\n", "y=gaussian(x)\n", "plot(x,y)\n", "xticks(range(-4,5))\n", "xlim(-4,4);" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To get a rough feeling for the fraction of the area under the curve, we can approximate the integral by doing a sum, with stepsize of .0001 (note that since the function is symmetric, we can integrate from 0 to `xmax` and double the result, rather than sum all the way from `-xmax` to `xmax`):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def q(xmax): return 2*.0001*sum(gaussian(arange(0,xmax,.0001)))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since it's a probability distribution, the total integral under the curve is 1, and the fractional values for +/- 1,2,3 standard deviations $\\sigma$ from the mean reproduce the\n", "\"[68\u201395\u201399.7](http://en.wikipedia.org/wiki/68\u201395\u201399.7_rule)\" rule:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "[(sigma,round(q(sigma),5)) for sigma in range(1,5)]" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 4, "text": [ "[(1, 0.68271), (2, 0.95453), (3, 0.99734), (4, 0.99998)]" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "These are related to \"z-values\" which can be found tabulated in\n", "standard tables. (Note that the value for $\\sigma=2$ rounds to .95 — the more accurate value for an exact .95 confidence interval is closer to $\\sigma=1.96$.)\n", "The above values are twice the cumulative from mean (0 to Z), since they include both above and below the mean." ] }, { "cell_type": "code", "collapsed": false, "input": [ "figure(figsize=(8,4))\n", "x=arange(-5,5.1,.1)\n", "y=gaussian(x)\n", "plot(x,y)\n", "fill_between(x,y,color='blue',where=abs(x)<3.01)\n", "fill_between(x,y,color='#00AAAA',where=abs(x)<2.01)\n", "fill_between(x,y,color='cyan',where=abs(x)<1.01)\n", "for i,iy,pct in (1,.235,' 68'),(2,.05,' 95'),(3,.02,'99.7'):\n", " annotate(pct+'%',xytext=(i+(.4 if i < 3 else .46),iy),xy=(.4,iy),\n", " arrowprops={'arrowstyle':\"<-\"},ha='center',va='center',fontsize=14)\n", " annotate('$\\pm'+str(i)+'\\sigma$',xy=(-(i+.05),iy),xytext=(0,iy),\n", " arrowprops={'arrowstyle':\"->\"},ha='center',va='center',fontsize=15)\n", "xticks(range(-4,5))\n", "xlim(-4,4);" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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rcjDBYXckKsBoglZ+48kXU6H7Lgh12h2K8ieXJMEVJ+jz7+N2R6ICjCZo5ReW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\nJN21a1deeuklRowYweDBgwk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"text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also define a function `p(xmax)` as the area under the curve for `x > xmax`\n", "(though note that `p(xmax)` could also be defined in terms of `q(xmax)` since by symmetry `q(xmax)+2p(xmax)=1` ):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def p(xmax): return .0001*sum(gaussian(arange(xmax,10,.0001)))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that values of `x` more than roughly 1.645 standard deviations above the mean correspond to the typically used p-value of .05:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "round(p(1.645),4)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 7, "text": [ "0.05" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This values can also be obtained from \"z-values\" tabulated in\n", "standard tables.\n", "The usual cumulative z-value gives the probability that a statistic is less than z,\n", "and the probability of being above that value is called the complementary cumulative,\n", "equal to 1 minus the cumulative z-value. (So the cumulative z-value of 1.645 is roughly .95,\n", "and the complementary cumulative z-value is .05 .)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "x=arange(-5,5.1,.05)\n", "y=gaussian(x)\n", "plot(x,y)\n", "fill_between(x,y,color='r',where=x>1.64)\n", "annotate('$1.645\\sigma$',xytext=(.5,.08),xy=(1.645,.08),\n", " arrowprops=dict(arrowstyle=\"->\"),ha='center',va='center',fontsize=15)\n", "annotate('5%',xytext=(2.5,.08),xy=(2,.02),\n", " arrowprops=dict(arrowstyle=\"->\"),ha='center',va='center',fontsize=14)\n", "xticks(arange(-4,5));\n", "xlim(-4,4);" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, a quick look at the Central limit theorem: consider a set of $N$ independent random variables $X_i$, with arbitrary probability distributions $p_i(x)$, with means $\\mu_i$ and variances $\\sigma_i^2$, \n", "If we consider the sum $X=\\sum_{i=1}^N X_i$, then for sufficiently large $N$, and whatever the other properties of those independent probability distributions, the probability of $X$ will tend to a gaussian (normal) distribution with $\\mu=\\sum_{i=1}^N \\mu_i$ and variance $\\sigma^2=\\sum_{i=1}^N\\sigma_i^2$.\n", "\n", "As an illustration, we'll consider a set of $N=100$ biased coins, given by the following array of probabilities for flipping heads:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "N=100\n", "ps=rand(N)\n", "print ps\n", "m=sum(ps)\n", "s=sqrt(sum(ps*(1-ps)))\n", "print '\\n sum = ',m,', sqrt of sum of variances=',s" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[ 5.69170818e-01 8.67317321e-01 6.66190386e-01 2.98615928e-01\n", " 2.16640163e-01 9.07432348e-01 4.65905118e-01 9.39634488e-01\n", " 8.18324716e-02 1.64673829e-01 5.10504864e-01 4.75663720e-01\n", " 5.86620705e-01 3.64455305e-01 1.45011289e-02 6.65831850e-01\n", " 4.47445059e-01 3.08840076e-01 3.30986516e-01 6.80786463e-01\n", " 4.10972309e-01 4.10871930e-01 7.42533642e-01 4.42544269e-02\n", " 1.36767530e-01 8.57689528e-01 9.48110526e-01 4.93218187e-01\n", " 4.32564845e-02 6.84323916e-01 3.93547920e-01 4.67854357e-01\n", " 1.06940518e-01 9.62199663e-01 3.08810573e-01 8.93916702e-01\n", " 7.62989511e-01 3.47333936e-01 3.91478825e-01 8.75229484e-01\n", " 4.19158208e-02 2.44692077e-01 7.56443810e-01 7.41190405e-02\n", " 4.58250678e-01 6.69825429e-01 7.80104641e-01 8.09701763e-01\n", " 8.95061322e-02 6.12538795e-01 6.70219482e-01 1.30603654e-02\n", " 5.00404869e-01 2.74923459e-01 1.68611343e-01 1.22729215e-01\n", " 6.90070397e-01 8.75336794e-01 6.42762121e-01 4.99318009e-01\n", " 2.26807512e-01 6.74476531e-01 5.42228998e-01 2.57091495e-01\n", " 1.34922835e-01 2.03299768e-01 9.91107966e-01 4.41279185e-01\n", " 2.15137178e-01 6.75745280e-01 6.29044299e-01 7.57308471e-01\n", " 1.13220670e-01 9.10278417e-01 2.81916025e-01 2.34356596e-01\n", " 6.60694147e-02 3.01243888e-01 1.94465041e-01 6.69993652e-01\n", " 1.17672630e-01 1.72354766e-01 3.61265695e-01 8.58106032e-01\n", " 2.59886545e-04 6.63260747e-01 8.28243635e-01 2.15207409e-01\n", " 5.05777674e-01 3.51390766e-01 2.79628882e-01 8.66800091e-01\n", " 4.26554671e-01 7.84459506e-01 5.67139702e-02 6.55103683e-01\n", " 1.91337638e-01 5.94856297e-01 6.77305971e-01 6.19628490e-01]\n", "\n", " sum = 46.5858044316 , sqrt of sum of variances= 4.1297714602\n" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "(Note that [`rand()`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.rand.html)\n", "is a \"convenience function\", for [`random.random()`](https://docs.python.org/2/library/random.html), and generates an array of random numbers valued between 0 and 1.)\n", " \n", "We flip each coin by picking a random number between 0 and 1, and if that number is less than the \"bias\" of the coin, then we record a flip of heads for the variable $X_i$. The variable $X$ is then the sum of the number of recorded heads for the 100 flips. We do 100000 trials of 100 flips apiece:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "trials = 100000\n", "def rtrial(): return sum(rand(N) < ps)\n", "results = [rtrial() for t in xrange(trials)]\n", "print mean(results),std(results)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "46.57011 4.11410556353\n" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see these compare well to the above values expected from the central limit theorem, and the probability distribution for $X$ is gaussian, even though it is composed from the sum of one hundred different (and non-gaussian) random variables:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "figure(figsize=(8,4))\n", "hist(results,arange(-.5,101),label='data');\n", "xg=arange(int(m-4*s)+1,int(m+4*s),.1)\n", "yg = gaussian(xg,m,s)\n", "plot(xg,trials*yg,'r-',label='gaussian',linewidth=1.5)\n", "plot((m,m),(0,trials*gaussian(m,m,s)),'y--')\n", "annotate('',xytext=(m-s,trials*gaussian(m-s,m,s)),xy=(m+s,trials*gaussian(m+s,m,s)),\n", " arrowprops=dict(arrowstyle=\"<->\"))\n", "title('data fit by gaussian with predicted mean={}, stdev={}'.format(round(m,1),round(s,1)))\n", "xlim(int(m-4*s)+1,int(m+4*s));" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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4Bjc3Cv54uCVtLFuG9n3t2xe+/x7OnoXKttt7Xwg9yDF0IWzBuXMQHAwjR0LD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\ntmxJZGQk7u7uALi7uxMREQEYC3qjRo2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"text": [ "" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that `N` needn't even be as large as 100 for the central limit theorem to work reasonably well -- retry the above, e.g., for `N=30`.\n", "\n", "For another example of arbitrary probability distributions combining to form a normal distribution, consider a population with bimodal heights: half this population has height of exactly 5.5 feet, and the other half has height of exactly 6 feet. We choose `N` people at random and measure the average of their heights. That average will be normally distributed, with a mean of 5.75 and a standard deviation of `sqrt(N/16.)/N` $=1/4\\sqrt N$.
\n", "(1/16. is the variance of the above distribution, since each of the possibilities is 1/4. from the mean, and the additional factor of `N` in the denominator is because we are considering the `mean` rather than the `sum`.)\n", "For `N=100`, the std is 1/40. = .025 .\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def height(n):\n", " return [6 if tall else 5.5 for tall in random.random(n) > .5]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "cell_type": "code", "collapsed": false, "input": [ "N=100\n", "trials=100000\n", "results = [mean(height(N)) for t in xrange(trials)]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "print mean(results),'expected mean',5.75\n", "print std(results),'expected std',1/40." ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "5.7500644 expected mean 5.75\n", "0.0250118542423 expected std 0.025\n" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that the mean and std of the distribution are as expected, and in addition the full probability distribution is well described by a normal distribution (where the average heights over 100 people can take fractional values in between 5.5 and 6 even though any individual height is one of those two extremes with equal probability):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "step=.005\n", "hist(results,arange(5.5-step/2,6.1,step),label='data');\n", "xg=arange(5.6,6,step/5)\n", "yg=step*trials*gaussian(xg,5.75,1./40)\n", "plot(xg,yg,'r')\n", "xlabel('average height of '+str(N)+' bimodal people')\n", "xlim(5.65,5.85);" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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GyujJThK4lwAOoLhJfnMuTrqJRJ00EVjHqDISgagLfsDMGVpyF5v0DkXUEZIM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\nOGmYWMtYvUNyKVarFavV6tQ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"text": [ "" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a last example, consider measuring samples of the population for some trait. Standard red/green color blindness affects roughly 8% of males (the relevant gene that codes for the pigment in the retinal cone cells is on the X chromosome, so is sex-linked, and only .6% of females are affected since that would require two of the variant X chromosomes, `.08*.08=.0064`). Suppose that 194 boys in an incoming class of high school students are tested for color blindness, what range of results is expected?\n", "\n", "We're considering drawing the sample at random from some large population, so the result of the test can be considered a random variable with a probability of .08 of testing positive. For a set of `n` samples with probability `p`, define:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def rn(p,n): return sum(random.random(n) <= p)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This will return the total number of `n` that test positive, if each has a probability of p of testing positive.\n", "\n", "These results should be roughly normally distributed, so if we do some large number of trials:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "trials=100000\n", "results = [rn(.08,194) for t in xrange(trials)]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "print mean(results),'expected',.08*194\n", "print std(results),'expected',sqrt(.08*.92*194)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "15.53083 expected 15.52\n", "3.77356191298 expected 3.77867701716\n" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that the distribution is roughly a normal distribution:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "hist(results,bins=arange(-.5,36))\n", "xg=arange(0.,35.,.1)\n", "yg=trials*gaussian(xg,.08*194,sqrt(.08*.92*194))\n", "plot(xg,yg,'r-')\n", "for b in 3.5,7.5,23.5,27.5: plot((b,b),(0,trials/20),'k--')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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Ibm4ue/bswW63ExISQlhYGOA6tZSVlUV5eTnZ2dnYbAYam0YYypXApRzBSdwF36tbPsO5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\nIAjfevtt3tadIeC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SExMNs73OZrLZbIDebXXmzBkGDRpEZGQks2bNwmw2a99ODWUCvdtp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\nwP8HivMSbmj2IucAAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "where the expected mean $\\pm 2$ and $\\pm 3$ standard deviations are indicated by dotted lines.
\n", "From the '68-95-99.7' rule, it's very likely (95%) that the results of a single measurement will fall between\n", "$15.52 \\pm 2\\cdot 3.78$, so from 8 to 23, and almost certain (99.7%) that they'll fall between $15.52 \\pm 3\\cdot 3.78$, so from 4 to 27 measured to have color blindness." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Additional note:** looking carefully at the above graph, systematic deviations from a normal distribution are evident (the actual data is shifted slightly to the left of the red line near the maximum). This is in part because the mean of 15.5 is small enough that it's better described by a Poisson distribution.\n", "On the other hand, if we're only trying to estimate the likely range of results as within three standard deviations from the mean, then from the graph using a normal distribution is clearly good enough.\n", "\n", "To compare instead with a Poisson distribution, the following code could be added:\n", "\n", " from scipy.misc import factorial\n", " def poisson(z,m): return exp(-z)*z**m/factorial(m)\n", " yg=trials*poisson(.08*194,xg)\n", " plot(xg,yg,'y-')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is also possible to draw from a normal distribution, the function `normalvariate()` generates random pulls from a specified mean and standard deviation:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from random import normalvariate\n", "normalvariate(0,1),normalvariate(100,20)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 20, "text": [ "(1.2483312609153183, 91.68889999099383)" ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Many such pulls these can be generated either via loop\n", "\n", " [normalvariate(0,1) for i in xrange(10)]\n", "(or with an equivalent function `norm()` from scipy.stats):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "trials=1000\n", "xdata=arange(-4,4,.1)\n", "plot(xdata,trials*.1*gaussian(xdata),'r')\n", "hist([normalvariate(0,1) for t in xrange(trials)],bins=xdata);\n", "#same thing:\n", "#hist(norm.rvs(0,1,trials),bins=xdata)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.stats import norm\n", "#http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.norm.html\n", "#has some other useful methods" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 22 }, { "cell_type": "code", "collapsed": false, "input": [ "norm.rvs(0,1,10)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 23, "text": [ "array([-0.26953132, 1.42484678, -0.19684232, -0.32139534, 0.43598938,\n", " 0.03922205, 0.09901545, 0.94656892, -0.3505457 , 0.81771766])" ] } ], "prompt_number": 23 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Addendum" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the end of class, we considered the example of a sample poll of n=1000 people, of whom k=550 answered 'Yes' to some question.\n", "Assuming the 1000 people are drawn at random from some very large population (for some details of the assumptions that go into this, see [sampling](http://terrytao.wordpress.com/2008/10/10/small-samples-and-the-margin-of-error/)), what is the likely percentage of 'Yes' voters in the full population? The idea is that the fraction p=550/1000=.55 is sampled from some much larger population whose overall fraction is some unknown value q. Each sampled person can be considered to be a Bernoulli process with probability q of success, and therefore the number of 'Yes' voters is normally distributed with mean q and standard deviation `sqrt(n*q(1-q))`. Say the range of q of interest is that from which we could have drawn with at least 90% probability the 550 'Yes' votes of 1000 and inferred p=.55 . We know that values of k within 1.645 standard deviations from the mean `n*q` will occur 90% of the time, so the range of interest is values of q such that p=k/n is within 1.645 standard deviations above or below. We don't know the real q so can't calculate its $1.645\\sigma$ precisely, but q will be close enough to p that we can estimate $\\sigma=$ `sqrt(p(1-p)/n)`, and $1.645\\sigma=1.645*\\sqrt{.45*.55/1000}\\approx.026$. (Note that the n has switched to the denominator inside the square root, because we are considering the standard deviation of k/n, hence divide the standard deviation of k by n.) The range of q that could give the inferred value of p=.55 with 90% likelihood is thus given by q between the values $.55\\pm.026$, as in the simulation below (where the mean of the normal distribution is animated to range between those two values, and the observed p is fixed at .55):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from urllib2 import urlopen\n", "from IPython.display import Image\n", "Image(urlopen('https://courses.cit.cornell.edu/info2950_2014fa/resources/ci90.gif').read())" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "png": 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ArSIBqk93vV2LEOeartooCNR4t+sKBDQXEjCLdrSppgdxB5OauewbCAw7EvHrd/O7ZFsJq6mb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\ns7SVF8k3y2P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RSTpNVB69VJ4agVzYmtHH1snRCBgUFrTZ8UjcibPIwXLddSUdfzm4Ir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\nPIYODr8AKK09Paxra4ZsbIhwcIx2d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SSVAHJCsIHBCxPTizHvsR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p3vmKfmgm1fKhB+mhCAWhBiMJTrYpQgdT6gtyiJeoDY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2qLETFLnlm7FC/QBg+QGonghcdatySB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\nLjLpBgSa0/Vz9SMAEUBVP5qqqnLhIwKrkRWwwNGaDE6u93wRLnf1h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VXMfpCcQRoceo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NXdi0bBd2moMqbdob\ne8o7zRJq/Wr9/BY0WtWkWJhYHbKv3RbSSdWynYNXXdsm3dd1kQfWu9t9iNa+rXqabRfnetrEjYN6\nfdySetts/vHXzU2Imgrd0Z3cdQGfQ2nWKDzRKNHa+zfacxGlZBnYz4rAK4HZ2Kzdc8HZ5w3MoK0S\n7G3R7v0WpT2U6I3apame9O3YoWkXMhvft0zb/y3awD0XjKvfydzbKFHfa33fbdG5v+ndvwsGHZAB\nLYdxCP7YdvEGXFndhQgGF2Bx1MjTda2le0HdIt6HJRCq0J3iIr0X+NvihKgB57YB2C3jPW0XkW3j\nhPjiylYBX7rj0n0VcgzkfUjiJm7kEp4W0NzdwAwGJaAAHHDiICHedsHjSc0XSXDCv7nfz2oBR8DI\nIwHhrv3kZ4GT3inmtFoDj/vgAG6cdHEHT0idbk6nNQDK/ieB5uOd4Gxxsnju0VqpxH0+5+U6FwM+\n6MlcAwZuEn6+5UdOFVPtnRaOwvLp4CSh5XXB5RCqF2N9xMxNq/Ec41Fx0I6VuzXt6RFx10qOyit7\n3DyO0Z9UNXnF6hDhyq9+jLHs21zO1A5DJINATLj+EIi967g8m7IuFTGNuIibuHTxh8jOik8grcs+\n0o5FNc9u027h6lKezJhaviXB6ZMeFDFNtsOuaicU1VXB4owOhaOerjY62CUR6W5BuknI5TFtd1Ub\n1lWhxQKa5wGPg0dwqyRh7+X+Eyvg1nCdu3NX7Awh7XpO6DgozPWO6Kx8Flv55hQ/guJ+5hjPrG+h\n6wPf/ug4SO8g3+EBzhZGoLzfDswe7MVZHvI6G+3DHY7xXtXV7usJ7xTePu2ovK5IDfHjOvIADfSk\neAd/W9tE/9u4DfBIz4pLi9VNj9xiffNR34eBUOpD3/NM8fNkeem/y8VVYOghQe5qrpt0segcb/LA\nu/QhgfBpr51uUc7PKvBt7riXzBFyT9417xZycOd33/FRGMQg0feA7rRsEeUT7/Y5+Ab+ffg0r/hp\nwd3yTvh+eASMLfkqT+dn8ePUydxi/7pmrd4z3/mJfhUbn/UE+fEhXfXZ3Ra9y/oEOdAuDftzGxW5\nTfsEqenEjPv+7hSdzPutGLqv7/VIkd/E34ogfPvI/n8UX/65oy/9hgjeG4H2fk/5VgEIlvy5bp7z\nxmjAa9wRiO/hnl8VVUAFLp/3jt+HsUv+k8+eaWGl6//ut2zAfvsR5b/yGQ8VviwIAJEH0ECCBQ0e\nRAioRkKGDR0SXMiQCZ5EFS1exJhRY8UaGz1+BBlS5EiSJUsCMJlS5UqWiVC2hBmT5RsqhQQ+xAkx\n586HERH+oSJFJsaOQ40eRSryZVKmTV06hRqzSR6bPHHesJr1IFaGd44kxRFV7NiTZM2qXHpWbcZC\nSBJV1RpXrtw8TiiuxZs3Zlq9fZ/6XRtHKNy5hQ03/PMnzhTAjR1r5Pv4bGTJTZ/cSSTo5mHOnQdS\n/kWCqPLovpRJQzV9WqahI6IJe4aNM9DPPInsqsZNNnXulQQaSJBwcTdvlYIrvo6d3OqfRHAYE4eO\ndHj0kQsIZZxOXeTtt5s5c1WOEDzC2m2PhtWOO3t6j9axs2cpKHRFzbB9hjd43+Cf2oma3JWpKPhG\nW29AjBb4bQvhDEyJJovq80w//AaSsCDmEpEjCqMEZNCxAjtMZJBECGkADYs+7HAJQCxCzrAKC5ot\nwob6MwQJQ4biEES/UNTxiixOpIGGK0zUcaM8mLioxcJeDI9J/iySIg4ci9QLDSFp4HHAQw5J5JAJ\niPyLSo+mgCNJ7w5jUrk0L0wEjyamFLO0OD3q/uO3Bq5YcM6MEDnixgfPNGy8CQcSdL+LklgxJvT0\nXCtLMR1NLw4NzRy0Uob6q6gKKhgdENIiPaWOCUyPA9TSSp9kcT5O0wMVxFaJ+2MJtko1FcbkRk0k\nCilX1e5VBn3NLcpZayUWEDYrOpLXXpUFCVjV2vKT0mItRfUiUZmFzln4tCWtiio0UnKuQiccl6Bj\nK5IDCpgWxRa1djfitjJEkBAEXFoZipGhNJNzEtdE5g2kpRzfTSreZTl9Qyh77SN2TY0SFphgdyXO\nU09Ekgh4YRlrTdPff+llaWCKhzKYupIbg2NSjSDsbN/Y+t2oiudUEnnkvWyu6GS/APYoXLlc/mb4\n0o14phln6YzWuS+IPWL5O2LLHejci95Q2SR2jb4Z56Tzaqvenu+dFrZq90QU6x2R1nOKTT/yOWyx\nP5LDCbPl1HpOQJKIdqO23e7M44uasGPuvLbOjXC1mpAjpL3lypdvvy36A2/BG0VbzDjkVhxsvg8b\neyMqZp5ct8qLlC9jkBa3CmrlVH/cootbz+jq0E0yXLXax3Ji18yDNtXhkPCQfKQKZmfpdtKMh6oK\ndUdqGs2GGercI+VJGp54tEbvEPi8T9c8K6AfatyhjklqokyRqreeduwNDAQJqdnu3qrvW0aMpEKS\nAPAj9NMnCfnK/EfK/TBTEtTxZH6cgRlJ/v7gvpDsj39KWR97DLEE8xEwfgQRQwIS4IOCiEEADnBA\nD24whwmAUAgFAQIA6gAbqEUPJHfgwQIW8COL6CEADbgTDuqEwwBooSJaQNAMHlixkQFwKIVgwrdS\nUsCBJMANgHBAGzD4gYLQYQ1/qEMCpAiIOXwgASs01ftAggAiAKIBfLAIGk6wEQKICA0mqMh1hpiz\nCFJHEEt4w0qa1xA1UBEQPuDgQPqYEBSIYSAoYMMXawW7jOjhBHdIQg/wVBE1auQME6iICsA0RzrW\nDT55QELiVsLELrhgIFyAAUHUMAAHfIANBXEDAvwACC+kUpHUYiRGyhCDRORhBy1I4wAk/nACNFok\nBj5MxJ0mMIE9cDJMRYRPFZIgRuZdkJamBAQqCVKHWYYhAQTxQwO8AAg/PGCFt6SWSXZZkS6EIAo3\n2lIizrAA1w1Ajg2QQSL2QE9OGtFD6fkDE6BQCJgwcZB/DKRBbvkBIAykDQNIAAEAgM5B5fIie4Bj\nIq6ABSokYYAVcY88M5qIExAppEP0Z2NSWpJARCEJgYsJE//gxD9EkSB0GMgaCICVFKTyIBQ9TAst\nehEC9CEREkBjQJVQmz0QAD0qIINFtECDRPSBAM5cqVnQIEMadpI3f4ACEuAgGpkwERAZJAAHhzAE\nQAgBhBFQQwXUAAAQOiAMBQGqi+pn/pKtLgBPW/BhEAxwAAjooSOHGACXLLKC32zygVklywJEhNQT\npZE0gajCEpigu5Q4llS8sxQFZnQUOzgBCTmg5lE8WyQ9VJZZeshoFiYZptkCBhBykEISklCFRLGE\nqsPamKkKMFqkCEIETDiCrgBB1qTUVky/feaq1pmILeTTq9A1CyIKAYg7xKEKUWCCEZggBTl4LSbO\n/WxwLTXchCSGKR0phB2m0AQjLAEKVYjDHf4giO2pBLvPda2ypksG67rEW28YgbcUvOAFU0FTmqJC\nhCX8uc9NwcJSkEIUoPAEJzSBCUtAghGOkIQmQIEKcLDDcpPy3+PcAQ95yMOLYRzj/hnLWMYwvrGN\n81ADHNe4xz7+8YyBrOMgFyDHNKbxezMSiDvAgQpQcAKIjYCEJCyhCU54QhSigGELq83BDPZWgsGs\n4DdUocxlNnOa33DmNJuZzWtms5sP/OY5q3nNbb6znON8ZwSX+Q+QFQtsK+KjEwHA0IdGdKIVvWhG\nN9rRj4Z0pCU9aUpX2tKXxnSmNb1pTnfa0e2SbDKL6UxSl5pZ8/SrqVW9ala32tWvFguq0bsHAJSh\nIr4BTnRQ3dVEoOGGEpjkGRgwQ13LsLZoEIAE7nTr3wQHOn3lda+VjUl5Dhu9qpE1RvSAQ2rj2tmr\nCjVlL2KCE9g6ESeFjmQRIe5E/jjyQH1AxBnTPdlRV5KocozOAowq7zg2QERy1PdRR52bcI+aRP8G\nKb5XJWiN1jYLWoiBuQmgcN4w/AqztXdF3J0I2RKH4VnoasYTTp2Pz3aqF7F4tE9jcZP/d+LMGnCB\nBwHHiIMUOAoiTsxrOIAGEDMROucN0NvNc5+f++bQEboMZDABCURV6LhJugxM0HSbSwDnnBK6Cpq5\nAnP/uwGtDTovE0Fgi8Rznj8XO9nDXhG1d4lLZw/RiL6e87QXeOmIIIS+n66apE/gEHk3qtfBzqiN\nExqkBCBAAAZwhovQQOWkKfy1rZNyj2f04hpBt+FzE3mLZOG3MSADRgf9+NEU5L6rl09EzS3ieF4F\nnN8XqXk8vbRa0oT69SLS51XPLaLXE3zfo8Z9U93epS+l+/cW4QMcDyHvgLPb96JGvvLPKPvir6qv\neNLC1VNv6x0uGzrZ3oKCtMBtsM+TAKQnzfWpK/5mt7b71z6N+sNfkSwgCJnZfraxE5H9QePQh++H\ntQAUwAEkwAI0wANEwARUwAVkwAZ0wAeEwAiUwAmkwAq0wAvEwAzUwA3kwA70wA8EwRAUwREkwRI0\nwRNEwRRUwRVkwRZ0wReEwRiUwRmkwRq0wRvEwRzUwR3kwR70wR8EwiAUwpUICAAh+QQAFAAAACwA\nAAAAsAEgAYcAAAAMDAwPDxISEhIdHR0APwAANBUcHCIQEC8iIiIkJC0tLS0mJjAqKjUuLjkyMjI7\nOzsAXwAAcxccRyMjTyMvTz8yUj8/Xz8II0YSLEczM0A5OUc9PUwjI2Q+Pn84UkZDQ0NLS0tDQ1RK\nSl1TU1NbW1tHYUdQUGRWVmxZWXBiYmJsbGx0dHR8fHwAgAA/nz9lpX9/v38ODpgAPYM/P4YtLZgj\nI6tra4ZsbIhxcY51dZN5eZgXF9YBAf8LC/8TE/8bG/8jI/8rK/8zM/88PP9iYsZDQ/9MTP9TU/9b\nW/9kZP9sbP9zc/99ff+Dg4OCgoiKioqJiZSTk5Obm5uCgqOKiq2MjLCPlbSSkreWlry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Y0evWOHKKAhd3w+hRoVIk+pVR1StfqUzw9Ad+AgBRtWrMilZpZkRZs2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\ng1JiFuMb3sgmI6dJTTlF95Yt3KEBY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"prompt_number": 25, "text": [ "" ] } ], "prompt_number": 25 }, { "cell_type": "markdown", "metadata": {}, "source": [ "For values of q either below .524 or above .576, the total probability of finding 550 'Yes' voters falls below 10%, because it is too large a fluctuation from the mean in a sample of 1000. (A hidden assumption in this has been that all values of q are a priori equally likely, which implied as well that the mean of the true normal distribution is equally likely to be above or below the observed sample mean.)\n", "\n", "There is an on-line \"margin of error\" [calculator](http://www.americanresearchgroup.com/moe.html), which calculates the 95% confidence interval (corresponding to $1.96\\sigma$) for an evenly split sample (probability p=.5).\n", "For a sample of n=1000 and a large overall population size, say N=1000000, the \"margin of error\" calculator should give \n", "`1.96* sqrt(.5*.5/1000)` = .031 (try it), and as long as the overall population size N is at least roughly 20 times larger than the sample size n, the calculator should continue to give a margin of error of `1.96/sqrt(2*2*n)=.98/sqrt(n)`, independent of the overall size of the population. As the sample size becomes comparable to the overall population size, however, the margin of error *decreases*, until the limiting value of sample size equal to population size, at which point the margin of error is zero (the entire population is sampled). The exact value for the standard deviation of drawing n samples (with no repeats) from a population of N is given by\n", "$\\sqrt{{1-n/N\\over 1-1/N}q(1-q)}$, where the first factor is known as the \"[finite population correction](http://en.wikipedia.org/wiki/Standard_error#Correction_for_finite_population)\"\n", "(and can be derived by considering the probability distribution for the [hypergeometric](http://en.wikipedia.org/wiki/Hypergeometric_distribution) distribution, the simple modification necessary for \"sampling without replacement\", i.e., not asking the same person more than once, which becomes more likely as the sample size becomes an appreciable fraction of the total population). Note that the extra factor is zero when n=N, and is irrelevant when N is large and much greater than n.
\n", "You can also check to see whether the above calculator has implemented this correction factor properly by considering the intermediate range of values of n/N.\n", "\n", "\n", "For reference, this is the code used to produce the plots for the above animated .gif:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "p=.55\n", "n=1000\n", "j=0 #png counter\n", "sigma=sqrt(p*(1-p)/n)\n", "ci90=1.645*sigma #90% confidence interval\n", "h=gaussian(0,0,sigma) #value of gaussian at peak\n", "x=arange(p-6.5*sigma,p+6.5*sigma,.001)\n", "for q in arange(p-ci90,p+ci90+.001,2*ci90/23):\n", " figure(figsize=(6,4))\n", " errorbar(p, 27.5, xerr=ci90, fmt='go', label='$\\pm1.645\\sigma$')\n", " y=gaussian(x,q,sigma) #centered at q, std=sigma\n", " plot(x,y,'b',label='$\\sigma={:.3f}$'.format(sigma))\n", " fill_between(x,y,color='b',alpha=.2,where=abs(x-q)<1.65*sigma)\n", " ch=gaussian(p,q,sigma) #value of q-centered gaussian at p=.55\n", " vlines((p-ci90,p,p+ci90),0,(h,ch,h),color='g',linestyles='--')\n", " plot(p,ch,'go')\n", " for pl in (.524, .576): text(pl,.5,pl,ha='center')\n", " xlim(p-6.5*sigma,p+6.5*sigma),ylim(0,29)\n", " title('90% confidence interval (n={0:}, k={1:}, $\\sigma=\\sqrt{{{2:}\\cdot{3:}/{0:}}}$)'.format(n,int(p*n),p,1-p))\n", " # label='' on the fill_between is broken, so a non-visible hack for the legend:\n", " fill((0,.1,0),(5.1,5.0,4.9), color='b', alpha=.2,label='$\\pm1.645\\sigma$') \n", " legend(numpoints=1)\n", " savefig('pngs/img{:03d}.png'.format(j))\n", " j+=1\n", " close();\n", "## then convert -delay 20 -loop 0 img*.png ci90.gif" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 24 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }