{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Введение в ТВиМС: практикум по описанию выборок №2\n", "\n", "*Алла Тамбовцева, НИУ ВШЭ*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Часть 1: построение гистограммы" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Вновь импортируем библиотеку `numpy` для работы с массивами и модуль `pyplot` из библиотеки `matplotlib` для отрисовки графиков:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from matplotlib import pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Создадим массив `turnout` со значениями явки из семинарского листочка:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[38.19 44.2 37.31 37.58 36.62 46.16 37.65 35.02 35.24]\n" ] } ], "source": [ "turnout = np.array([38.19, 44.20, 37.31, 37.58, \n", " 36.62, 46.16, 37.65, 35.02, 35.24])\n", "print(turnout)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Построим для этой выборки гистограмму:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(array([2., 1., 4., 0., 0., 0., 0., 0., 1., 1.]),\n", " array([35.02 , 36.134, 37.248, 38.362, 39.476, 40.59 , 41.704, 42.818,\n", " 43.932, 45.046, 46.16 ]),\n", " )" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# color: цвет заливки\n", "# edgecolor: цвет границ, k – сокращение от black\n", "\n", "plt.hist(turnout, color = \"hotpink\", edgecolor = \"k\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Функция `hist()` сама подбирает шаг и нужное число столбцов для гистограммы. Но выставить желаемое число столбцов или шаг тоже можно, воспользовавшись аргументом `bins`. В него можно подставить нужное число столбцов:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(array([7., 0., 2.]),\n", " array([35.02 , 38.73333333, 42.44666667, 46.16 ]),\n", " )" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAD4CAYAAADFAawfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAMsklEQVR4nO3df4zk9V3H8eerd4dAbSUpG1M5jquK1FoDNBvUkDT1UHPQS/UPjRBLggXXoFXamNSamJg2mtSYVEyj15yUSlJaWvmRNJcWJVHSEit0D66G6xVFAr2D6i1R0p7GEuDtHzN3bPdmb7+7N9+Zz+4+H8mG+fG9ufcnwz758p3vzKSqkCS16zXTHkCSdHqGWpIaZ6glqXGGWpIaZ6glqXFb+3jQ888/v3bu3NnHQ0vShnTgwIHnq2pm1H29hHrnzp3Mz8/38dCStCEleWa5+zz0IUmNM9SS1DhDLUmNM9SS1DhDLUmNM9SS1LgVQ53kkiQHF/18O8n7JjGcJKnDedRV9QRwGUCSLcCzwH09zyVJGlrtoY+rgH+vqmVPzJYkjddq35l4LfCZUXckmQPmAHbs2LHmgXZu38Ezzx5Z85/X+F10wYU8ffSb0x5D2rTS9RtekpwFPAf8RFX95+m2nZ2drbW+hTwJdfP+Nf1Z9SN79+A3AUn9SnKgqmZH3beaQx9XA4+uFGlJ0nitJtTXscxhD0lSfzqFOsm5wM8D9/Y7jiRpqU4vJlbV/wJv6HkWSdIIvjNRkhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcYZakhpnqCWpcV2/hfy8JHcn+UaSw0l+pu/BJEkDnb6FHPgL4P6q+uUkZwHn9jiTJGmRFUOd5PXA24EbAKrqReDFfseSJJ3Q5dDHDwMLwCeTPJbktiSvXbpRkrkk80nmFxYWxj6oJG1WXUK9FXgbsLeqLgf+B/jg0o2qal9VzVbV7MzMzJjHlKTNq0uojwJHq+rh4fW7GYRbkjQBK4a6qv4DOJLkkuFNVwFf73UqSdJJXc/6+B3gzuEZH08Bv97fSJKkxTqFuqoOArM9zyJJGsF3JkpS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4zp9C3mSp4HvAC8DL1WV30guSRPSKdRDP1tVz/c2iSRpJA99SFLjuoa6gL9PciDJ3KgNkswlmU8yv7CwML4JJWmT6xrqK6vqbcDVwG8nefvSDapqX1XNVtXszMzMWIeUpM2sU6ir6rnhP48B9wFX9DmUJOlVK4Y6yWuTvO7EZeAXgMf7HkySNNDlrI8fBO5LcmL7T1fV/b1OJUk6acVQV9VTwKUTmEWSNIKn50lS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDWuc6iTbEnyWJL9fQ4kSfpeq9mjvgU43NcgkqTROoU6yXbgncBt/Y4jSVqq6x71rcAHgFeW2yDJXJL5JPMLCwtjGU6S1CHUSfYAx6rqwOm2q6p9VTVbVbMzMzNjG1CSNrsue9RXAu9K8jRwF7Aryad6nUqSdNKKoa6qP6iq7VW1E7gW+Ieqenfvk0mSAM+jlqTmbV3NxlX1IPBgL5NIkkZyj1qSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalxhlqSGrdiqJOcneSRJF9LcijJhyYxmCRpYGuHbb4L7Kqq40m2AQ8l+WJV/XPPs0mS6BDqqirg+PDqtuFP9TmUJOlVnY5RJ9mS5CBwDHigqh4esc1ckvkk8wsLC+OeU5I2rU6hrqqXq+oyYDtwRZK3jthmX1XNVtXszMzMuOeUpE1rVWd9VNULwIPA7l6mkSSdostZHzNJzhtePgf4OeAbfQ8mSRroctbHG4E7kmxhEPbPVdX+fseSJJ3Q5ayPfwEun8AskqQRfGeiJDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS41YMdZILk/xjksNJDiW5ZRKDSZIGVvwWcuAl4Peq6tEkrwMOJHmgqr7e82ySJDrsUVfVt6rq0eHl7wCHgQv6HkySNLCqY9RJdgKXAw/3MYwk6VRdDn0AkOT7gXuA91XVt0fcPwfMAezYsWNsA0oabef2HTzz7JFpj6FFLrrgQp4++s2xP26nUCfZxiDSd1bVvaO2qap9wD6A2dnZGtuEkkZ65tkj1M37pz2GFsnePb08bpezPgJ8AjhcVR/tZQpJ0rK6HKO+Erge2JXk4PDnmp7nkiQNrXjoo6oeAjKBWSRJI/jORElqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklqnKGWpMYZaklq3IqhTnJ7kmNJHp/EQJKk79Vlj/pvgN09zyFJWsaKoa6qLwH/NYFZJEkjjO0YdZK5JPNJ5hcWFsb1sJK06Y0t1FW1r6pmq2p2ZmZmXA8rSZueZ31IUuMMtSQ1rsvpeZ8BvgJckuRokhv7H0uSdMLWlTaoqusmMYgkaTQPfUhS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4wy1JDXOUEtS4zqFOsnuJE8keTLJB/seSpL0qhVDnWQL8JfA1cBbgOuSvKXvwSRJA132qK8Anqyqp6rqReAu4Bf7HUuSdMLWDttcABxZdP0o8FNLN0oyB8wNrx5P8sRah8rePWv9o6t1PvD8pP6yCRr7upKM8+HOhM/ZIhP8XTkTG/U5gxFrO4PflYuWu6NLqEf9rXXKDVX7gH2rGGrqksxX1ey05xi3jbou2Lhr26jrAtc2Dl0OfRwFLlx0fTvwXD/jSJKW6hLqrwIXJ3lTkrOAa4HP9zuWJOmEFQ99VNVLSd4L/B2wBbi9qg71PtlkrKtDNauwUdcFG3dtG3Vd4NrOWKpOOdwsSWqI70yUpMYZaklq3KYIdZKzkzyS5GtJDiX50PD2JPmTJP+a5HCS3532rKt1mrVdleTRJAeTPJTkR6c961ok2ZLksST7h9fflOThJP+W5LPDF7jXpRFru3P4UQ2PJ7k9ybZpz7gWS9e16PaPJTk+rbnGYcRzNpGGbIpQA98FdlXVpcBlwO4kPw3cwODUwzdX1Y8zeNflerPc2vYCv1ZVlwGfBv5wijOeiVuAw4uu/ynw51V1MfDfwI1TmWo8lq7tTuDNwE8C5wA3TWOoMVi6LpLMAudNZ5yxWrq2G5hAQzZFqGvgxH/Jtw1/CrgZ+HBVvTLc7tiURlyz06ytgNcPb/8B1uG570m2A+8EbhteD7ALuHu4yR3AL01nujOzdG0AVfWF4fNZwCMM3rOwroxa1/Dzgv4M+MC05hqHUWtjQg3ZFKGGk//LchA4BjxQVQ8DPwL8apL5JF9McvF0p1ybZdZ2E/CFJEeB64GPTHPGNbqVwS/3K8PrbwBeqKqXhtePMviIg/Vo6dpOGh7yuB64f9JDjcGodb0X+HxVfWs6I43NqLVNpCGbJtRV9fLwMMB24IokbwW+D/i/4VtA/xq4fZozrtUya3s/cE1VbQc+CXx0mjOuVpI9wLGqOrD45hGbrrvzS5dZ22J/BXypqr48wbHO2Kh1Jfkh4FeAj01tsDE4zXM2kYZ0+ayPDaWqXkjyILCbwR7ZPcO77mMQtHVr0dquBi4d7lkDfJb1t3d2JfCuJNcAZzM4jHMrcF6SrcO96vX6cQanrC3Jp6rq3Un+CJgBfnOqE67NqOfsEIPXUZ4cfljRuUmerKr19uL2yOeMSTWkqjb8D4N/8c8bXj4H+DKwh8HhgPcMb38H8NVpzzrGtT0P/Njw9huBe6Y96xms8R3A/uHlvwWuHV7+OPBb055vjGu7Cfgn4JxpzzXOdS25/fi0ZxvzczaRhmyWPeo3AncMX9R4DfC5qtqf5CHgziTvB46zPl9lX25tvwHck+QVBmdHvGeaQ47R7wN3Jflj4DHgE1OeZ5w+DjwDfGW493lvVX14uiNpBR9hAg3xLeSS1LhN82KiJK1XhlqSGmeoJalxhlqSGmeoJalxhlqSGmeoJalx/w+0KNCUzFcNTQAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# например, 3\n", "plt.hist(turnout, bins = 3, color = \"hotpink\", edgecolor = \"k\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Также в этот аргумент можно подставить список или массив с границами столбцов. Попробуем воспроизвести те гистограммы, которые мы строили вручную. Зафиксируем стартовую и конечную точку. Стартовая точка была по условию равна 35, а конечную посчитаем самостоятельно – округлим максимальное значение в меньшую сторону и возьмем следующее за ним :" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "start = 35\n", "end = int(turnout.max()) + 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Теперь сформируем набор чисел от `start` до `end` с шагом 2 – это и будет шаг гистограммы:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# помним про range(), не забываем +1 \n", "\n", "plt.hist(turnout, bins = range(start, end + 1, 2), \n", " color = \"hotpink\", edgecolor = \"k\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Теперь построим гистограмму с шагом 4:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist(turnout, bins = range(start, end + 1, 4), \n", " color = \"hotpink\", edgecolor = \"k\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Часть 2: выборочная дисперсия и стандартное отклонение" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Посчитаем выборочную оценку дисперсии:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "13.36433333333333\n" ] } ], "source": [ "print(turnout.var())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Важно:** Python коварен, он по умолчанию считает смещенную оценку дисперсии (которая с $n$ в знаменателе, а не с $n-1$). Чаще всего нас это не устраивает, поэтому запросим несмещенную оценку – попросим вычитать 1 в знаменателе:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "15.034874999999996\n" ] } ], "source": [ "# аргумент ddof = 1\n", "\n", "print(turnout.var(ddof = 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Со стандартным отклонение то же самое:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3.877483075398266\n" ] } ], "source": [ "print(turnout.std(ddof = 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Часть 3: предельные теоремы" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Для этой части нам потребуется модуль `stats` из библиотеки `scipy` – будем генерировать случайные выборки:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "from scipy import stats" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Давайте визуализируем действие закона больших чисел. \n", "\n", "> **Закон больших чисел.** С увеличением размера выборки, среднее значение выборки становится ближе к среднему значению генеральной совокупности." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Возьмем выборки разного размера из нормального распределения со средним значением 168 и стандартным отклонением 2 (можно считать, что мы извлекаем выборки из генеральной совокупности, которая представляет собой набор значений роста женщин в сантиметрах).\n", "\n", "Начнем с небольшой выборки в 10 наблюдений." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[168.820987 167.57992973 173.85091224 166.2265467 168.11245187\n", " 170.21872813 168.88667925 171.9283559 165.92869349 169.15312952]\n" ] } ], "source": [ "# norm: нормальное распределение со средним loc и ст отклонением scale\n", "# rvs: выборка, от random variable sample\n", "\n", "sample01 = stats.norm.rvs(loc = 168, scale = 2, size=10)\n", "print(sample01)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Теперь сгенерируем выборки из 100 и 1500 наблюдений:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "sample02 = stats.norm.rvs(loc = 168, scale = 2, size=100)\n", "sample03 = stats.norm.rvs(loc = 168, scale = 2, size=1500)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Сравним средние выборок:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "169.07064138215762\n", "168.27520438134118\n", "168.02119805846888\n" ] } ], "source": [ "print(sample01.mean())\n", "print(sample02.mean())\n", "print(sample03.mean())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Действительно, самое близкое к среднему генеральной совокупности 168 среднее последней, самой большой выборки.\n", "\n", "Заодно построим гистограммы для каждой выборки и убедимся, что с увеличением объема выборки распределение выборки все больше похоже на распределение генеральной совокупности, то есть на нормальное в нашем случае:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist(sample01, color = \"cornflowerblue\", edgecolor = \"white\", alpha=0.5);" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist(sample02, color = \"cornflowerblue\", edgecolor = \"white\", alpha=0.5);" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist(sample03, color = \"cornflowerblue\", edgecolor = \"white\", alpha=0.5);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Теперь визуализируем действие центральной предельной теоремы.\n", "\n", "> **Центральная предельная теорема.** Пусть у нас есть генеральная совокупность со средним значением $\\mu$ и стандартным отклонением $\\sigma$. Если мы извлечем из этой совокупности все возможные выборки достаточно большого размера $n$ ($n \\geqslant 30$) и посчитаем по каждой выборке среднее значение, то эти средние значения будут примерно нормально распределены со средним $\\mu$ и стандартным отклонением $\\frac{\\sigma}{\\sqrt{n}}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Сформируем генеральную совокупность – 10000 значений из равномерного распределения на отрезке от 0 до 1:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "population = stats.uniform.rvs(size = 10000)\n", "plt.hist(population, color = \"cornflowerblue\", edgecolor = \"white\", alpha=0.5);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Видно, что распределение на нормальное совсем не похоже, это ожидаемо – оно равномерное :)\n", "Зафиксируем среднее и стандартное отклонение генеральной совокупности:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean: 0.49889629993662493\n", "St Deviation: 0.2865068970135882\n" ] } ], "source": [ "print(\"Mean: \", population.mean())\n", "print(\"St Deviation: \", population.std(ddof = 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Теперь выберем из этой генеральной совокупности 10000 выборок по 1000 наблюдений (все возможные выборки не будем, очень много, можете посчитать $C_{10000}^{1000}$) и зафиксируем среднее по каждой выборке. \n", "\n", "Для псевдослучайного выбора воспользуемся `random.choice()` из `numpy`:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "means = []\n", "\n", "# выбираем 10000 раз разные выборки с n=1000 из population\n", "# считаем среднее по каждой и добавляем в means\n", "\n", "for i in range(10000):\n", " sample = np.random.choice(population, size = 1000)\n", " avg = sample.mean()\n", " means.append(avg)\n", " \n", "# преобразуем в массив для удобства\n", "\n", "means = np.array(means)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "А теперь магия (действие центральной предельной теоремы)! Построим гистограмму для набора из полученных выборочных средних:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist(means, color = \"cornflowerblue\", edgecolor = \"white\", alpha=0.5);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Распределение средних, действительно, оказалось нормальным, хотя выборки мы извлекали из равномерного распределения, которое на нормальное совсем не похоже. Проверим параметры полученного распределения:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean: 0.498860747704185\n", "St Deviation: 0.009054966571835136\n" ] } ], "source": [ "print(\"Mean: \", means.mean())\n", "print(\"St Deviation: \", means.std(ddof = 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Среднее, полученное по набору выборочных средних («среднее средних») практически совпадает со средним генеральной совокупности `population`. А вот стандартное отклонение, по идее, должно примерно совпадать со стандартным отклонением `population`, уменьшенным в $\\sqrt{1000}$ раз, так как мы извлекали выборки объема $n = 1000$:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.009060143599102326\n" ] } ], "source": [ "print(population.std(ddof = 1) / np.sqrt(1000)) # есть такое" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Всё!" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 2 }