{"id":82389,"date":"2021-04-05T15:40:30","date_gmt":"2021-04-05T19:40:30","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=82389"},"modified":"2022-11-21T09:47:19","modified_gmt":"2022-11-21T14:47:19","slug":"central-limit-theorem-explained-in-python-with-examples-part-iv","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-iv\/","title":{"rendered":"Central Limit Theorem explained in Python (with examples) \u2013 Part IV"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>See&nbsp;<a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples\/\">Part I<\/a>,&nbsp;&nbsp;<a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-ii\/\">Part II<\/a>&nbsp;<em>and<\/em><\/em> <em><a href=\"\/campus\/ibkr-quant-news\/central-limit-theorem-explained-in-python-with-examples-part-iii\/\">Part III<\/a><\/em> <em>to get started.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In case 2, we will repeat the above process, but with a much larger sample size (n=500):<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n# drawing 50 random samples of size 500<br>\nsample_size=500<br><br>\n\ndf500 = pd.DataFrame()<br><br>\n\nfor i in range(1, 51):<br>\n    exponential_sample = np.random.exponential((1\/rate), sample_size)<br>\n    col = f&#8217;sample {i}&#8217;<br>\n    df500[col] = exponential_sample<br><br>\n\n\ndf500_sample_means = pd.DataFrame(df500.mean(),columns=[&#8216;Sample means&#8217;])<br>\nsns.distplot(df500_sample_means);\n<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"386\" height=\"248\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/sampling-distribution-quantinsti.png\" alt=\"\" class=\"wp-image-82408 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/sampling-distribution-quantinsti.png 386w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/sampling-distribution-quantinsti-300x193.png 300w\" data-sizes=\"(max-width: 386px) 100vw, 386px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 386px; aspect-ratio: 386\/248;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The sampling distribution looks much more like a normal distribution now as we have sampled with a much larger sample size (n=500).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us now check the mean and the standard deviation of the 50 sample means:<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n#The first 5 values from the 50 sample means<br>\ndf500_sample_means.head()\n<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td><\/td><td><strong>Sample means<\/strong><\/td><\/tr><tr><th>sample 1<\/th><td>4.061534<\/td><\/tr><tr><th>sample 2<\/th><td>4.052322<\/td><\/tr><tr><th>sample 3<\/th><td>3.823000<\/td><\/tr><tr><th>sample 4<\/th><td>4.261559<\/td><\/tr><tr><th>sample 5<\/th><td>3.838798<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">We can observe that the mean of all the sample means is quite close to the population mean&nbsp;(\u03bc=4)(\u03bc=4).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly, we can observe that the standard deviation of the 50 sample means is quite close to the value stated by the CLT, i.e.,&nbsp;(\u03c3\/\u221an)(\u03c3\/n)&nbsp;= 0.178.<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n# An estimate of the mean of the sampling distribution can be obtained as:<br>\nnp.mean(df500_sample_means).values[0]\n<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n0.18886796530269118\n\n<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n# The above value is very close to the value stated by the CLT, which is:<br>\nsd\/ np.sqrt(sample_size)\n\n<\/p>\n\n\n\n<p style=\"background-color:#fcfcdb;font-size:11px\" class=\"has-background\">    \n0.17888543819998318\n\n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, we observe that the mean of all sample means is very close in value to the population mean itself. Also, we can observe that as we increased the sample size from 2 to 500, the distribution of sample means increasingly starts resembling a normal distribution, with mean given by the population mean\u00a0\u03bc\u03bc\u00a0and the standard deviation given by\u00a0(\u03c3\/\u221an)(\u03c3\/n), as stated by the Central Limit Theorem.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Stay tuned for the next installment, in which Ashutosh will go over the code for binomially distributed population<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Visit QuantInsti for additional details and to download full code:&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/central-limit-theorem\/\">https:\/\/blog.quantinsti.com\/central-limit-theorem\/<\/a><\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><small>Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.<\/small><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ashutosh Dave demonstrates how to do exponential sampling in Python and offers ready-to-use code.<\/p>\n","protected":false},"author":295,"featured_media":19357,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,343,349,338,350,341,344],"tags":[806,4941,865,595],"contributors-categories":[13654],"class_list":["post-82389","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-programing-languages","category-python-development","category-ibkr-quant-news","category-quant-asia-pacific","category-quant-development","category-quant-regions","tag-data-science","tag-financial-mathematics","tag-github","tag-python","contributors-categories-quantinsti"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v28.0) - 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