{"id":274,"date":"2007-01-15T20:46:57","date_gmt":"2007-01-15T20:46:57","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/01\/15\/basics-normal-distributions\/"},"modified":"2007-01-15T20:46:57","modified_gmt":"2007-01-15T20:46:57","slug":"basics-normal-distributions","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/01\/15\/basics-normal-distributions\/","title":{"rendered":"Basics: Normal Distributions"},"content":{"rendered":"

In general, when we gather data, we expect to see a particular pattern to
\nthe data, called a normal distribution<\/em>. A normal distribution is one
\nwhere the data is evenly distributed around the mean in a very regular way,
\nwhich when plotted as a
\nhistogram will result in a bell curve<\/em>. There are a lot of ways of
\ndefining “normal distribution” formally, but the simple intuitive idea of it
\nis that in a normal distribution, things tend towards the mean – the closer a
\nvalue is to the mean, the more you’ll see it; and the number of values on
\neither side of the mean at any particular distance are equal.<\/p>\n

<\/p>\n

If you plot that a set of data with a normal distribution
\non a graph, you get something that looks like a bell, with the hump of the
\nbell positioned at the mean.<\/p>\n

\"bell-curve.jpg\"<\/p>\n

For example, here’s a graph that I generated using random numbers. I
\ngenerated 1 million random numbers between 1 and 10; divided them into groups
\nof ten; and then took the sum of each group of 10. The height of each point in
\nthe graph at each x coordinate is the number of times the sum was was that
\nnumber. The mean came out to approximately 55, the mode was 55, and the median
\nwas 55 – which is what you’d hope for in a normal distribution. The number of
\ntimes that 55 occurred was 432,000. 54 came up 427,000 times; 56 came up
\n429,000. 45 came up 245,000 times; 35 came up 38,000 times; and so on. The
\ncloser a value is the the mean, the more often it occurs in the population;
\nthe farther it is from the mean, the less often in occurs.<\/p>\n

In a perfectly normal distribution, you’ll get a perfectly smooth bell
\ncurve. In the real world, we don’t see perfect normal distributions, but most
\nof time in things like surveys, we expect<\/em> to see something close. Of
\ncourse, that’s also the key to how a lot of statistical misrepresentation is
\ncreated – people exploit the expectation that there’ll be a normal
\ndistribution, and either don’t mention, or don’t even check, whether the
\ndistribution is<\/em> normal. If it is not<\/em> normal, then many of
\nthe conclusion that you might want to draw don’t make sense.<\/p>\n

For example, the salary example from the mean, median, and mode<\/a> post is also using this. The reason that the median is so different from the mean is because the distribution is severely skewed away from a normal distribution. (Remember, in a proper normal distribution, the number of values included at the same distance either side of the mean should be equal. But in
\nthis example, the mean was 200,000; if you went plus 100,000, you’d get one value; if you went minus 100,000, you’d also get one – which looks good.
\nBut if you went plus 200,000, you’d still get just one; if you went minus 200,000, you’d get twenty-one values!) But it’s a common rhetorical trick to take a very abnormal distribution, not mention that it’s abnormal, and
\nquote something about the mean in order to support an argument.<\/p>\n

For example, the last round of tax cuts put through by the Bush administration was very<\/em> strongly biased towards wealthy people. But during the last presidential election, in speech after speech, ad after ad, we heard about how much the average<\/em> American taxpayer saved as a result of the tax cuts. In fact, most people didn’t get much; a fair number of people saw an effective increase<\/em> because of the AMT; and a small number of people got huge<\/em> cuts.<\/p>\n

\"bimodal.jpg\"<\/p>\n

For a different example, the high school that I went to in New Jersey was considered one of the best schools in the state for math. But the vast majority of the math teachers there were just horrible – they had three or four really great teachers, and a dozen jackasses who should never have been allowed in front of a classroom. But the top performing math students in the school did so<\/em> well that we significantly raised the mean for the school, making it look as the the typical<\/em> student in the school was good at math. In fact, if you looked at a graph of the distribution of scores, what you would see would be what’s called a bimodal<\/em> distribution: there would be two<\/em> bells side by side – a narrow bell toward the high end of the scores (corresponding to the scores of that small group of students with the great teachers), and a shorter wide bell well to its left, representing the rest of the students.<\/p>\n","protected":false},"excerpt":{"rendered":"

In general, when we gather data, we expect to see a particular pattern to the data, called a normal distribution. A normal distribution is one where the data is evenly distributed around the mean in a very regular way, which when plotted as a histogram will result in a bell curve. There are a lot […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74,61],"tags":[],"class_list":["post-274","post","type-post","status-publish","format-standard","hentry","category-basics","category-statistics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4q","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/274","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=274"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/274\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=274"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=274"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}