R Sampling - Revision history

Rb: /* Demonstration of the Central Limit Theorem (CLT) */

2016-02-03T09:04:17Z

‎Demonstration of the Central Limit Theorem (CLT)

2016-02-03T09:01:35Z

‎Demonstration of the Central Limit Theorem (CLT)

2016-02-03T08:59:10Z

‎Demonstration of the Central Limit Theorem (CLT)

2016-02-03T08:58:41Z

‎Demonstration of the Central Limit Theorem (CLT)

2016-02-03T08:55:49Z

‎Demonstration of the Law of Large Numbers (LLN)

2016-02-03T08:55:33Z

‎Demonstration of the Law of Large Numbers (LLN)

2016-02-02T23:29:10Z

‎Sampling Examples

2016-02-02T23:28:48Z

‎Sampling Examples

2016-02-02T23:26:41Z

‎Sampling Examples

2016-02-02T23:26:30Z

‎Sampling Examples

@@ Line 76: / Line 76: @@
 <pre class="r">library(ggplot2)  # to make density plots</pre>
-<pre>## Warning: package 'ggplot2' was built under R version 3.1.3</pre>
 To get this idea bedded in, let's draw 1000 samples of size two from our UK population function and then let's look at the resulting distribution.

@@ Line 97: / Line 97: @@
 [[File:Sampling_dist1.jpeg|frame|none]]
-Clearly this is not a normal distribution. Let's see what happens if we increase the sample size to <code>n=10</code>.
+The differences with the normal distribution are actually quite subtle, but especially the left hand tail does not match. In our sample of 2 we get too few sample averages in the tail.. Let's see what happens if we increase the sample size to <code>n=10</code>.
 <pre class="r">nsamples &lt;- 10000  # Number of samples to draw

@@ Line 95: / Line 95: @@
 #Plot.
 ggplot(dat, aes(x = s_means, fill = lines)) + geom_density(alpha = 0.5) + ggtitle(&quot;Sample Size = 2&quot;)</pre>
-[[File:Sampling_files/figure-html/unnamed-chunk-6-1.png|frame|none]]
+[[File:Sampling_dist1.jpeg|frame|none]]
 Clearly this is not a normal distribution. Let's see what happens if we increase the sample size to <code>n=10</code>.
@@ Line 114: / Line 114: @@
 #Plot.
 ggplot(dat, aes(x = s_means, fill = lines)) + geom_density(alpha = 0.5)+ ggtitle(&quot;Sample Size = 10&quot;)</pre>
-[[File:Sampling_dist1.jpeg|frame|none]]
+[[File:Sampling_dist2.jpeg|frame|none]]
 And we are bang on! Here, after only 10 observations the CLT has kicked in and has delivered a sampling distribution for the sample mean that is basically a normal distribution. And that happened despite the distribution from which we sample being very non-normal.
 However, our age distribution had an important feature that makes this convergence to the normal distribution pretty quick (here already after a sample size of 5); it is fairly symmetric.

@@ Line 114: / Line 114: @@
 #Plot.
 ggplot(dat, aes(x = s_means, fill = lines)) + geom_density(alpha = 0.5)+ ggtitle(&quot;Sample Size = 10&quot;)</pre>
-[[File:Sampling_files/figure-html/unnamed-chunk-7-1.png|frame|none]]
+[[File:Sampling_dist1.jpeg|frame|none]]
 And we are bang on! Here, after only 10 observations the CLT has kicked in and has delivered a sampling distribution for the sample mean that is basically a normal distribution. And that happened despite the distribution from which we sample being very non-normal.
 However, our age distribution had an important feature that makes this convergence to the normal distribution pretty quick (here already after a sample size of 5); it is fairly symmetric.

@@ Line 65: / Line 65: @@
 # set up the plot
 plot(x, y, type=&quot;l&quot;, main=&quot;Law of Large Numbers&quot;, xlab = &quot;Sample Size (n)&quot;, ylab = &quot;Sample mean&quot;)</pre>
-[[File:Sampling_LLN.jpeg|frame|none]]
+[[File:Sampling LLN.jpeg|frame|none]]
 As you can see the sample mean can deviate substantially from the population mean for small sample sizes, but as we increase the sample size we will get closer to the population mean of 40. The LLN basically states that we can get arbitrarily close if we just increase the sample size sufficiently.

@@ Line 3: / Line 3: @@
 In this document we demonstrate the workings of the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT). The example we are working with is that of sampling from the population of the UK and asking for the respondent's age. here is an illustration of the UK's age distribution:
-[[File:UKagedistribution.png|frame|none|200px]]
+[[File:UKagedistribution.png|frame|none|100px]]
 As we do not have the actual age data for the entire distribution available (although one could use census data and get pretty close) we will simulate a set 65 Million (app. size of the UK population) ages.