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Sampling Sampling: Experiment Show that the distribution of averages from a non-normal distribution tends to follow a normal distribution. Obtained 20 or so uniform random numbers, sum them and compute the average. Compute 100,000 of these averages. Plot a histogram of the averages. What do you see? What you observed is an effect based on the Central Limit Theorem 2 Central Limit Theorem (CLT) The Central Limit Theorem is a statement about the characteristics of the sampling the distribution of means of random samples from a given population. The Central Limit Theorem consists of three statements: [1] The mean of the sampling distribution of means is equal to the mean of the population from which the samples were drawn. [2] The variance of the sampling distribution of means is equal to the variance of the population from which the samples were drawn divided by the size of the samples. [3] If the original population is distributed normally (i.e. it is bell shaped), the sampling distribution of means will also be normal. If the original population is not normally distributed, the sampling distribution of means will increasingly approximate a normal distribution as sample size increases (i.e. when increasingly large samples are drawn). 3 Standard Error (SE) Population π₯3 Samples π₯2 π₯1 π₯4 π₯5 π₯6 4 Compute the average weight (π₯π ) for each rabbit in the sample Standard Error (SE) π₯1 π₯2 π₯3 π₯4 π₯5 π₯6 β¦. 1. Because of the central limit theorem, the averages are normally distributed. 2. The mean of the averages converges to the mean of the population 3. The standard deviation of the average is called the standard error 4. where ππ₯ is the population standard deviation. 5 Central Limit Theorem (CLT) The normal distribution arises repeatedly in biology. Gauss and Laplace noticed that measurement errors tend to follow a normal distribution. Quetelet and Galton observed that the normal distribution fits data on the heights and weights of human and animal populations. This holds true for many other characters as well. SE can be proved by using arguments based on error propagation which we will return to later. 6 Standard Error Question: Assume you sample from a normal distribution. Will the standard error be smaller, the same or larger than the population standard deviation? 7 Standard Error 8 Standard Error The standard error of the mean (SE of the mean) estimates the variability between sample means that you would obtain if you took multiple samples from the same population. The standard error of the mean estimates the variability between samples The standard deviation measures the variability within a single sample. This is much more interesting 9 Standard Error The standard error equation also states that if we want to reduce the error of our estimate of a mean value by a factor of ten, we have to gather one hundred times more data. 10 Standard Error Example Three measurements of gene expression yield the values 1.34, 3.23, and 2.11 Find the mean and standard deviation of these values. Find the standard deviation of the mean Note that the three values represent a sample of expression values. Since we donβt have the population standard deviation we must use the sample standard deviation to estimate the standard error. The large N the better the approximation. 11 Standard Error The cholesterol levels in a sample of 25 men aged 55 to 60 was found to have a mean of 5.2 mmol/L. The standard deviation of the same was 0.6 mmol/L. Compute the standard error of the means from the sample? SE ~ 0.6/square Root (25) = 0.12 mmol/L What does this mean? Form up in pairs. Each pair write a sentence that explains what this means. 12 What you should use in your graphs when plotting data? 13 Common ways to describe variability in data 14