# Central Limit Theorem

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26 November 2007

Central Limit Theorem

Introduction

When we speak about probability theory and its theorems , the Central Limit Theorem is one of those which occur to our minds . Doubtlessly , the Central Limit Theorem is one of the central theorems in probability theory . Moreover , its uses and applications we notice in our everyday activities without even knowing it . This work will explain what the Central Limit Theorem is , how it is applied and how it is useful within the theoretical framework

of probability theory , statistics and mathematics

The Central Limit Theorem (CLT

The Central Limit theorem consists of the three statements

1 . The mean of the sampling distribution of means is equal to the mean of the population from which the samples are 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 . Of the original population is not normally distributed , the sampling distribution of means will increasingly approximate a normal distribution as sample size increases . That is , when increasingly large samples are drawn

Let 's imagine the situation when we throw a die several times . For example , we throw the die ten times . However , each of us will notice that there will hardly be ones only . What we will see is that there will always be approximately constant amount of all numbers , which will range from one to six . Of course , it is probable that some of us will get different results Fig .1 . Traditional representation of CLT in the form of a histogram (Bellhouse 353 (for example , five sixes , but these results will be relatively rare . If we sum up the results aftermath , the most probable number we will acquire is 30 , or 40 . The minimal number in this case will be equal to 10 , while the maximal number will be 60 . Why do we tend to have 30 or 40 , rather than 10 or 60 ? The reason is simple : the combinations which create 30 or 40 are more numerous than those , which result in 10 or 60 Thus , one can get the middle values in many more different ways than the extremes (Dudley 94 . Consequentially , throwing one die there are equal probabilities that we will get any of six numbers throwing several dice the probability of extreme values is less than of middle ones

CLT finds one of its basic applications in statistics . It is one of the most helpful means of calculating certain parameters within the specific sample (Adams 24 . For example , statistics is used to measure some common characteristics in people to draw practical conclusions and recommendations . CLT seems not to be related to this issue at all however the situation is completely different , and it appears that we will hardly be able to...

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