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BIO 500 GC Discussion Important Central Limit Theorem

BIO 500 GC Discussion Important Central Limit Theorem

Understanding the Central Limit Theorem (CLT)
According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question, as the sample size increases, notwithstanding the actual distribution of the data. In other words, the data is accurate whether the distribution is normal or aberrant.

As a general rule, sample sizes of around 30-50 are deemed sufficient for the CLT to hold,3 meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples one takes, the more the graphed results take the shape of a normal distribution. Note, however, that the central limit theory will still be approximated in many cases for much smaller sample sizes, such as n=8 or n=5.3

The central limit theorem is often used in conjunction with the law of large numbers, which states that the average of the sample means and standard deviations will come closer to equaling the population mean and standard deviation as the sample size grows, which is extremely useful in accurately predicting the characteristics of populations.

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The Central Limit Theorem in Finance
The CLT is useful when examining the returns of an individual stock or broader indices, because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors of all types rely on the CLT to analyze stock returns, construct portfolios, and manage risk.

Say, for example, an investor wishes to analyze the overall return for a stock index that comprises 1,000 equities. In this scenario, that investor may simply study a random sample of stocks to cultivate estimated returns of the total index. To be safe, use at least 30-50 randomly selected stocks across various sectors, should be sampled for the central limit theorem to hold. Furthermore, previously selected stocks must be swapped out with different names to help eliminate bias.

The central limit theorem is a result from probability theory. This theorem shows up in a number of places in the field of statistics. Although the central limit theorem can seem abstract and devoid of any application, this theorem is actually quite important to the practice of statistics.

So what exactly is the importance of the central limit theorem? It all has to do with the distribution of our population. This theorem allows you to simplify problems in statistics by allowing you to work with a distribution that is approximately normal.

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Statement of the Theorem
The statement of the central limit theorem can seem quite technical but can be understood if we think through the following steps. We begin with a simple random sample with n individuals from a population of interest. From this sample, we can easily form a sample mean that corresponds to the mean of what measurement we are curious about in our population.

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BIO 500 GC Discussion Important Central Limit Theorem

BIO 500 GC Discussion Important Central Limit Theorem

A sampling distribution for the sample mean is produced by repeatedly selecting simple random samples from the same population and of the same size, and then computing the sample mean for each of these samples. These samples are to be thought of as being independent of one another.

The central limit theorem concerns the sampling distribution of the sample means. We may ask about the overall shape of the sampling distribution. The central limit theorem says that this sampling distribution is approximately normal—commonly known as a bell curve. This approximation improves as we increase the size of the simple random samples that are used to produce the sampling distribution.

There is a very surprising feature concerning the central limit theorem. The astonishing fact is that this theorem says that a normal distribution arises regardless of the initial distribution. Even if our population has a skewed distribution, which occurs when we examine things such as incomes or people’s weights, a sampling distribution for a sample with a sufficiently large sample size will be normal.

Central Limit Theorem in Practice
The unexpected appearance of a normal distribution from a population distribution that is skewed (even quite heavily skewed) has some very important applications in statistical practice. Many practices in statistics, such as those involving hypothesis testing or confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statistics course is that the populations that we work with are normally distributed.

The assumption that data is from a normal distribution simplifies matters but seems a little unrealistic. Just a little work with some real-world data shows that outliers, ​skewness, multiple peaks and asymmetry show up quite routinely. We can get around the problem of data from a population that is not normal. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.

Thus, even though we might not know the shape of the distribution where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal. Of course, in order for the conclusions of the theorem to hold, we do need a sample size that is large enough. Exploratory data analysis can help us to determine how large of a sample is necessary for a given situation.

 

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