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Class notes Statistics computer science Applied Statistics Using SPSS, Statistica, Matlab and R
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STA1510 ASSIGNMENT 3 SEMESTER 2 2023
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Central limit theoremCentral limit theorem is a statistical theory which states that when the large sample size has a finite
variance, the samples will be normally distributed and the mean of samples will be approximately equal
to the mean of the whole population.
In other words, the central limit theorem states that for any population with mean and standard deviation,
the distribution of the sample mean for sample size N has mean μ and standard deviation σ / √n .
As the sample size gets bigger and bigger, the mean of the sample will get closer to the actual population
mean. If the sample size is small, the actual distribution of the data may or may not be normal, but as the
sample size gets bigger, it can be approximated by a normal distribution. This statistical theory is useful in
simplifying analysis while dealing with stock indexes and many more.
The CLT can be applied to almost all types of probability distributions. But there are some exceptions. For
example, if the population has a finite variance. Also, this theorem applies to independent, identically
distributed variables. It can also be used to answer the question of how big a sample you want. Remember
that as the sample size grows, the standard deviation of the sample average falls because it is the
population standard deviation divided by the square root of the sample size. This theorem is an important
topic in statistics. In many real-time applications, a certain random variable of interest is a sum of a large
number of independent random variables. In these situations, we can use the CLT to justify using the
normal distribution.
Central Limit Theorem Statement
The central limit theorem states that whenever a random sample of size n is taken from any distribution
with mean and variance, then the sample mean will be approximately normally distributed with mean and
variance. The larger the value of the sample size, the better the approximation to the normal.
Assumptions of Central Limit Theorem
The sample should be drawn randomly following the condition of randomization.
The samples drawn should be independent of each other. They should not influence the other
samples.
When the sampling is done without replacement, the sample size shouldn’t exceed 10% of the
total population.
The sample size should be sufficiently large.
, The formula for the central limit theorem is given below:
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