Central Limit Theorem (CLT) Definition
The central limit theorem states that with the assumption that all samples are equal in size, the example six gets larger, the distribution of same means approximates that of a normal distribution. In other terms, CLT is a statistical theory that states that given a large sample size from a population that has finite variance level, then all samples’ mean from the same population will approximately equal to the population’s mean.
A Little More on What is Central Limit Theorem
The Central Limit Theorem has an essential component where your sample’s means will also be the mean of the population. It means that if you add together the mean from all of your sample population, you will be able to find the average, which will also be your real population mean. Likewise, if in your sample, you can get all of the standard deviations, then you will be able to find your population’s actual standard deviation. CLT is a valuable phenomenon that can help you to predict a population’s characteristics correctly.
What are the Assumptions Behind the CLT?
CLT, as a technique, has various assumptions. They are as follows:
- The sampling of data must be selected in a random manner
- The samples collected should be independent of each other in that one sample should not influence others.
- When there is a selection of samples without replacement, then all samples should not exceed ten percent of the population.
- The size of the sample has to be large enough. Note that the size of the sample will depend on the population. Where the population happens to be symmetric, then a sample of 30 can work well.
How Central Limit Theorem works
Let’s assume that in a university science department, there are 15 sections where each section hosts close to 100 students. You have the task of calculating the student’s average weight in that science department. In this case, you need to calculate the average. Below is the procedure for calculating the average:
- First, you will have to measure all the student’s weight in the science department
- Secondly, you add all the weights
- Lastly, you will divide the weights’ total sum with the students’ total number to get the average
However, if the data is humongous CLT approach won’t make sense at all. The reason is that it will be tiresome and a long process to measure the weight of all the students. The following is an alternative approach to this.
You first create groups of students selected randomly from the class. Take this to be your sample. You will then draw multiple samples of 30 students each. You will then do the following:
- Calculate these samples’ mean
- Calculate the means of the samples’ mean
- The value will give you students’ approximate mean weights in the science department
Finance and the Central Limit Theorem
CLT can be used to simplify a significant number of analysis procedures. For instance, all types of investors can use it to assess their stock’s returns, manage risk, and construct portfolios. Note that in statistics, statisticians consider CLT to be the basis for sampling. What this means is that it holds the foundation for sampling as well as statistical analysis in finance.
Let’s assume that John wants to analyze his overall stock index return consisting of 1,000 stocks. He can take random stock samples from the index to get an estimate of the total index’s gains. For CTL to hold, John decides to randomly pick the samples and makes sure that he evaluates at least 30 stocks in each sample. For him to avoid bias in the sample selection, he ensures that he replaces the previously selected stocks for selecting in other examples.
Generally, the average returns from the selected samples approximate those of the profits for the whole index, including that of normal distribution. Note that the approximation will still hold even if there is no normal distribution of the entire index’s actual returns.
Central Limit Theorem significance
There various statistical significance and practical applications when you used the central limit theorem.
Statistical significance of CLT
To analyze data, you will have to apply statistical methods such as hypothesis testing as well as constructing confidence intervals. According to this method, there is the assumption that there is a distribution of a population. Where there are non-normal distributions or distributions that are not known, we treat the sampling distribution as normal.
Also, the sample means standard deviation is bound to decrease if you increase the samples taken from the population. With this, you will be able to estimate the population mean with much accuracy. You can also use the sample mean to come up with a range of values called the confidence interval.
Practical Application of Central Limit Theorem
Political analysts can apply CLT on the election or political polls. They use the CLT to calculate political polls. For instance, CLT helps them to estimate the percentage of those who support a particular political party or candidate. You can also use the CLT to calculate confidence intervals such as calculation of mean family income for a specific region.