Central Limit Theorem
The Central Limit Theorem (CLT) is an important topic in mathematics. In this article, we will look at the central limit definition, along with all the major concepts that one needs to know about this topic. The central limit theorem can be explained as the mean of all the given samples of a population. This is an approximation if the sample size is large enough and has finite variation.
The central limit theorem can also be explained as the distribution of a sample mean which approximated the normal distribution. This is applicable when the sample size becomes larger. The assumption here is that all the samples are similar and the shape of the population distribution could possibly be anything.
To understand the central limit theorem better, let’s look at an example. Let’s assume that there are ten teams of cricket in your school. Every team has a total of 100 students in it. Now, if we want to measure the average height of all the students in the sports teams, then that would be a humongous task.
The best way to go about this task is to find the heights of all the students and add the values. After that, the sum of heights can be divided by the total number of students. The answer that we arrive at after this step will be the value of average height. This method will be quite tiresome in this example. But can you imagine how hard and extensive a similar task would be with 2000 people?
Due to this reason, this method is not always possible. Thankfully, one can use the central limit theorem to make his or her calculations easy. In the method of the central limit theorem, the individual would start by randomly picking students from different teams. Every sample will include only 20 students. After that, the individual can follow the steps that are mentioned below.
Take all the samples and try to find the mean for every single sample.
Once that is done, then find the mean of the means of the samples.
The value you receive will be the approximate mean height of all the students in the sports teams.
If you plot this data, then you will get a bell curve shape. Check this fact by plotting a histogram of a large sample of mean heights.
Once you are done with the plotting of the histogram, then the shape of your graph should be similar to the image of the graph that is attached below.
(Image will be uploaded soon)
It is important for readers to remember that the sample one takes for plotting the histogram should be sufficiently large. This is because when the sample size is larger, the sample mean distribution will also show normality. This will happen as one goes about calculating the value by using repeated sampling.
Central Limit Theorem Formula
Till now, we have discussed the meaning of the central limit theorem along with what it dictates. The next is to focus on the central limit theorem formula. But before we discuss the formula, it should be noted that the central limit theorem is valid for a large sample size only (n ≥ 30). The formula for the central limit theorem is mentioned below.
μx = μ
σx = σ / √n
In this formula, μ = population mean
σ = population standard deviation
μx = sample mean
σx = sample standard deviation
n = sample size
The Statement and Assumptions of Central Limit Theorem
When it comes to the central limit theorem, then there is a specific statement that students must memorize. The statement is:
The central limit theorem mentions that whenever a random sample of size n is taken from any particular distribution, with mean and variance, then the sample mean would be the approximate normally distributed with both variance and mean. It should be noted that the larger the value of the sample size, the better quality the approximation to the normal.
There are several assumptions that are made in the central limit theorem. Do you know what those assumptions are? If you don’t, then go through the list of those assumptions that are mentioned below.
The final sample should ideally be drawn in a random manner after the condition of randomization.
The sample that is drawn should also be independent of one another. This means that the sample should not influence other samples.
If the sampling is done without any replacement, then the sample size should not exceed 10% of the total population.
The sample size should also be quite large.
Fun Facts About the Central Limit Theorem Application
Do you know about the central limit importance? Do you know about all the different applications of the central limit theorem? According to experts, there are several applications of the central limit theorem. We have created a table of those applications and that table is mentioned below.
FAQs on What is Central Limit Theorem?
1. What is the Central Limit Theorem (CLT) in simple terms?
The Central Limit Theorem (CLT) is a fundamental principle in statistics which states that if you take a sufficiently large number of random samples from any population, the distribution of the sample means will be approximately a normal distribution (a bell curve), regardless of the original population's distribution. This holds true as long as the sample size is large enough, typically n ≥ 30.
2. What are the key conditions required for the Central Limit Theorem to be valid?
For the Central Limit Theorem to provide a reliable approximation, several conditions must be met:
- Randomisation: The samples must be selected randomly from the population.
- Independence: Each sample drawn must be independent of the others. The selection of one sample should not influence the selection of another.
- Sample Size: The sample size should be 'sufficiently large'. While n ≥ 30 is a common rule of thumb, the necessary size can vary depending on the skewness of the original population's distribution.
- 10% Condition: When sampling without replacement, the sample size (n) should not be more than 10% of the total population size (N) to ensure independence.
3. What is the formula associated with the Central Limit Theorem?
The Central Limit Theorem provides formulas for the mean and standard deviation of the sampling distribution. The formulas are:
- Mean of the sample means (μx) = Population mean (μ)
- Standard deviation of the sample means (σx), also known as the standard error = σ / √n
Here, μ is the population mean, σ is the population standard deviation, and n is the sample size.
4. Can you provide a real-world example of the Central Limit Theorem?
Imagine you want to find the average height of all students in a very large school. Measuring everyone is impractical. Instead, you can use the CLT. You take multiple random samples of 30 students each and calculate the average height for each sample. According to the CLT, if you plot a histogram of these sample averages, the graph will form a bell-shaped curve. The mean of these sample averages will be a very close estimate of the true average height of all students in the school.
5. Why is the Central Limit Theorem so important in statistics?
The Central Limit Theorem is extremely important because it allows us to make inferences about a population using sample data, even if we don't know the shape of the population's distribution. It forms the basis for many statistical procedures, such as hypothesis testing and constructing confidence intervals. It enables statisticians and researchers to use the properties of the normal distribution, which is well-understood, to analyse data from populations that are not normally distributed.
6. Does the original population have to be normally distributed for the Central Limit Theorem to work?
No, and this is one of the most powerful aspects of the theorem. The original population's distribution can be skewed, uniform, binomial, or any other shape. As long as you take sufficiently large random samples, the distribution of the sample means will tend to approximate a normal distribution. The more skewed the original population, the larger the sample size you will need for the approximation to be accurate.
7. How does increasing the sample size affect the outcome of the Central Limit Theorem?
Increasing the sample size (n) has two main effects. First, it makes the approximation of the sampling distribution to a normal distribution more accurate. Second, as seen in the formula for standard error (σ / √n), a larger 'n' decreases the standard deviation of the sample means. This means the sample means will be clustered more tightly around the true population mean, leading to a more precise estimate of the population parameter.
8. What are some practical applications of the Central Limit Theorem in fields like research and polling?
The Central Limit Theorem has numerous applications. In election polling, it's used to estimate the percentage of voters who support a candidate and to calculate the margin of error. In manufacturing, it's used for quality control by analysing samples of products to determine if the entire production line is within acceptable limits. In scientific research, it is used to analyse experimental data and determine if the results are statistically significant.
