Central Limit Theorem formula proof and solved examples
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 Central Limit Theorem Explained in Statistics
1. What is the Central Limit Theorem?
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the population’s original distribution. In simple terms, even if the population is not normal, the distribution of sample means becomes approximately normal when n is large (typically n ≥ 30).
- The population can be skewed or uniform.
- The samples must be independent and identically distributed.
- The theorem applies to sample means (and sums).
2. What is the formula for the Central Limit Theorem?
The key formula in the Central Limit Theorem standardizes the sample mean as Z = (X̄ − μ) / (σ / √n). Here:
- X̄ = sample mean
- μ = population mean
- σ = population standard deviation
- n = sample size
3. Why is the Central Limit Theorem important?
The Central Limit Theorem is important because it allows us to use the normal distribution to make inferences about population means, even when the population is not normally distributed. This makes statistical methods like:
- Confidence intervals
- Hypothesis testing
- Prediction and estimation
4. What conditions are required for the Central Limit Theorem to apply?
The Central Limit Theorem applies when certain key conditions are satisfied. The main requirements are:
- Samples are independent.
- Observations come from the same distribution.
- The sample size is sufficiently large (commonly n ≥ 30).
5. What does the Central Limit Theorem say about sample size?
The Central Limit Theorem states that as the sample size (n) increases, the sampling distribution of the mean becomes more normal. In practice:
- For small samples, the distribution may not look normal.
- For n ≥ 30, the distribution is usually approximately normal.
- Larger sample sizes reduce the standard error (σ/√n).
6. Can you give an example of the Central Limit Theorem?
Yes, the Central Limit Theorem can be illustrated with a simple example involving sample means. Suppose:
- Population mean μ = 50
- Population standard deviation σ = 10
- Sample size n = 25
- Mean = 50
- Standard deviation = 2
7. What is the difference between population distribution and sampling distribution in CLT?
The population distribution describes individual data values, while the sampling distribution describes the distribution of sample means. Key differences include:
- Population distribution: Shape can be normal, skewed, uniform, etc.
- Sampling distribution: Becomes approximately normal as n increases (by CLT).
- Sampling distribution has mean μ and standard deviation σ/√n.
8. What is the standard error in the Central Limit Theorem?
The standard error is the standard deviation of the sampling distribution of the mean and is given by σ / √n. It measures how much the sample mean varies from the population mean.
- As n increases, the standard error decreases.
- Smaller standard error means more precise estimates.
9. Does the Central Limit Theorem apply to non-normal populations?
Yes, the Central Limit Theorem applies to non-normal populations as long as the sample size is sufficiently large. Even if the population is:
- Skewed
- Uniform
- Exponential
10. What are common mistakes when using the Central Limit Theorem?
Common mistakes when applying the Central Limit Theorem include ignoring its assumptions and misinterpreting its scope. Frequent errors are:
- Using small sample sizes when the population is highly skewed.
- Confusing the population distribution with the sampling distribution.
- Forgetting that CLT applies to sample means or sums, not individual values.
- Using σ instead of the standard error (σ/√n) in calculations.
