Central Limit Theorem formula proof and solved examples
If we talk about the central limit theorem meaning, it means that the mean value of all the samples of a given population is the same as the mean of the population in approximate measures, if the sample size of the population is fairly large and has a finite variation. The central limit theorem is one of the important topics when it comes to statistics. In this article, we will be learning about the central limit theorem standard deviation, the central limit theorem probability, its definition, formula, and examples.
Central Limit Theorem Definition
Let us first define the central limit theorem.
The Central Limit Theorem states that the overall distribution of a given sample mean is approximately the same as the normal distribution when the sample size gets bigger and we assume that all the samples are similar to each other, irrespective of the shape of the total population distribution.
Central Limit Theorem Statistics Example
To understand the Central Limit Theorem better, let us consider the following example.
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Assume that you have 10 different sports teams in your school and each team consists of 100 students. Now, we need to find out the average height of all these students across all the teams. How will we do it when there are so many teams and so many students?
Well, the easiest way in which we can find the average height of all students is by determining the average of all their heights. To do so, we will first need to determine the height of each student and then add them all. Then, we will need to divide the total sum of the heights by the total number of the students and we will get the average height of the students. Well, this method to determine the average is too tedious and involves tiresome calculations. So, how do we calculate the average height of the students? We can do so by using the Central Limit Theorem for making the calculations easy.
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In this method of calculating the average, we will first pick the students randomly from different teams and determine a sample. Every sample would consist of 20 students. Then, we would follow the steps mentioned below:
First, we will take all the samples and determine the mean of each sample individually.
Then, we will determine the mean of these sample means.
This way, we can get the approximate mean height of all the students who are a part of the sports teams.
If we find the histogram of all these sample mean heights, we will obtain a bell-shaped curve.
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Note: It is important to remember that the samples that are taken should be enough by size. When we take a larger sample size, the sample mean distribution becomes normal when we calculate it by repeated sampling.
Central Limit Theorem Formula
Now that we learned how to explain the central limit theorem and saw the example, let us take a look at what is the formula of the Central Limit Theorem.
We can apply the Central Limit Theorem for larger sample size, i.e., when n ≥ 30.
The formula of the Central Limit Theorem is given below.
μx = μ
𝜎x= 𝜎/√n
Here,
μ is the population mean
𝜎 is the standard deviation of the population
μx is the sample mean
𝜎x is the sample standard deviation
n is the sample size
FAQs on Central Limit Theorem Explained for Statistics
1. What is the Central Limit Theorem in simple terms?
The Central Limit Theorem (CLT) states that the distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the population’s original distribution. In simple terms, if you repeatedly take large random samples from any population and compute their means, those means will form a normal distribution.
- Applies to independent and identically distributed random variables.
- Works even if the population is skewed or non-normal.
- Becomes more accurate as sample size (n) increases.
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).
- x̄ = sample mean
- μ = population mean
- σ = population standard deviation
- n = sample size
3. Why is the Central Limit Theorem important in statistics?
The Central Limit Theorem is important because it allows us to use the normal distribution to make inferences about sample means, even when the population is not normal.
- Forms the basis of confidence intervals.
- Justifies hypothesis testing using Z-scores.
- Enables probability calculations for large samples.
4. What conditions are required for the Central Limit Theorem?
The Central Limit Theorem requires independent observations and a sufficiently large sample size.
- Samples must be random and independent.
- The population should have a finite mean (μ) and finite variance (σ²).
- Typically, n ≥ 30 is considered large enough.
5. How does sample size affect the Central Limit Theorem?
As the sample size increases, the sampling distribution of the mean becomes more normally distributed and less spread out.
- Larger n makes the normal approximation more accurate.
- The spread decreases because the standard error = σ / √n.
- Doubling n reduces variability by a factor of √2.
6. What is the difference between the Law of Large Numbers and the Central Limit Theorem?
The Law of Large Numbers states that the sample mean converges to the population mean, while the Central Limit Theorem describes the distribution of the sample mean.
- Law of Large Numbers: x̄ → μ as n increases.
- Central Limit Theorem: The distribution of x̄ becomes approximately normal.
7. What is the sampling distribution of the mean under the Central Limit Theorem?
Under the Central Limit Theorem, the sampling distribution of the mean is approximately normal with mean μ and standard deviation σ/√n.
- Mean of sampling distribution: μ
- Standard deviation (standard error): σ / √n
- Shape: Approximately normal for large n
8. Can you give an example of the Central Limit Theorem?
Yes, for example, if a population has mean μ = 50 and standard deviation σ = 10, then for n = 25, the sampling distribution of the mean is approximately normal with standard error 2.
- Given: μ = 50, σ = 10, n = 25
- Standard error = 10 / √25 = 10 / 5 = 2
- Sampling distribution: Mean = 50, SD = 2
9. Does the Central Limit Theorem apply to small sample sizes?
The Central Limit Theorem works best for large samples, but it can apply to small samples if the population is normally distributed.
- If population is normal, CLT works for any n.
- If population is non-normal, usually n ≥ 30 is recommended.
- Highly skewed data may require even larger samples.
10. What is the standard error in the Central Limit Theorem?
The standard error in the Central Limit Theorem is the standard deviation of the sampling distribution, given by σ / √n.
- σ = population standard deviation
- n = sample size
- Measures variability of sample means
