Central Limit Theorem for Proportions
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
1. What is the Central Limit Theorem (CLT) in simple terms?
The Central Limit Theorem is a fundamental principle in statistics which states that if you take a sufficiently large number of samples from a population, the distribution of the sample means will be approximately a normal distribution (a bell-shaped curve). This holds true regardless of the original shape of the population's distribution.
2. What is the formula for the Central Limit Theorem and what do its components represent?
The Central Limit Theorem provides formulas to describe the distribution of sample means. The key formulas are:
- Mean of the Sample Means (μₓ) = μ (The mean of all sample means is equal to the population mean).
- Standard Deviation of the Sample Means (σₓ) = σ/√n (This is also known as the standard error).
Here, μ is the population mean, σ is the population standard deviation, and n is the sample size. The formula shows that as the sample size 'n' increases, the standard error decreases.
3. What are the key conditions required for the Central Limit Theorem to apply?
For the Central Limit Theorem to be valid, three main conditions must be met:
- Randomization: The samples must be drawn from the population in a random manner to avoid bias.
- Independence: The sample values must be independent of each other. The selection of one individual should not influence the selection of another.
- Sample Size: The sample size should be 'sufficiently large'. A common rule of thumb is a sample size (n) of 30 or greater. However, if the original population is already close to normal, smaller samples may suffice.
4. Why is the Central Limit Theorem so important in statistics?
The Central Limit Theorem is crucial because it allows us to use normal distribution probabilities to make inferences about a population mean, even if the population's own distribution is not normal or is unknown. It forms the foundation for many statistical procedures, such as constructing confidence intervals and conducting hypothesis tests, which are essential for data analysis and research.
5. Can you provide a real-world example of the Central Limit Theorem?
Imagine an e-commerce company wants to find the average weight of all packages it ships. It would be impossible to weigh every single package. Instead, they can use the CLT. They can take multiple random samples of, say, 50 packages each and calculate the average weight for each sample. According to the CLT, the distribution of these sample averages will form a bell curve around the true average weight of all packages, allowing the company to make a very accurate estimate.
6. Does the Central Limit Theorem state that the original data becomes normally distributed?
No, this is a common misconception. The Central Limit Theorem does not change the distribution of the original population data. It specifically describes the distribution of the sample means. The original population's distribution can be skewed, uniform, or any other shape, but the distribution of the means of its large samples will still approximate a normal distribution.
7. How does increasing the sample size affect the outcome of the Central Limit Theorem?
Increasing the sample size (n) has two significant effects:
- The distribution of the sample means becomes a closer approximation of a perfect normal distribution.
- The standard error (σ/√n) decreases. This means the sample means will be clustered more tightly around the true population mean, leading to a more precise and reliable estimate of the population mean.
