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| Meta Title | Basics Of Central Limit Theorem. The central limit theorem states that… | by Dhaval Raval | Medium |
| Meta Description | Basics Of Central Limit Theorem The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the … |
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May 29, 2023
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The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the population isn’t normally distributed.
The central limit theorem relies on the concept of a sampling distribution, which is the probability distribution of a statistic for a large number of samples taken from a population.
Imagining an experiment may help you to understand sampling distributions:
· Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean.
· Now you draw another random sample of the same size, and again calculate the mean.
· You repeat this process many times, and end up with a large number of means, one for each sample.
The distribution of the sample means is an example of a sampling distribution.
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.
Sample Size and the Central Limit Theorem: -
The sample size
(n)
is the number of observations drawn from the population for each sample. The sample size is the same for all samples.
The sample size affects the sampling distribution of the mean in two ways.
1. Sample size and normality: -
The larger the sample size, the more closely the sampling distribution will follow a normal distribution.
When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”
By convention, we consider a sample size of 30 to be “sufficiently large.”
· When n < 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.
· When n ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.
2. Sample size and standard deviations: -
The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is).
· When n is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.
· When n is high, the standard deviation is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.
Conditions of the central limit theorem: -
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Photo by
MedicAlert UK
on
Unsplash
The central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions:
1. The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30.
2. The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the sampling is random.
3. The population’s distribution has finite variance. Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.
Example of Central Limit Theorem: -
Suppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this:
Press enter or click to view image in full size
Image source: — Google Image
Solution:
— Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.
Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired.
Example: sample of n = 5
68, 73, 70, 62, 63
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.
So,
Mean = (68 + 73 + 70 + 62 + 63) / 5
Mean = 67.2 years.
Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean.
Means of small samples;
60.8, 57.8, 62.2, 68.6, 67.4, 67.8, 68.3, 65.6, 66.5, 62.1
If you repeat the procedure many more times, a histogram of the sample means will look something like this:
Press enter or click to view image in full size
Image source: — Google Image
Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew.
Notice also that the spread of the sampling distribution is less than the spread of the population.
This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.
Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.
So, sample of n = 50
73, 49, 62, …, 64, 68 (till 50 retirees randomly)
The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large.
Mean of a large sample,
Mean = 64.8 years.
Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:
Press enter or click to view image in full size
Image source: — Google Image
In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.
Conclusion: -
In conclusion, the Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the sampling distribution of the mean of a large number of independent and identically distributed (i.i.d.) random variables approaches a normal distribution, regardless of the shape of the original population distribution.
Understanding the Central Limit Theorem helps statisticians and data analysts make accurate and reliable inferences about population parameters. It demonstrates the power of sampling and the convergence of sample means to a normal distribution, even when the population distribution may be non-normal. |
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# **Basics Of Central Limit Theorem**
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[Dhaval Raval](https://medium.com/@dhaval.sony.504?source=post_page---byline--f7c4d05ac6ac---------------------------------------)
6 min read
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May 29, 2023
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**The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the population isn’t normally distributed.**
The central limit theorem relies on the concept of a sampling distribution, which is the probability distribution of a statistic for a large number of samples taken from a population.
**Imagining an experiment may help you to understand sampling distributions:**
*· Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean.*
*· Now you draw another random sample of the same size, and again calculate the mean.*
*· You repeat this process many times, and end up with a large number of means, one for each sample.*
The distribution of the sample means is an example of a sampling distribution.
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.
## **Central Limit Theorem formula: -**
We can describe the sampling distribution of the mean using this notation:
![]()
## **Sample Size and the Central Limit Theorem: -**
The sample size *(n)* is the number of observations drawn from the population for each sample. The sample size is the same for all samples.
The sample size affects the sampling distribution of the mean in two ways.
**1\. Sample size and normality: -**
The larger the sample size, the more closely the sampling distribution will follow a normal distribution.
When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”
> ***By convention, we consider a sample size of 30 to be “sufficiently large.”***
*· When n \< 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.*
*· When n ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.*
**2\. Sample size and standard deviations: -**
The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is).
*· When n is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.*
*· When n is high, the standard deviation is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.*
## **Conditions of the central limit theorem: -**
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The central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions:
*1\. The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30.*
*2\. The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the sampling is random.*
*3\. The population’s distribution has finite variance. Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.*
## **Importance of the central limit theorem: -**
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![]()
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- *The central limit theorem is useful because it allows one to assume that the sampling distribution of the mean will be normally — distributed in most cases. This allows easier statistical analysis and inference.*
- *This theorem states that if you take a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from that population will be roughly equal to the population mean.*
- *Central limit theorem helps you balance the time and cost of collecting all the data you need to draw conclusions about the population.*
## **Example of Central Limit Theorem: -**
Suppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this:
Press enter or click to view image in full size
![]()
Image source: — Google Image
*Solution:* — Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.
Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired.
Example: sample of n = 5
*68, 73, 70, 62, 63*
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.
So,
*Mean = (68 + 73 + 70 + 62 + 63) / 5*
*Mean = 67.2 years.*
Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean.
Means of small samples;
*60\.8, 57.8, 62.2, 68.6, 67.4, 67.8, 68.3, 65.6, 66.5, 62.1*
If you repeat the procedure many more times, a histogram of the sample means will look something like this:
Press enter or click to view image in full size
![]()
Image source: — Google Image
Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew.
Notice also that the spread of the sampling distribution is less than the spread of the population.
This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.
Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.
So, sample of n = 50
*73, 49, 62, …, 64, 68 (till 50 retirees randomly)*
The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large.
Mean of a large sample,
*Mean = 64.8 years.*
Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:
Press enter or click to view image in full size
![]()
Image source: — Google Image
In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.
## **Conclusion: -**
![]()
In conclusion, the Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the sampling distribution of the mean of a large number of independent and identically distributed (i.i.d.) random variables approaches a normal distribution, regardless of the shape of the original population distribution.
Understanding the Central Limit Theorem helps statisticians and data analysts make accurate and reliable inferences about population parameters. It demonstrates the power of sampling and the convergence of sample means to a normal distribution, even when the population distribution may be non-normal.
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6 min read May 29, 2023
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Image source: — Google Image
**The central limit theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the population isn’t normally distributed.**
The central limit theorem relies on the concept of a sampling distribution, which is the probability distribution of a statistic for a large number of samples taken from a population.
**Imagining an experiment may help you to understand sampling distributions:**
*· Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean.*
*· Now you draw another random sample of the same size, and again calculate the mean.*
*· You repeat this process many times, and end up with a large number of means, one for each sample.*
The distribution of the sample means is an example of a sampling distribution.
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough. Regardless of whether the population has a normal, Poisson, binomial, or any other distribution, the sampling distribution of the mean will be normal.
## **Sample Size and the Central Limit Theorem: -**
The sample size *(n)* is the number of observations drawn from the population for each sample. The sample size is the same for all samples.
The sample size affects the sampling distribution of the mean in two ways.
**1\. Sample size and normality: -**
The larger the sample size, the more closely the sampling distribution will follow a normal distribution.
When the sample size is small, the sampling distribution of the mean is sometimes non-normal. That’s because the central limit theorem only holds true when the sample size is “sufficiently large.”
> ***By convention, we consider a sample size of 30 to be “sufficiently large.”***
*· When n \< 30, the central limit theorem doesn’t apply. The sampling distribution will follow a similar distribution to the population. Therefore, the sampling distribution will only be normal if the population is normal.*
*· When n ≥ 30, the central limit theorem applies. The sampling distribution will approximately follow a normal distribution.*
**2\. Sample size and standard deviations: -**
The sample size affects the standard deviation of the sampling distribution. Standard deviation is a measure of the variability or spread of the distribution (i.e., how wide or narrow it is).
*· When n is low, the standard deviation is high. There’s a lot of spread in the samples’ means because they aren’t precise estimates of the population’s mean.*
*· When n is high, the standard deviation is low. There’s not much spread in the samples’ means because they’re precise estimates of the population’s mean.*
## **Conditions of the central limit theorem: -**
Press enter or click to view image in full size
Photo by [MedicAlert UK](https://unsplash.com/@medicalertuk?utm_source=medium&utm_medium=referral) on [Unsplash](https://unsplash.com/?utm_source=medium&utm_medium=referral)
The central limit theorem states that the sampling distribution of the mean will always follow a normal distribution under the following conditions:
*1\. The sample size is sufficiently large. This condition is usually met if the sample size is n ≥ 30.*
*2\. The samples are independent and identically distributed (i.i.d.) random variables. This condition is usually met if the sampling is random.*
*3\. The population’s distribution has finite variance. Central limit theorem doesn’t apply to distributions with infinite variance, such as the Cauchy distribution. Most distributions have finite variance.*
## **Example of Central Limit Theorem: -**
Suppose that you’re interested in the age that people retire in the United States. The population is all retired Americans, and the distribution of the population might look something like this:
Press enter or click to view image in full size
Image source: — Google Image
*Solution:* — Age at retirement follows a left-skewed distribution. Most people retire within about five years of the mean retirement age of 65 years. However, there’s a “long tail” of people who retire much younger, such as at 50 or even 40 years old. The population has a standard deviation of 6 years.
Imagine that you take a small sample of the population. You randomly select five retirees and ask them what age they retired.
Example: sample of n = 5
*68, 73, 70, 62, 63*
The mean of the sample is an estimate of the population mean. It might not be a very precise estimate, since the sample size is only 5.
So,
*Mean = (68 + 73 + 70 + 62 + 63) / 5*
*Mean = 67.2 years.*
Suppose that you repeat this procedure 10 times, taking samples of five retirees, and calculating the mean of each sample. This is a sampling distribution of the mean.
Means of small samples;
*60\.8, 57.8, 62.2, 68.6, 67.4, 67.8, 68.3, 65.6, 66.5, 62.1*
If you repeat the procedure many more times, a histogram of the sample means will look something like this:
Press enter or click to view image in full size
Image source: — Google Image
Although this sampling distribution is more normally distributed than the population, it still has a bit of a left skew.
Notice also that the spread of the sampling distribution is less than the spread of the population.
This sampling distribution of the mean isn’t normally distributed because its sample size isn’t sufficiently large.
Now, imagine that you take a large sample of the population. You randomly select 50 retirees and ask them what age they retired.
So, sample of n = 50
*73, 49, 62, …, 64, 68 (till 50 retirees randomly)*
The mean of the sample is an estimate of the population mean. It’s a precise estimate, because the sample size is large.
Mean of a large sample,
*Mean = 64.8 years.*
Again, you can repeat this procedure many more times, taking samples of fifty retirees, and calculating the mean of each sample:
Press enter or click to view image in full size
Image source: — Google Image
In the histogram, you can see that this sampling distribution is normally distributed, as predicted by the central limit theorem.
## **Conclusion: -**
In conclusion, the Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the sampling distribution of the mean of a large number of independent and identically distributed (i.i.d.) random variables approaches a normal distribution, regardless of the shape of the original population distribution.
Understanding the Central Limit Theorem helps statisticians and data analysts make accurate and reliable inferences about population parameters. It demonstrates the power of sampling and the convergence of sample means to a normal distribution, even when the population distribution may be non-normal. |
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