The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of the sampling distribution of the sample mean. It is a key principle underlying many statistical techniques and hypothesis-testing procedures.
FORMULA FOR CLT:
Let’s explore some example problems that demonstrate the application of the Central Limit Theorem (CLT) in various scenarios:
Consider that we are given a dataset to find the average age of people purchasing the product, but the dataset has large numbers of data in it. Hence we can use CLT to calculate the average age by using a random sample of data above 30 and at last add the total average mean occured.
The above dataset contains nearly 2000 data.
only select the necessary columns,
the mean of the whole column[‘Age’] can be found using the (mean()) from numpy. Also, it is found that the average mean of age is 48.96.
now using the CLT, we should find a similar average by calculating it from different random sampling.
we have taken the random sampling as 1200 with a sample size of 66. Also, plot this to find the difference.
From the above visualization, we can find that the distribution of the sample mean age is in the form of a bell curve also the mean is 47.68, which is similar to the original mean given.
RULES FOR CLT:
Here are the key rules for the Central Limit Theorem (CLT) in a concise form:
1. Random Sampling: The samples must be selected randomly and independently from the population.
2. Sample Size: As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.
3. Mean and Standard Deviation: The mean of the sampling distribution of the sample means is equal to the population mean, and the standard deviation (standard error) is equal to the population standard deviation divided by the square root of the sample size.
4. Independence: The samples should be selected independently, meaning that the selection of one item does not influence the selection of another.
5. Large Sample Size: The CLT tends to work well with sample sizes of at least 30, but larger sample sizes provide better approximations to normality.
6. Population Shape: The CLT applies regardless of the shape of the population distribution.
These rules highlight the fundamental principles of the Central Limit Theorem, which is a critical concept in statistics for making inferences about populations based on sample data.