What is the Central limit theorem?

The Central Limit theorem states that the distributions of the sample mean of the identically, independent, randomly selected distributions from any population, converge towards the normal distributions (a bell-shaped curve) as the sample size increases, even if the population distribution is not normally distributed.

Or

Let be independent and identically distributed (i.i.d.) random variables drawn from the populations with the common mean and variance . Then, as the sample size n approaches infinity i.e. , the sampling distribution of the sample mean will converge to a normal distribution with mean and variance .

Assumptions for Central Limit Theorem

The key assumptions for the Central Limit Theorem (CLT) are as follows:

• Independence: The random variables in the sample must be independent of each other.
• Identical Distribution: The random variables are identically distributed means each observation is drawn from the same probability distribution with the same mean and the same variance .
• Sample Size: Sample Size (n) should be sufficiently large, typically , for the Central Limit Theorem CLT to provide accurate approximations.

Properties of Central limit theorem

Some of the key properties of the CLT are as follows:

• Regardless of the shape of the population distribution, the sampling distribution of the sample mean approaches a normal distribution as the sample size n increases.
  
• As the sample size n increases, The sampling distribution’s variance will be equal to the variance of the population distribution divided by the sample size:
  

Applications of Central Limit Theorem

• The CLT is widely used in scenarios where the characteristics of the population have to be identified but analysing the complete population is difficult.
• In data science, the CLT is used to make accurate assumptions of the population in order to build a robust statistical model.
• In applied machine learning, the CLT helps to make inferences about the model performance.
• In statistical hypothesis testing the central limit theorem is used to check if the given sample belongs to a designated population.
• CLT is used in calculating the mean family income in a particular country.
• The concept of the CLT is used in election polls to estimate the percentage of people supporting a particular candidate as confidence intervals.
• The CLT is used in rolling many identical, unbiased dice.
• Flipping many coins will result in a normal distribution for the total number of heads or, equivalently, total number of tails.

The Central Limit Theorem (CLT) is like a special rule in statistics. It says that if you gather a bunch of data and calculate the average, even if the original data doesn’t look like a neat bell-shaped curve, the averages of those groups will start to look like one if you have enough data.