"The central limit theorem (CLT) establishes that, in many situations, for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed."
A theorem which states that the sampling distribution of the sample means approaches a normal distribution.
Probability: Basic understanding of probability is essential to understand Central Limit Theorem. It is a measure of the likelihood of an event occurring.
Statistical Distributions: Familiarity with statistical distributions such as binomial, Poisson, uniform, exponential, normal, etc., is critical as these distributions play a significant role in Central Limit Theorem.
Sampling: Sampling is the process of selecting data from a larger population. The size of the sample and the method of sampling are essential in determining the characteristics of the sample.
Frequency Distribution: Frequency distribution is a table that displays the number of occurrences of a particular characteristic or value.
Standard Deviation: Standard deviation is a measure of the variability of a set of data.
Mean: Mean is the average of a set of data and is calculated by adding all the numbers together and then dividing by the number of data points.
Hypothesis Testing: Hypothesis testing involves making a statistical inference about a population based on sample data.
Confidence Intervals: Confidence intervals are used in estimation to determine the range of values that are likely to include the population value with a certain degree of confidence.
Normal Distribution: Normal distribution is a continuous probability distribution with a bell-shaped curve.
Law of Large Numbers: The law of large numbers states that as the sample size increases, the sample mean will approach the population mean.
Central Limit Theorem: Central Limit Theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, provided that the sample size is large enough.
Skewed Distribution: Skewed distribution is a probability distribution that is not symmetrical and has a long tail on one side.
Non-Normal Distribution: Non-normal distribution is a probability distribution that does not follow a normal distribution.
Z-score: Z-score is a standardized value that shows how many standard deviations a data point is away from the mean.
Margin of Error: Margin of error is the amount of error expected in the results of a statistical survey or experiment.
"The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions."
"...this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory."
"Let X₁, X₂, ..., Xₙ denote a random sample of n independent observations from a population with overall expected value (average) μ and finite variance σ², and let X̅ₙ denote the sample mean of that sample (which is itself a random variable). Then the limit as n → ∞ of the distribution of (X̅ₙ - μ) / σₙ, where σₙ = σ / √(n), is the standard normal distribution."
"The limit as n → ∞ of the distribution of (X̅ₙ - μ) / σₙ represents where the probability distribution of these averages will closely approximate a normal distribution."
"In its common form, the random variables must be independent and identically distributed (i.i.d.)."
"Convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions."
"The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem."
"for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed."
"The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions."
"...this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory."
"X̅ₙ denote the sample mean of that sample (which is itself a random variable)."
"If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution."
"Convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions."
"σₙ = σ / √(n)"
"The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem."
"μ denotes the overall expected value (average) of the population."
"σ² denotes the finite variance of the population."
"If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution."
"The central limit theorem establishes that, in many situations, for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed."