Enter your population mean (μ), population standard deviation (σ), and sample size (n) to calculate the sample mean and sample standard deviation (standard error) using the Central Limit Theorem. Results show how the sampling distribution tightens as your sample grows, along with a breakdown of key CLT outputs.
Sample Mean (μx̄)
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Sample Standard Deviation / Standard Error (σx̄)
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Variance of Sampling Distribution (σ²/n)
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CLT Approximation
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The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This holds true as long as the samples are independent and identically distributed. It is one of the most fundamental results in statistics.
According to the CLT, the sample mean (μx̄) equals the population mean (μ). So if you know the population mean, your sample mean is simply μx̄ = μ. This makes the sample mean an unbiased estimator of the population mean.
The sample standard deviation, also called the standard error, is calculated using the formula σx̄ = σ / √n, where σ is the population standard deviation and n is the sample size. As the sample size increases, the standard error decreases, meaning sample means cluster more tightly around the population mean.
A commonly used rule of thumb is that a sample size of n ≥ 30 is sufficient for the CLT to apply and for the sampling distribution of the mean to be approximately normal. However, if the population is already normally distributed, even smaller samples will work. For heavily skewed populations, a larger sample size may be needed.
The CLT is significant because it allows statisticians to make inferences about population parameters using sample data, even when the population distribution is unknown or non-normal. It underpins many statistical methods including hypothesis testing, confidence intervals, and regression analysis.
Samples can be selected using various methods including simple random sampling (every member has an equal chance), stratified sampling (population divided into subgroups), cluster sampling (random groups are chosen), and systematic sampling (every k-th member is selected). Simple random sampling is generally considered the most accurate method for applying the CLT.
Once you know the sample mean (μx̄ = μ) and standard error (σx̄ = σ/√n), you can standardize to a z-score using z = (x̄ − μ) / (σ/√n) and then use a standard normal distribution table or z-score calculator to find the probability P(X̄ ≤ x). This lets you answer questions like 'what is the probability the sample mean exceeds a certain value?'
The CLT requires that (1) samples are drawn randomly and independently from the population, (2) the sample size is sufficiently large (typically n ≥ 30), and (3) the population has a finite mean and finite variance. When these conditions are met, the sampling distribution of the mean will be approximately normal regardless of the population's shape.