Enter your population mean (μ), population standard deviation (σ), and sample size (n) to compute the sample mean and sample standard deviation using the Central Limit Theorem. Optionally provide a specific sample mean value (x̄) to calculate the Z-score and the probability that a sample mean falls below that value.
Sample Standard Deviation (σx̄)
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Sample Mean (μx̄)
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Z-Score
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P(X̄ ≤ x̄)
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P(X̄ > x̄)
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The Central Limit Theorem states that the sampling distribution of the sample mean approximates a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This holds true for sample sizes of n ≥ 30 in most practical cases.
According to the Central Limit Theorem, the sample mean (μx̄) equals the population mean (μ). So if the population mean is known, the expected value of any sample mean drawn from that population is the same value.
The sample standard deviation (also called the standard error) is calculated as σx̄ = σ / √n, where σ is the population standard deviation and n is the sample size. As sample size increases, the standard error decreases, meaning sample means cluster more tightly around the population mean.
A common rule of thumb is that n ≥ 30 is sufficient for the CLT to apply, producing a roughly normal sampling distribution. For populations that are already normally distributed, even smaller samples work well. Highly skewed populations may require larger sample sizes.
A Z-score measures how many standard errors a specific sample mean value is away from the population mean. It is calculated as Z = (x̄ − μ) / (σ / √n). The Z-score is then used with the standard normal distribution to find the probability of observing a sample mean at or below that value.
Larger sample sizes reduce the standard error, making the sampling distribution narrower and more concentrated around the population mean. This means larger samples give more precise and reliable estimates of the population mean. The CLT guarantees normality of the sampling distribution improves with larger n.
Confidence intervals rely directly on the Central Limit Theorem. Because CLT guarantees a normal sampling distribution for large enough samples, statisticians can use Z-scores to construct intervals that capture the true population mean with a specified level of confidence (e.g. 95%).
The three main conditions are: (1) the samples must be drawn randomly from the population, (2) each observation must be independent, and (3) the sample size should be sufficiently large — typically n ≥ 30. When these conditions are met, the sampling distribution of the mean will be approximately normal.