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Central Limit Theorem Probability:
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The Central Limit Theorem (CLT) states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough. This calculator computes probabilities based on this normal approximation.
The calculator uses the normal approximation formula:
Where:
Explanation: The calculator converts your problem to a standard normal distribution using z-scores, then calculates the probability using the normal CDF.
Details: The CLT is fundamental in statistics as it allows us to make inferences about population parameters using sample statistics, even when the population distribution is not normal.
Tips: Enter the population mean and standard deviation, sample size, and your value(s) of interest. Select the type of probability calculation you need (less than, greater than, or between two values).
Q1: How large should my sample size be for CLT?
A: Typically n ≥ 30 is considered sufficient, but this depends on how non-normal your population distribution is.
Q2: What if my population isn't normally distributed?
A: The CLT still applies as long as sample sizes are large enough. For very skewed distributions, larger samples may be needed.
Q3: What's the difference between σ and σ/√n?
A: σ is population standard deviation, while σ/√n is the standard error of the mean (standard deviation of the sampling distribution).
Q4: When shouldn't I use this approximation?
A: For small samples from non-normal populations, exact methods or non-parametric tests may be more appropriate.
Q5: How accurate is the normal approximation?
A: For large samples, very accurate. For smaller samples, consider t-distribution for means when population σ is unknown.