Central Limit Theorem Reference Cheat Sheet

A printable reference covering the Central Limit Theorem, sampling distributions, standard error, z-scores, and normal approximation for grades 10-12.

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The Central Limit Theorem explains why normal curves appear so often in statistics, even when the original population is not normal. This cheat sheet helps students recognize when sample means or sample proportions can be modeled with an approximately normal distribution. It is useful for solving probability problems, checking conditions, and preparing for inference topics like confidence intervals and hypothesis tests.

Key Facts

For sample means, the Central Limit Theorem says that if $n$ is large enough, the sampling distribution of $\bar{x}$ is approximately normal with mean $\mu_{\bar{x}} = \mu$ and standard deviation $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ .
The standard error of the sample mean is $SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ , which measures the typical distance between $\bar{x}$ and $\mu$ .
A common rule of thumb is that the sampling distribution of $\bar{x}$ is approximately normal when $n \ge 30$ , especially if the population is not strongly skewed.
If the original population is normal, then the sampling distribution of $\bar{x}$ is normal for any sample size $n$ .
For sample proportions, the sampling distribution of $\hat{p}$ is approximately normal with mean $\mu_{\hat{p}} = p$ and standard error $SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ when $np \ge 10$ and $n(1-p) \ge 10$ .
To standardize a sample mean, use $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ when the population standard deviation $\sigma$ is known.
Larger sample sizes make the standard error smaller because $\frac{\sigma}{\sqrt{n}}$ decreases as $n$ increases.
The Central Limit Theorem describes the distribution of sample statistics, not the shape of the original population data.

Vocabulary

Central Limit Theorem: A theorem stating that the sampling distribution of a mean or proportion becomes approximately normal as the sample size becomes large enough.
Sampling Distribution: The probability distribution of a statistic, such as $\bar{x}$ or $\hat{p}$ , calculated from many samples of the same size.
Sample Mean: The average value from a sample, written as $\bar{x}$ .
Standard Error: The standard deviation of a sampling distribution, such as $SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ for sample means.
Sample Proportion: The fraction of a sample with a certain characteristic, written as $\hat{p}$ .
Normal Approximation: The use of a normal distribution to estimate probabilities for a sampling distribution when the required conditions are met.

Common Mistakes to Avoid

Confusing the population distribution with the sampling distribution is wrong because the Central Limit Theorem describes the behavior of statistics like $\bar{x}$ , not the original data values.
Using $\sigma$ instead of $\frac{\sigma}{\sqrt{n}}$ for sample mean problems is wrong because sample means vary less than individual observations.
Assuming $n \ge 30$ always guarantees accuracy is wrong because strong skewness or extreme outliers may require a larger sample size.
Forgetting the success-failure condition for proportions is wrong because $\hat{p}$ is not safely normal unless $np \ge 10$ and $n(1-p) \ge 10$ .
Thinking a larger sample size changes the mean of the sampling distribution is wrong because $\mu_{\bar{x}} = \mu$ stays the same while $SE_{\bar{x}}$ gets smaller.

Practice Questions

1 A population has mean $\mu = 80$ and standard deviation $\sigma = 12$ . For samples of size $n = 36$ , find $\mu_{\bar{x}}$ and $SE_{\bar{x}}$ .
2 A population has $p = 0.40$ and sample size $n = 100$ . Check whether the normal approximation for $\hat{p}$ is reasonable, then find $SE_{\hat{p}}$ .
3 A population has mean $\mu = 50$ and standard deviation $\sigma = 10$ . For $n = 25$ , calculate the z-score for a sample mean of $\bar{x} = 54$ using $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ .
4 Explain why increasing the sample size makes the sampling distribution of $\bar{x}$ narrower but does not change its center.

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