Central Limit Theorem Formula

The Formula

\bar{X} \sim N\left(\mu,\; \frac{\sigma}{\sqrt{n}}\right)

When to use: Roll a single die and the outcomes are flat (uniform). But average the rolls of 30 dice and the result looks like a bell curve every time. No matter how weird the original data looks—skewed, bimodal, flat—the averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.

Quick Example

Population of die rolls: uniform on \{1, 2, 3, 4, 5, 6\}. Take samples of n = 35 rolls and compute \bar{x} each time. \bar{X} \sim N\left(3.5,\; \frac{1.71}{\sqrt{35}}\right) \approx N(3.5,\; 0.289)

Notation

\frac{\sigma}{\sqrt{n}} is called the standard error of the mean.

What This Formula Means

For sufficiently large sample size (n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean \mu and standard deviation \frac{\sigma}{\sqrt{n}}, regardless of the shape of the population distribution.

Roll a single die and the outcomes are flat (uniform). But average the rolls of 30 dice and the result looks like a bell curve every time. No matter how weird the original data looks—skewed, bimodal, flat—the averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.

Formal View

If X_1, \ldots, X_n are i.i.d. with mean \mu and variance \sigma^2, then \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0, 1) as n \to \infty

Worked Examples

Example 1

medium
A highly skewed population (times between bus arrivals) has \mu=15 min and \sigma=8 min. For samples of n=64, describe the shape, mean, and SD of the sampling distribution of \bar{X}, and find P(\bar{X} < 14).

Solution

  1. 1
    CLT: despite skewed population, with n=64 \geq 30, \bar{X} is approximately normally distributed
  2. 2
    Mean: \mu_{\bar{X}} = 15 min; SE: \sigma_{\bar{X}} = \frac{8}{\sqrt{64}} = \frac{8}{8} = 1 min
  3. 3
    \bar{X} \sim N(15, 1)
  4. 4
    P(\bar{X} < 14) = P(Z < \frac{14-15}{1}) = P(Z < -1) = 0.1587

Answer

\bar{X} \sim N(15, 1); P(\bar{X} < 14) \approx 0.159 despite non-normal population.
The CLT's power: even with a skewed population, sample means become normally distributed with large enough n. This allows us to use normal-distribution methods (z-scores, standard tables) for any population shape, which is why CLT is central to statistical inference.

Example 2

hard
A fair die (μ=3.5, σ=1.71) is rolled n=100 times. By CLT, find the approximate probability that the sample mean is between 3.3 and 3.7.

Common Mistakes

  • Thinking n \geq 30 is a hard rule—highly skewed populations may need larger samples for the CLT to kick in.
  • Confusing \sigma (population SD) with \frac{\sigma}{\sqrt{n}} (standard error)—the spread of sample means shrinks by \sqrt{n}.
  • Applying the CLT to individual observations rather than to the sample mean or sample sum.

Why This Formula Matters

Without the CLT, we'd need to know the exact population distribution before doing any inference. The CLT lets us use normal-based methods (z-tests, confidence intervals) even when the population is non-normal.

Frequently Asked Questions

What is the Central Limit Theorem formula?

For sufficiently large sample size (n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean \mu and standard deviation \frac{\sigma}{\sqrt{n}}, regardless of the shape of the population distribution.

How do you use the Central Limit Theorem formula?

Roll a single die and the outcomes are flat (uniform). But average the rolls of 30 dice and the result looks like a bell curve every time. No matter how weird the original data looks—skewed, bimodal, flat—the averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.

What do the symbols mean in the Central Limit Theorem formula?

\frac{\sigma}{\sqrt{n}} is called the standard error of the mean.

Why is the Central Limit Theorem formula important in Math?

Without the CLT, we'd need to know the exact population distribution before doing any inference. The CLT lets us use normal-based methods (z-tests, confidence intervals) even when the population is non-normal.

What do students get wrong about Central Limit Theorem?

The CLT applies to sample means (and sums), not to individual observations. A single data point from a skewed population is still skewed.

What should I learn before the Central Limit Theorem formula?

Before studying the Central Limit Theorem formula, you should understand: sampling distribution, normal distribution.

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