• The Central Limit Theorem (CLT) is a key Statistics concept.
• It states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal.
• This property allows us to use the normal distribution to approximate probabilities

• Conditions to use CLT:

  1) Samples are random and independent.

  2) The sample size (n) is sufficiently large (n ≥ 30).

\[ \bar{X}_n \approx N\left(\mu, \frac{\sigma^2}{n} \right) \]

• Where:
  • \(\mu\) is the population mean
  • \(\sigma^2\) is the population variance
  • \(n\) is the sample size
• Essentially, the sample mean will be approximately normally distributed.

• Defined by mean \(\mu\) and standard deviation \(\sigma\).
• Probability density function:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

• Using standard deviations:
  • \(68\%\) within \(\mu \pm \sigma\)
  • \(95\%\) within \(\mu \pm 2\sigma\)
  • \(99.7\%\) within \(\mu \pm 3\sigma\)

• Let’s start with a non-normal population distribution. Below is an Exponential Distribution.

• Now, we’ll take random samples from this non-normal distribution and compute their means. Notice how as sample size increases, they become more and more normal.

• First will be a sample with n=5.

• Next will be a sample with n=100.

ggplot(df5, aes(x = mean)) +
  geom_histogram() +
  labs(title="Sample Size 5 (<=30)",
       x = "Sample Mean",
       y = "Frequency")

ggplot(df100, aes(x = mean)) +
  geom_histogram() +
  labs(title="Sample Size 100 (>=30)",
       x = "Sample Mean",
       y = "Frequency")

• Next, we’ll look at a plot that displays how the density of a sample mean distribution increases as sample size (n) increases.

• The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches normality as sample size increases, regardless of the original population’s shape.
• The CLT’s main use is that it allows us to use normal probability models, even when starting with non-normal populations.