• The Central Limit Theorem (CLT) is a fundamental concept in statistics that approximates the normal distribution of a sample as the sample size gets larger.

• It explains why sample means tend to follow a normal distribution, even if the population is not normal.

• While the CLT holds true to abnormal sample sizes, it is highly recommended to use samples of 30 or more.

• In real world applications, many sample sizes are abnormal or skewed.

• Applying the CLT allows us to:

  • Use the normal distribution for inference.
  • Build confidence intervals.
  • Conduct hypothesis tests.

• Extremely useful in predicting behaviors of large populations.

ggplot(penguins, aes(x = flipper_length_mm)) +
  geom_histogram(bins = 30, fill = "skyblue", color = "black") +
  ggtitle("Flipper Length Distribution (Original Population)")

The data is abnormal

• Samples must be collected randomly.

• Samples should be independent from one another i.e one sample should not have a participant from another sample.

• The larger the sample size, the more normal the distribution becomes

Let \(X_1, X_2, \ldots, X_n\) be random variables:

Sample mean: \[ \bar{X}_n = \frac{\sum_{i=1}^{n} X_i}{n} \]

Population mean: \[ \mu = \frac{\sum_{i=1}^{N} X_i}{N} \]

• \(\ n\) : Sample size
• \(\ N\) : Population size

As \(n \to \infty\): \[ \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]

• \(\sigma\): Standard Deviation

• \(\bar{X}\) - \(\mu\): The difference between Sample and Population Means

• \(\frac{\sigma}{\sqrt{n}}\) : Sample Standard Deviation

• When the sample mean is standardized the sample follows a normalized distribution.

The distribution of repeated samples approach normality.

• Population vs. Sample Mean: the sample mean takes a portion of a population to calculate the mean whearas a population take the mean of an entire population.

• Works for any population with finite \(\mu\) and \(\sigma\).

• Sample size \(n \geq 30\) is usually sufficient.

  • For very skewed populations, you may need a larger \(n\).
  • Population must have finite variance.

Visualization of the relationship between flipper length, bill length, body mass and species of penguins.