Lecture 8: The central limit theorem

BIOS 600 - Spring 2026

Announcements

  • HW 3 due Thursday 2/12 at 11:59pm. You’ll get feedback on HW3 by Monday 2/16.

  • Lab 4 due Friday 2/13 at 11:59pm. Feel free to use lab time to ask about lab or exam prep.

  • Thursday 2/12 class will be a review session.

  • Tuesday 2/17: Exam 01 in-class. All you need is something to write with.

  • Please email me if you have an accommodation letter for extra time on the exam, even if we’ve talked about it already.

  • Today’s lecture not on exam 1.

Lab 3

  • Good job across the board
  • Be sure to double check answering all parts of the question
  • Go from this:
ggplot(data = cdc, aes(x = Region, y = InfantMortalityRate)) + 
  geom_boxplot() + labs(title = "Infant Mortality Rate (%) by Region", xlab = "Region", ylab = "Infant Mortality Rate (%)")

to this:

ggplot(data = cdc, aes(x = Region, 
                       y = InfantMortalityRate)) + 
  geom_boxplot() +
  labs(title = "Infant Mortality Rate (%) by Region", 
       xlab = "Region", 
       ylab = "Infant Mortality Rate (%)")

Overview

  • Central Limit Theorem

  • Application Exercise

  • Why is the Central Limit Theorem important?

Reading

A roadmap ahead

  • In just a few more lectures, we’ll have learned enough tools to perform a pretty wide range of analyses and answer associated questions.

  • In the latter half of this course, we’ll emphasize specific scientific questions of interest, statistical methods for testing such questions, and translating those results to real-world conclusions.

A roadmap for today

  • Today’s topic is fundamental to making sound statistical inferences and conclusions.

  • The basic idea is called the central limit theorem, which states that for any distribution with a well-defined mean and variance, the distribution of the means computed from samples of size n will be approximately Gaussian.

Recall: What is statistical inference?

  • Statistical inference is the act of generalizing from a sample in order to make conclusions regarding a population while quantifying the degree of certainty we have.

  • We are interested in population parameters, which we do not observe.

  • Instead, we must calculate statistics from our sample in order to learn about the parameters.

The sampling distribution of the mean

Suppose we’re interested in the resting heart rate of students in BIOS 600, and are able to do the following:

  1. Take a random sample of size n from this population
  • Calculate the mean resting heart rate in this sample, \(\bar x_1\)
  1. Put the sample back, take a second random sample of size n
  • Calculate the mean resting heart rate from this new sample, \(\bar x_2\)
  1. Put the sample back, take a third random sample of size n
  • Calculate the mean resting heart rate from this sample, too…

… and so on

The sampling distribution of the mean

After repeating this many times, we have a dataset that has the sample averages from the population: \(\{\bar x_1, \bar x_2, \ldots \}\)

  • Can we say anything about the distribution of these sample means?

The central limit theorem

The central limit theorem states that for a population with mean \(\mu\) and standard deviation \(\sigma\), these three properties hold for the distribution of sample averages \(\bar X\):

  1. The mean of the sampling distribution is identical to the population mean \(\mu\)

  2. The standard deviation of the distribution of the sample averages is \(\sigma / \sqrt{n}\), or the standard error (SE) of the mean

  3. For n large enough (in the limit, as \(n \rightarrow \infty\)), the shape of the sampling distribution of means is approximately normal

What is this really saying?

  • The central limit theorem tells us that sample averages are normally distributed, if we have enough data.

  • This is true even if our original variables are not normally distributed.

Application Exercise

Application Exercise

For today’s participation, go to the course website and find instructions and template for AE 02.

AE 02 is due on Canvas on Friday 2/20 at 11:59pm.

Why the CLT matters

  • In many situations, we can base statistical inference on the N(0,1) distribution
  • Since there are few assumptions on the underlying distribution, this applies to many types of statistical estimates
  • Greatly simplifies statistical analysis

Example: IQ tests

  • IQ tests are designed to have a probability distribution with \(\mu\) = 100 and \(\sigma\) = 15.

  • Suppose we draw samples of size \(n\) = 20 from this population.

  • From the central limit theorem, the distribution of the sample averages will be approximately normal with mean 100 and standard deviation \(15/\sqrt{20}\)

IQ tests

  • If the population distribution is normal to begin with, then the distribution of the sample averages will also be exactly normal.

  • If the population distribution is not normal, then the rule of thumb is that we need at least \(n = 30\) for the central limit theorem to kick in for approximate normality.

Example

  • Suppose I give a random sample of \(n = 30\) BIOS 600 students an IQ test*, and the sample average score is 120.

  • Does this mean that BIOS 600 students are smarter than average?

  • I know there are many problems with IQ and IQ testing…bear with me here!

Example

  • The central limit theorem tells us that the distribution of means of samples of size 30 from this population is also normal, with mean \(\mu = 100\) and \(SE = \sigma/ \sqrt{n} = 15 \sqrt{30} \approx 2.7\).

  • \(Z = \frac{\bar{X} - \mu}{SE}\) is a standard normal random variable, and here \(Z \approx 7.3\).

  • The probability of a z-score greater than this is extremely small.

  • What does this mean? (p-values coming soon…)

  • What are the upper and lower limits that enclose 95% of the means for samples of size \(n\) drawn from the population? (Confidence intervals coming soon…)

Recap

  • The central limit theorem is the foundation of statistical inference.

  • The Central Limit Theorem tells us that when you take the average of a large enough sample, that average will follow a normal distribution, even if the original data don’t.

  • This is important because many statistical methods, like confidence intervals and hypothesis tests, rely on normal distributions.

  • Thanks to the CLT, we can apply these methods to real-world data, even if the data itself isn’t normally distributed, as long as the sample size is large enough.

Next class

  • Review for Exam 01