Author: Lauren Semester: 2024 Fall B

This presentation will provide an overview of sampling distributions, which is a critical topic in statistics across all scientific fields. In addition, a couple of visualizations of sampling distributions will be provided in the form of graphs.

Most of the examples will pertain to topics in Psychology. Statistics help in designing experiments and studies in psychology by determining sample sizes, selecting appropriate methods, and controlling variables to ensure reliable results.

The sampling distribution of sample means can be described by its shape, its center, and how much it spreads out.

The example below shows a normal distribution, which looks like a bell curve and has one peak in the middle, with two tails that stretch out symmetrically on each side:

## High Sample Size Normal Distribution

The law of large numbers means that the larger our sample size, the more likely our sample mean will accurately reflect the true population mean. In simple terms, it tells us that bigger samples provide more reliable results.

Larger sample sizes create narrower sampling distributions, which increases the probability that a sample mean will be close to the center and decreases the probability that it will be in the tails. See an example below of a normal distribution with a higher sample size:

In statistics, a uniform distribution means every outcome has the same chance of happening. These distributions help predict the likelihood of future events. For example, a coin toss has a uniform distribution since the chances of getting heads or tails are the same.

A uniform distribution has a flat, horizontal line. For a coin flip, the probability of getting heads or tails is 50% (p = 0.50), which would be shown as a straight line at 0.50 on the y-axis of a chart, as shown on the next slide:

A negatively skewed distribution, or left-skewed distribution, is a type of distribution where more values are concentrated on the right side of the graph. This results in the left tail of the graph being longer, indicating that there are fewer values on the lower end of the scale.

The general relationship between the central tendency measures in a negatively skewed distribution can be expressed using the equation: Mode > Median > Mean

A positively skewed distribution, also known as right-skewed, is characterized by most values being concentrated on the left side of the graph, with a long tail extending to the right. This indicates that there are a few extremely high values that pull the mean above the median, creating the skew.

The exponential distribution is often used to visualize the waiting time until a specific event happens.

For instance, in psychology, if a population grows at a constant percentage rate per year, the size of the population over time can be modeled with an exponential growth graph:

Binomial distributions look similar to normal, however the binomial distribution is discrete and best for scenarios with fixed trials and two outcomes, while the normal distribution is continuous and describes data clustering around a mean with a characteristic bell curve.

In the field of psychology, researchers can use binomial distribution to model outcomes (i.e., successes) in a treatment group compared to the control group. They can then perform statistical tests to determine if the treatment is significantly more effective than no treatment:

n <- 1000
p <- 0.3333

probabilities <- dbinom(0:n, size = n, prob = p)

barplot(probabilities, names.arg = 0:n, col = "skyblue", 
        main = "Example of a Binomial Distribution", 
        xlab = "Number of Successes", ylab = "Probability", 
        border = "blue")