Essay on Normal, T, Chi-Square, and F Distributions

Understanding various statistical distributions, and the assumptions that underpin those distributions, is foundationial to performing various tests and analyses. Knowledge of the assumptions that play into each distribution allow us to more easily determine the appropriateness of different tests. On the other hand, knowledge of how these distributions are interconnected can allow us to spot patterns we may not have otherwise. By considering several several distributions fundamental to statistical analysis, the normal distribution, t distribution, chi-square distribution, and F distribution we will hopefully come to a greater understanding of their conditions and uses.

The normal distribution plays a crucial role in statistics and is frequently used to both model data perform hypotheses tests. It’s a symmetric bell-shaped distribution in which the values are clustered around the center. In order to be used as a tool to analyze a dataset, the observations within the dataset must be randomly sampled from some population and independent of one another, meaning that the value of one observation should not affect the value of another. It also assumes a population mean of \(\mu\) and variance of \(\sigma^2\). Finally, the distribution of the sample means must follow a normal distribution. This assumption met if either the population the observations are sampled from follows a normal distribution or there are enough observations in the dataset that the Central Limit Theorem can be used. The Central Limit Theorem states that as the sample size (\(n\)) approaches infinity, the distribution of the sample means starts to approximate a normal distribution. In practice this typically means that the Central Limit Theorem is applied to samples greater than 30.

The t distribution is similar to the normal distribution in shape, with the main difference being the flatter and heavier tails indicating that there is a greater probability of more extreme values in relation to the mean. Much like the normal distribution it assumes that the observations must be randomly sampled from some population and independent of one another. However, unlike the normal distribution it does not necessarily assume knowledge of a population’s mean and standard deviation. Finally, the distribution of sample means must follow a normal distribution. The t distribution depends on the parameter \(\nu\), also known as the degrees of freedom, which is equal to one less than the sample size \(n\) and determines the shape of the distribution. Interestingly, as the sample size increases, the t-distribution converges to the standard normal distribution, with the standard normal distribution being a normal distribution where \(\mu=0\) and \(\sigma^2=1\). This is one of many of examples of the interconnectedness between the normal, t, chi-square, and F distributions.

The chi-square distribution is a special case of the gamma distribution which follows a right-skewed distribution. Once again, the observations must be randomly sampled from some population and independent of one another. In cases where the chi-square distribution is being used to test the variance of a population, the population being sampled must follow a normal distribution. Furthermore, the chi-square distribution can be derived from the normal distribution and can be used to represent the distribution of the sum of squares, the sum of squares being the squared differences between individual datapoints and the mean, of standard normal variables.
The normal distribution, t distribution, and chi-square distribution relate quite directly to each other in an interesting way. The t distribution is formed by taking the ratio of the standard normal distribution over the square root of the chi-square distribution divided by its degrees of freedom. This connection between distributions continues with the introduction of the F-distribution.

The F-distribution is also a right-skewed distribution and can be used to analyze the differences between two sample variances. The observations must be randomly sampled from two populations and independent of one another. Furthermore, each sample must come from a normally distributed population. The shape of the F-distribution is dependent on two sets of degrees of freedom, compared to the t-distribution and the chi-square distribution which only rely on one. These degrees of freedom correspond with the numerator and the denominator of F-distribution formula, which is made up of a ratio of two independent chi-square distributions. Similar to the t-distribution, the F-distribution also converges to a standard normal distribution as the degrees of freedom increase.

As been highlighted here, not only are these distributions fundamental to statistical analyses, and serve many purposes in visualizing and testing data, they are also incredibly connected with one another. While understanding this is interesting on its own, it also serves a practical purpose. For instance, knowledge that the chi-square test can be used to represent the distribution of the sum of squares of standard normal variables provides us with an avenue for additional analysis. Overall, it is worth while to consider these distributions due to both the interest and real world implications.

---
title: "Assignment 2: Sampling Distributions"
author: "Grace Lippert"
date: " Due: 2/10/2026 "
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# Essay on Normal, T, Chi-Square, and F Distributions

Understanding various statistical distributions, and the assumptions that underpin those distributions, is foundationial to performing various tests and analyses.  Knowledge of the assumptions that play into each distribution allow us to more easily determine the appropriateness of different tests.  On the other hand, knowledge of how these distributions are interconnected can allow us to spot patterns we may not have otherwise.  By considering several several distributions fundamental to statistical analysis, the normal distribution, t distribution, chi-square distribution, and F distribution we will hopefully come to a greater understanding of their conditions and uses.  

The normal distribution plays a crucial role in statistics and is frequently used to both model data perform hypotheses tests.  It’s a symmetric bell-shaped distribution in which the values are clustered around the center.  In order to be used as a tool to analyze a dataset, the observations within the dataset must be randomly sampled from some population and independent of one another, meaning that the value of one observation should not affect the value of another.  It also assumes a population mean of $\mu$ and variance of $\sigma^2$.  Finally, the distribution of the sample means must follow a normal distribution.  This assumption met if either the population the observations are sampled from follows a normal distribution or there are enough observations in the dataset that the Central Limit Theorem can be used.   The Central Limit Theorem states that as the sample size ($n$) approaches infinity, the distribution of the sample means starts to approximate a normal distribution.  In practice this typically means that the Central Limit Theorem is applied to samples greater than 30. 

The t distribution is similar to the normal distribution in shape, with the main difference being the flatter and heavier tails indicating that there is a greater probability of more extreme values in relation to the mean.  Much like the normal distribution it assumes that the observations must be randomly sampled from some population and independent of one another.  However, unlike the normal distribution it does not necessarily assume knowledge of a population’s mean and standard deviation.  Finally, the distribution of sample means must follow a normal distribution.  The t distribution depends on the parameter $\nu$, also known as the degrees of freedom, which is equal to one less than the sample size $n$ and determines the shape of the distribution.  Interestingly, as the sample size increases, the t-distribution converges to the standard normal distribution, with the standard normal distribution being a normal distribution where $\mu=0$ and $\sigma^2=1$.  This is one of many of examples of the interconnectedness between the normal, t, chi-square, and F distributions. 

The chi-square distribution is a special case of the gamma distribution which follows a right-skewed distribution.  Once again, the observations must be randomly sampled from some population and independent of one another.  In cases where the chi-square distribution is being used to test the variance of a population, the population being sampled must follow a normal distribution.  Furthermore, the chi-square distribution can be derived from the normal distribution and can be used to represent the distribution of the sum of squares, the sum of squares being the squared differences between individual datapoints and the mean, of standard normal variables.  
The normal distribution, t distribution, and chi-square distribution relate quite directly to each other in an interesting way.  The t distribution is formed by taking the ratio of the standard normal distribution over the square root of the chi-square distribution divided by its degrees of freedom.  This connection between distributions continues with the introduction of the F-distribution.  

The F-distribution is also a right-skewed distribution and can be used to analyze the differences between two sample variances.  The observations must be randomly sampled from two populations and independent of one another.  Furthermore, each sample must come from a normally distributed population.  The shape of the F-distribution is dependent on two sets of degrees of freedom, compared to the t-distribution and the chi-square distribution which only rely on one.  These degrees of freedom correspond with the numerator and the denominator of F-distribution formula, which is made up of a ratio of two independent chi-square distributions.  Similar to the t-distribution, the F-distribution also converges to a standard normal distribution as the degrees of freedom increase.

As been highlighted here, not only are these distributions fundamental to statistical analyses, and serve many purposes in visualizing and testing data, they are also incredibly connected with one another.  While understanding this is interesting on its own, it also serves a practical purpose.  For instance, knowledge that the chi-square test can be used to represent the distribution of the sum of squares of standard normal variables provides us with an avenue for additional analysis.  Overall, it is worth while to consider these distributions due to both the interest and real world implications.

# Sources Used:

https://pengdsci.github.io/STA506/w03/03-SamplingDistributions.html

https://www.statology.org/normal-distribution-vs-t-distribution/

https://sixsigmastudyguide.com/chi-square-distribution/



