A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.

A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).

All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference.

Could you explain each sentence in simple terms?
user
Statistical modeling is the process of applying statistical analysis to a dataset.

A statistical model is usually specified as a mathematical relationship between random and non-random variables. 

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.
assistant
Great! Now could you re-write this as a bulleted list using shorthand? Try write it like a rushed student's lecture notes.
user
Sure, here you go:

* Stats modeling = applying stats to dataset
* Stat model = math relationship between random/non-random variables
* Stat inference = using data to infer underlying distribution
assistant
Please write these points as sentences. I want each sentence to be as short as is possible while describing the topic. If your sentence is above 20 words, please try to rewrite it. Thank you!
user
Here are simplified, minimal sentences on the topics:

Statistical modeling: Applying statistics to data.
Statistical model: Math relates random and non-random variables.
Statistical inference: Inferring underlying probability distribution from data.

Does this help summarize the key points? Let me know if you would like me to elaborate on any of the sentences or provide additional context.
assistant