Studentt #md

Help me understand the Student's t-distribution.

I've studied it but I don't remember it.

Of course, I do want to understand the procedure to use when trying to estimate the variance of a population from a single sample. I understand that this is its main use.

But I mainly want to have a terse, but 100% conceptually crisp, understanding of the key concepts necessary to understand why this procedure is what it is. FOr me to remember things, it is important that they be very coherent. So I care a lot about precision in terminology, e.g., to distinguish what is treated as a random variable, what is a parameter of a distribution, over what range of values an expectation is calculated, etc..

🤖🤖🤖

Of course. Here is a terse, conceptual explanation.

The Core Problem

We want to make inferences about a population's mean (μ). The Central Limit Theorem tells us that if we take many samples, the distribution of their sample means (x̄) will be normal. We can standardize any given x̄ using a Z-statistic:

Z = (x̄ - μ) / (σ / √n)

where:

x̄ is the random variable (it changes with each sample).
μ and σ are fixed parameters of the population.
n is the sample size.

This Z follows a perfect N(0,1) normal distribution. The problem? We almost never know the true population standard deviation, σ.

The Inevitable Substitution

We must substitute σ with our best guess: the sample standard deviation, s.

s = sqrt( Σ(xᵢ - x̄)² / (n - 1) )

This creates a new statistic, the t-statistic:

t = (x̄ - μ) / (s / √n)

The Conceptual Leap

The key insight is this: s is not a fixed parameter like σ. s is also a random variable. If you take a different sample, you get a different x̄ and a different s.

The Z-statistic has only one source of randomness: the numerator (x̄). The t-statistic has two: the numerator (x̄) and the denominator (s). This additional uncertainty from s means the resulting distribution is no longer Normal N(0,1). It has "fatter tails," meaning extreme values are more likely. This new distribution is the Student's t-distribution.

Degrees of Freedom (df)

The t-distribution is not a single curve but a family of curves, indexed by degrees of freedom (df).

Definition: df = n - 1.
Concept: df quantifies how reliable our estimate s is.
- With a small sample (low df), s can vary wildly from the true σ. The t-distribution is wide with fat tails to account for this high uncertainty.
- With a large sample (high df), s becomes a very good estimate of σ. The uncertainty from s diminishes, and the t-distribution converges to the standard normal Z-distribution.

Summary: Conceptual Crispness


Concept	Z-statistic	t-statistic
——————-	————————————————-	———————————————-
Random Variables	`x̄`	`x̄` and `s`
Denominator	Based on a fixed parameter, `σ`	Based on a random variable, `s`
	(The Standard Error).	(The Estimated Standard Error).
Distribution	Normal N(0,1). One curve.	Student's t. A family of
		curves indexed by `df`.
Core Idea	Normalizing a sample	Normalizing a sample mean
	mean with a known variance.	with an estimated variance.

Procedure: Confidence Interval for a Mean

To estimate the range where the true population mean μ likely lies:

From your single sample, calculate the sample mean x̄ and sample standard deviation s.
Choose a confidence level (e.g., 95%).
Determine your degrees of freedom: df = n - 1.
Find the critical t-value (t_crit) for your confidence level and df. This value marks the boundaries on the t-distribution that capture the central area (e.g., 95%) of the probability.
Calculate the margin of error: E = t_crit * (s / √n).
The confidence interval is x̄ ± E.

This interval is wider than one you'd calculate with a Z-statistic, correctly reflecting the added uncertainty of not knowing σ.

Thanks, this is great. I'd like to query one particular piece of terminology you used.

You described the "z-statistic". I am more used to seeing "z-score".

Do these mean the same thing?

Disregarding convention, which is more conceptually correct and why?

🤖🤖🤖

Excellent question. The terms are often used interchangeably, but they have distinct meanings.

They do not mean the same thing.

A z-score measures the distance of a single data point from its population mean, in units of population standard deviations.
- Formula: z = (x - μ) / σ
- Purpose: Descriptive. It tells you where an individual observation x stands within its population.
A z-statistic (or test statistic) measures the distance of a sample mean from the population mean, in units of standard errors.
- Formula: z = (x̄ - μ) / (σ / √n)
- Purpose: Inferential. It's used to test a hypothesis about a population based on a sample.

Which is more conceptually correct?

In the context of our discussion (estimating a population parameter from a sample), z-statistic is the more conceptually correct term.

Here is why:

Object of Study: We are analyzing a sample mean (x̄), not an individual data point (x).
"Statistic" vs. "Score": A statistic is, by definition, a quantity computed from a sample. Since x̄ is a summary of our sample, any value derived from it is a statistic. A "score" is a more general term that can apply to a single data point.
Purpose: We are using this value to make an inference or test a hypothesis (e.g., "Is it plausible that my sample came from a population with mean μ?"). This makes it a test statistic.

In short, you use a z-score to describe one person's height relative to their country's average. You use a z-statistic to test if the average height of a classroom is significantly different from the country's average.

One thing which confuses me is the following.

When we are trying to determine the population variance, based only on an observation of a single sample, then obviously we do not know the population variance. So there is epistemic uncertainty regarding the population variance.

Yet the population variance is not treated as a random variable. It is described as a fixed parameter of the distribution, one which we merely do not happen to know.

In common language, it would be natural to talk about whether it is more or less "likely" that the population variance is actually big or small. But in the precise language of probability, it seems, it would be incorrect to translate such talk into statements about the "probability" of the population variance having a large vs small value, because we do not treat the population variance as a random variable, so we do not assign probabilities regarding its potential values.

In other words, I am confused by the mismatch between (what I understand to be) the formally correct and precise way to talk about hypothesis testing, confidence intervals, and the modelling of uncertainty, and my intuitive sense that we use probabilities to reflect anything which we do not know certainly.

Can you help me understand things better so that I am not so confused. Is my concern typical? Is it based on some basic misunderstand which, once clarified, will make the theoretically correct view feel more intuitive coherent?

🤖🤖🤖 Yes. Your concern is not only typical, it is the single most important conceptual question in statistics. You have hit upon the central philosophical tension between the two major schools of thought: Frequentism and Bayesianism.

The framework you have learned is the Frequentist one. Your intuition is pulling you toward the Bayesian one.

Let's clarify the two views.

1. The Frequentist View (The one you've been taught)

This is the school of t-tests and confidence intervals.

Definition of Probability: Probability is the long-run frequency of an outcome if an experiment were repeated infinitely. It's about repeatable events.
Parameters (like μ and σ): Are considered fixed, unknown constants. There is one true value for the population variance, σ². It doesn't change. Since it is not the outcome of a repeatable random experiment, it is not a random variable.
The Source of "Randomness": The only things that are random are the data we sample. Therefore, any quantity calculated from the data (like x̄ or s) is a random variable.
Handling Uncertainty: This is the counter-intuitive part. Since you can't assign a probability to the fixed parameter σ², you make probabilistic statements about the procedure.

A 95% confidence interval does not mean "there is a 95% probability that the true μ is in this interval."

It means: "If we were to repeat this entire sampling and interval-creation procedure 100 times, 95 of the intervals we create would contain the true, fixed μ." The randomness is in the intervals, which change with each sample, not in the parameter.

Your intuition is rebelling because this feels like a convoluted linguistic trick to avoid saying the obvious. And in a way, it is.

2. The Bayesian View (The one that matches your intuition)

This view directly addresses your confusion.

Definition of Probability: Probability is a measure of the degree of belief or certainty about a proposition, given the available information. It is a tool for quantifying epistemic uncertainty.
Parameters (like μ and σ): Since we are uncertain about their true values, it is perfectly natural and correct to treat them as random variables. We can—and should—assign probability distributions to them.
Handling Uncertainty: The process is more direct.
1. You start with a prior distribution for the parameter (e.g., a belief about what σ² might be before seeing the data).
2. You collect data (your sample).
3. You use Bayes' Theorem to update your belief, resulting in a posterior distribution for the parameter. This posterior distribution directly expresses your updated knowledge: "Given the data, it is most likely that σ² is small, and very unlikely that it is large."
The Result: You can construct a 95% credible interval, which means exactly what you think it should: "Given the data, there is a 95% probability that the true value of σ² lies within this interval."

Summary: Your Confusion is Justified

Topic	Frequentist View (Standard teaching)	Bayesian View (Your intuition)
What is a parameter (σ²)?	A fixed, unknown constant. Not a	A random variable, because its true
	random variable.	value is unknown to us.
What is probability?	Long-run frequency of a repeatable	A degree of belief or confidence in
	event.	a statement.
Can we say P(σ² > 10)?	No. This is meaningless, as σ² is	Yes. This is the probability that
	a fixed constant.	the true value of σ² is > 10.
How we talk about it	"We are 95% confident that this interval	"There is a 95% probability that the
	captures the true value."	true value is in this interval."

Your confusion arises because you were taught a formal system (Frequentism) whose philosophical foundation is at odds with the intuitive way humans think about uncertainty. The system is logically coherent within its own rules, but those rules (especially that parameters are not random variables) can feel unnatural.

Once you realize there are two different, valid systems for modeling uncertainty, the Frequentist approach becomes less confusing. You can see it as one possible, albeit indirect, way to be precise about uncertainty.

Thank you this is very helpful.

I'd like to follow up on one particular point, which is the frequentist interpretation of the meaning if a confidence interval.

You say two things regarding what it means:

"We are 95% confident that this interval captures the true value"
"If we were to repeat this entire sampling and interval-creation procedure 100 times, 95 of the intervals we create would contain the true, fixed μ."

The first thing relies on a notion of degree of confidence measured in percentage. This seems to me unrigorous, and a roundabout way of avoiding talking about probability, which is what the second thing tries to do more forthrightly. Yes?

The second thing is odd because it seems to be saying, "Here is a particular confidence interval, defined by particular bounds (e.g., A and B). This the 95% confidence interval. And what that means is that, 95 times out of 100, if you took a random sample, and made a different confidence interval in the same way but using that different sample, then that different confidence interval with different bounds (e.g., C and D) would also contain the true value $\mu$.

Is that right? That seems wrong.

I am confused by the contrast between the fact that a confidence interval is defined by one particular pair of bounds which was computed from one particular sample, and the claim that should it be interpreted as a statement of the fraction of other confidence intervals which have the property of including the true parameter value.

I want to ask, which is it? Is one confidence interval a statement about itself, derived from one sample? Or is it a statement about the collection of confidence intervals produced by a constant procedure in the face of random inputs to that procedure? I do not understand how it can be both.

🤖🤖🤖 You are absolutely correct on all points. Your analysis is precise, and your confusion is the natural result of grappling with the very strange nature of the frequentist confidence interval.

Let's break it down.

On "Confidence" vs. "Probability"

> The first thing relies on a notion of degree of confidence… This seems to me unrigorous, and a roundabout way of avoiding talking about probability… Yes?

Yes, exactly. "Confidence" is a carefully chosen term of art. It is a deliberate linguistic maneuver to create a framework that feels like a probability statement without violating the core frequentist rule: a fixed parameter cannot have a probability distribution. You are 95% "confident" in the procedure's reliability, not in the specific result.

On a Single Interval vs. a Collection of Intervals

This is the heart of the matter. Your paraphrase is spot on:

> …what that means is that, 95 times out of 100, if you took a random sample, and made a different confidence interval… that different confidence interval… would also contain the true value μ. Is that right? That seems wrong.

It is right. And it feels wrong because it's deeply counter-intuitive.

Let's use an analogy to clarify.

The Hoop-Tossing Analogy

Imagine a machine that tosses hoops. Somewhere in a field is a small, fixed peg (the true population mean, μ). You can't see the peg's exact location.

The Procedure: The machine is designed so that when it tosses a hoop, the hoop has a 95% chance of landing around the peg. The machine is the "confidence interval procedure." Its 95% success rate is a property of the machine itself, established over many test throws.
Your Sample: You press a button, and the machine makes one toss. This is like taking your one sample and calculating your one interval.
Your Interval: The hoop lands on the field. It has a specific location and diameter. This is your calculated interval, [A, B].

Now, look at the hoop lying on the grass. It either contains the peg, or it does not. The probability that this specific hoop contains the peg is now either 1 or 0. The "95%" seems to have vanished.

So, when you say "I have a 95% confidence interval," you are not talking about the hoop on the ground. You are talking about the machine that threw it. You are saying: "I used a machine that succeeds 95% of the time. This is the hoop it gave me. I therefore have 95% confidence in the process that led to this result."

Which is it? It cannot be both.

You are 100% correct. It cannot be both.

A confidence interval is not a statement about itself.

It is a statement about the reliability of the method that produced it. The 95% is a property of the procedure in the abstract, considered over all possible random samples you could have drawn.

Once you draw your one sample and calculate your one interval [A, B], the probability is gone. The statement "μ is in [A, B]" is either true or false. You don't know which, but you trust the procedure that got you there. This is why the Bayesian approach, which allows you to say "there is a 95% probability the true value is in this interval," feels more natural to so many people. It aligns with our intuitive desire to make a probabilistic statement about the result we actually have.

Okay. Thank you, this analogy is very helpful and clear. You are really helping me drill into the core of the concepts and my confusion. Let met follow up on this.

If I understand you correctly then, the statement "I have a 95% confidence interval" should be understood to mean, "I have one interval, produced from one sample, by a process which 95% of the time produces intervals which include the true value". It is like saying, "That is one hoop on the ground, produced by one throw, from my hoop-throwing machine which 95% throws a hoop around the target".

Is that right?

If it is, then it seems to me that any single interval cannot be the 95% confidence interval. It can only be a 95% interval, which means it is one interval, produced from one sample and the 95% confidence interval-making process.

Is that 100% correct?

🤖🤖🤖 Yes. That is 100% correct.

Your understanding is now perfectly aligned with the rigorous, frequentist interpretation.

The statement: "I have a 95% confidence interval of [A, B]."
Means precisely: "I have generated one specific interval, [A, B], using a procedure that, when applied to all possible samples, successfully captures the true parameter 95% of the time."

Therefore, your conclusion is the correct one:

> …it seems to me that any single interval cannot be the 95% confidence interval. It can only be a 95% interval…

This is the most precise and conceptually pure way to state it.

In practice, you will often hear statisticians and scientists refer to "*the/ 95% confidence interval for our sample." This is common shorthand. When they say this, what they mean is "the single 95% confidence interval that was generated from the one sample we actually have." It is a pragmatic, but slightly imprecise, way of speaking.

Your insistence on this distinction is not pedantry. It is the key to correctly understanding the entire concept and avoiding the intuitive-but-wrong interpretation that "there's a 95% probability the true mean is in this specific interval." The moment you say "*a/ 95% confidence interval," you are implicitly acknowledging the existence of all the other intervals you could have gotten, which is the very foundation of the frequentist idea.

algal/studentt.md