Question on Essential Math for Data Science (page 279 of 511)

where does this t-value formula come from

Excellent question. This formula isn't just pulled out of thin air; it's a standard statistical tool derived to solve a specific problem. Understanding its components is key to understanding its purpose.

In simple terms, the formula is a classic example of a signal-to-noise ratio.

Let's break it down:

t = Signal / Noise

The goal is to see if the "signal" (the correlation you found) is strong enough to stand out from the "noise" (the random variability you'd expect in your data).

1. The Numerator: r (The Signal)¶

The top part of the fraction is simply r, the correlation coefficient you calculated from your sample data.

This is your "effect size" or "signal."
It represents how strong the linear relationship appears to be in the data you collected.
A value of r close to 1 or -1 is a strong signal. A value close to 0 is a weak signal.

2. The Denominator: sqrt((1 - r²) / (n - 2)) (The Noise)¶

The bottom part of the fraction represents the standard error of the correlation coefficient. This is a more complex concept, so let's break it down further. Think of it as a measure of how much your calculated r would likely bounce around due to random chance if you were to take many different samples.

This "noise" term is influenced by two key factors:

a) The Scatter in Your Data (1 - r²)¶

You might remember that r² (the "coefficient of determination") tells you the proportion of the variance in one variable that is predictable from the other variable. For example, if r² is 0.90, it means 90% of the movement in Y is "explained" by the movement in X.
Therefore, 1 - r² represents the proportion of the variance that is unexplained. It's the "error" or the random scatter of your data points around the regression line.
If your correlation r is very strong (close to 1), then 1 - r² is very small. This means there's very little unexplained scatter, so the "noise" is low.
If your correlation r is weak (close to 0), then 1 - r² is large. This means there's a lot of unexplained scatter, so the "noise" is high.

b) Your Sample Size (n - 2)¶

n is your sample size.
This term is called the "degrees of freedom." The reason it's n-2 is that to define a line, you need at least two points. Those first two points don't provide any information about the error around the line (since a line will always pass through two points perfectly). Every data point after the first two contributes one "degree of freedom" to help you estimate the random error.
The effect is simple: As your sample size n gets bigger, n-2 also gets bigger. Since it's in the denominator of the denominator, it makes the overall "noise" term smaller. This makes perfect sense: a correlation found in a large sample is much more believable and less likely to be random "noise" than the same correlation found in a tiny sample.

Putting It All Together¶

The formula t = r / sqrt((1 - r²) / (n - 2)) is essentially asking:

"How many units of 'standard noise' is my 'signal'?"

You get a large t-value when your signal ® is strong and/or your noise (sqrt(...)) is low (because the scatter is small or your sample size is large). A large t-value suggests your finding is significant.
You get a small t-value when your signal ® is weak and/or your noise is high (because the scatter is large or your sample size is small). A small t-value suggests your finding could easily be due to random chance.

This calculated t-value is then compared to a known statistical distribution (the t-distribution) to determine the exact probability that you would see a correlation this strong just by luck.