Signal and Error
Info sheet · Statistics for Psychology & Neuroscience
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What you’ll get from this sheet
The “noise” idea, written as an equation. By the end you should be able to:
- Decompose data into Signal + Error.
- See why the F‑statistic, R², and t‑statistic are all the same ratio in disguise.
Every model writes the data as Data = Signal + Error: the part the model explains, plus the leftover it can’t. Almost every statistic is a ratio of those two — F = explained ÷ unexplained variance, R² = explained ÷ total, t = difference ÷ its standard error.
Data = Signal + Error
In nearly every model we build, the data split into two parts:
\[\text{Data} = \text{Signal} + \text{Error}\]
The signal (the systematic or explained part) is what our model accounts for — the effect of decade on danceability, of surgery on quality of life, of study time on recall. The error (the residual or unexplained part) is everything left over after the model has done its best. Crucially, error isn’t mistake: it’s the part of the world the model doesn’t capture, either because it’s genuinely random or because it comes from variables we didn’t (or couldn’t) measure. Every model is an admission of partial ignorance, and the error term is where that ignorance lives.
One ratio, many names
Almost every statistic we compute is a ratio of these two quantities — explained versus unexplained. Slide the signal strength and watch the total variance split into an explained part (the model) and a residual (the error), with R² and the F‑ratio reading straight off that split:
Turn the signal up and the green (explained) slice of the total‑variance bar grows while the residual stubs on the scatter shrink — R² climbs toward 1 and F climbs with it. That’s the whole game in one picture: the F‑statistic in an ANOVA is literally explained ÷ unexplained variance; R² in regression is the proportion of total variance explained; a t‑statistic is a difference (signal) ÷ its standard error (noise). Once you see the pattern, the alphabet soup of tests stops looking like unrelated recipes and starts looking like variations on one theme.
The recurring three-step
So throughout the book we do the same three things: (1) build a model that proposes a signal, (2) measure how much variation that signal explains, and (3) ask whether that explained variation is large enough, relative to the leftover error, to take seriously. Hold onto that framing — it’s the thread tying every chapter together.
See it in code
TipCheck your understanding
What do R² and the F‑statistic have in common, and how do they differ?
Both are built from the same split — explained variance versus unexplained (residual) variance. R² is a proportion: explained ÷ total variance (bounded 0–1, “how much of the wobble did we account for?”). The F‑statistic is a ratio: explained ÷ unexplained variance (adjusted for degrees of freedom), used to ask whether the explained part is big enough to be more than chance. Same ingredients, arranged for two different questions.
“Error” doesn’t mean you made a mistake — it’s just the variation your model didn’t capture, some of it genuinely random, some of it from variables you never measured. So a large residual isn’t necessarily a bad model; it may be an honest one facing a noisy world. And a high R² isn’t automatically good (it can signal overfitting), just as a low one isn’t automatically bad — always read the explained and the unexplained together.
Where this shows up next
This ratio is the backbone of ANOVA (the F‑ratio), regression (R²), and the t‑test. See those sheets — you’ll now recognise them all as the same move.