Mathematical Notation & Key Formulas

Info sheet · Statistics for Psychology & Neuroscience

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• Built from the opening “Mathematical Notation & Key Formulas” review segment (the RM1/RM2 recap): letter conventions (X/Y/Z, i/j/k/n, α/β/ε), population vs sample, H0/H1, Y∼N(μ,σ²), the GLM Y=β0+β1X+ε, and the t-test-as-linear-model example.
• Confirm the symbols exactly match your slides (e.g. whether you write \(s\) or \(\hat{\sigma}\) for the sample SD).

What you’ll get from this sheet

A decoder ring for the symbols and formulas used across the book — a recap of the conventions from RM1 and RM2. By the end you should be able to:

1. Interpret standard mathematical notation, and tell population from sample statistics.
2. Read the notation for a normal random variable and a general linear model.

Two habits unlock most of it: population quantities get Greek letters (\(\mu\), \(\sigma\)), their sample estimates get Latin letters (\(\bar{x}\), \(s\)); and the letters carry conventions — \(\alpha\) is the significance level, \(\beta\) a coefficient (or power), \(\varepsilon\) the error in a model. The big \(\Sigma\) just says add these up.

Letter conventions

A few habits make formulas readable:

• Coordinates: \(X\) (horizontal), \(Y\) (vertical), \(Z\) (depth). Because of this, avoid using \(X, Y, Z\) as variable names unless you really mean locations.
• Counting indices: \(i, j, k, n\) are the counters in formulas — \(x_i\) is “the \(i\)-th value”, \(n\) is how many there are.
• \(\alpha\) (alpha): the significance level, typically 0.05 or 0.01.
• \(\beta\) (beta): a regression coefficient — or the power of a test, depending on context.
• \(\varepsilon\) (epsilon): the error term in a linear model.

Population vs sample — the distinction to learn first

Almost every symbol comes in a pair: the true value in the population (which we never see) and our estimate of it from a sample. Descriptive statistics summarise data (mean, median, mode, SD); inferential statistics let us generalise (t-values, p-values). The parameters and their estimates:

Quantity	Population (parameter)	Sample (estimate)
Mean	\(\mu\) (mu)	\(\bar{x}\) (x-bar)
Standard deviation	\(\sigma\) (sigma)	\(s\)
Variance	\(\sigma^2\)	\(s^2\)
Size	\(N\)	\(n\)

The pattern: Greek = the truth we’re after; Latin (or a hat) = our best guess from data. A statistic like \(\bar{x}\) estimates the parameter \(\mu\); a hat marks an estimate or prediction (\(\hat{y}\), \(\hat{\beta}\)).

Hypotheses and the decision rule

Inference is framed as a contest between two hypotheses:

• Null hypothesis (\(H_0\)): no effect or difference.
• Alternative hypothesis (\(H_1\)): an effect or difference exists.

The rule: reject \(H_0\) if the p-value \(< \alpha\); otherwise fail to reject \(H_0\) (we never “accept” it). A dependent variable (DV) is the outcome whose value depends on one or more independent variables (IVs).

Summation: the \(\Sigma\)

The most common symbol. \(\displaystyle\sum_{i=1}^{n} x_i\) means “add up the \(x\) values, from the first (\(i=1\)) to the \(n\)-th.” So the mean is a summation in disguise:

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.\]

Drag the slider to watch the sum (and the mean) build term by term:

md`**n (how many values to add)** — currently **${+nSum}**`

viewof nSum = html`<input type="range" min="2" max="10" step="1" value="5" style="width:300px">`

sumWidget = {
  const data=[4,7,3,8,5,6,9,2,7,5];
  const n=+nSum, xs=data.slice(0,n);
  const S=xs.reduce((a,b)=>a+b,0), mean=S/n;
  const box=document.createElement("div");
  box.style.cssText="background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px;padding:12px 14px;font-size:15px";
  box.innerHTML=`<div>x = [ ${xs.join(", ")} ]  <span style="color:#94a3b8">(so x<sub>1</sub> = ${xs[0]}, x<sub>2</sub> = ${xs[1]}, …)</span></div>`+
    `<div style="margin-top:.55rem">&#8721;<sub>i=1</sub><sup>${n}</sup> x<sub>i</sub> = ${xs.join(" + ")} = <b style="color:#1F4E79">${S}</b></div>`+
    `<div style="margin-top:.35rem">x&#772; = (1/n) &#8721; x<sub>i</sub> = ${S}/${n} = <b style="color:#1E7A34">${mean.toFixed(2)}</b></div>`;
  return box;
}

Once you see that \(\Sigma\) is “just a loop that adds,” the intimidating formulas (variance, the regression slope, the F-ratio) become readable: they’re sums of things, divided by other sums of things.

Normal random variables

We write “\(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\)” as:

\[Y \sim N(\mu, \sigma^2)\]

reading piece by piece: \(Y\) the variable, \(\sim\) “is distributed as”, \(N\) normal, \(\mu\) the mean, \(\sigma^2\) the variance. (The binomial and Poisson use the same shape — \(X \sim \text{Binomial}(n,p)\), \(X \sim \text{Poisson}(\lambda)\).)

The general linear model in symbols

The skeleton behind most of the book:

\[Y = \beta_0 + \beta_1 X + \varepsilon\]

with \(Y\) the outcome (DV), \(\beta_0\) the intercept, \(\beta_1 X\) a coefficient times an independent variable, and \(\varepsilon\) the error. It’s the school-maths line \(y = b + mx\) wearing statistical clothes — \(\beta_0\) is the intercept \(b\), \(\beta_1\) is the slope \(m\). A larger \(|\beta_1|\) means that variable contributes more to \(Y\).

NoteA t-test is a linear model

An independent-samples t-test is this same equation with the group coded 0/1. Comparing apples and watermelons:

• Apples (group 0): \(\;Y = \beta_0 + \beta_1 \cdot 0 + \varepsilon\;\) → the mean of the apples is \(\beta_0\).
• Watermelons (group 1): \(\;Y = \beta_0 + \beta_1 \cdot 1 + \varepsilon\;\) → the mean of the watermelons is \(\beta_0 + \beta_1\).

So \(\beta_0\) is one group’s mean and \(\beta_1\) is the difference between the groups — testing “\(\beta_1 = 0\)” is exactly the t-test. This is why the whole book keeps returning to the general linear model: the familiar parametric tests are special cases of it.

A note on names: a General Linear Model (GLM) assumes the variables are normally distributed; a Generalized Linear Model relaxes that, allowing other outcome distributions (e.g. binomial for logistic regression).

See it in code

The notation maps directly onto one-word functions:

R

Python

MATLAB / Octave

x = [4 7 3 8 5];
sum(x)      % ∑ xᵢ
mean(x)     % x̄
std(x)      % s   (divides by n − 1 by default)
var(x)      % s²
numel(x)    % n

The R and Python tabs run live; MATLAB is a static reference.

Quick quiz: population or sample?

Is each symbol a true population parameter, or a sample statistic we compute from data?

NoteQuiz

Answer each question — the feedback appears beneath it.

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  const why=["The population mean — the true average we're trying to estimate.","The sample mean — our estimate of μ from data (a bar = a mean).","The population standard deviation.","The sample standard deviation — our estimate of σ.","The population variance.","The sample variance — our estimate of σ².","A true regression coefficient in the population.","An estimated coefficient from a fitted model (a hat = an estimate)."];
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    const fb=document.createElement("div"); fb.style.cssText="font-size:14px;min-height:1.2rem;color:#94a3b8"; fb.textContent="Click an answer to check.";
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Watch the context-dependent letters: \(\alpha\) is the significance level here but a reliability coefficient elsewhere; \(\beta\) is a regression coefficient and the Type II error rate (power context); \(\varepsilon\) is the model error but the sphericity index in repeated-measures ANOVA. Don’t mix the population (Greek) and sample (Latin) symbols — writing \(\sigma\) when you mean the sample \(s\) quietly claims you know the truth. And don’t press \(X, Y, Z\) into service as arbitrary variable names — they’re reserved for coordinates.

Where this shows up next

This notation runs through every sheet. The general-linear-model equation is the backbone of the regression and ANOVA chapters; the population/sample split underlies Describing Data; and the bold, transpose, and \(A^{-1}\) conventions of linear algebra live in the Matrix Algebra appendix.

Related topics

• Signal and Error — Data = Signal + Error, the same GLM idea.
• The General Linear Model — Y = β₀ + β₁X + ε in depth.
• Describing Data — where x̄, s, and ∑ do their work.
• Normal Random Variables — reading and combining N(μ, σ²).
• Matrix Algebra Basics — vector and matrix notation.