Before you can infer anything, you have to describe it honestly.

What statistics is actually for

Statistics exists because of one gap: you can never measure everyone you want to talk about. You want to say something about adolescents with social anxiety, or about working memory in healthy adults, or about how a drug affects rats. You measure forty of them. Every technique in the next 111 days is machinery for crossing that gap responsibly — and for stating honestly how wide the gap still is.

The group you want to conclude about is the population. The group you actually measured is the sample. A number describing a population is a parameter — μ, σ, ρ, π, written in Greek. A number computed from your sample is a statistic — M, s, r, p, written in Latin. You almost never see a parameter. You compute statistics and use them to estimate parameters, and the entire discipline is the study of how wrong that estimate is likely to be.

One more distinction to fix now, because it decides which half of this course you are in. A variable is anything that takes different values across cases: depression score, condition, reaction time, handedness. A constant does not vary, and cannot correlate with anything. If your manipulation produced no variability — everyone scored the ceiling on your task — no statistical technique in existence can rescue the study. Most 'my analysis failed' problems are design problems arriving late.

Scales of measurement, and what each one permits

Stevens' four scales are the most reliably examined content in any psychology statistics paper, and the most commonly reduced to memorised labels. The point of the taxonomy is not classification. It is permission: the scale of a variable determines which arithmetic operations mean anything, and therefore which statistics are legitimate.

The interval-vs-ratio line matters less in practice than the textbook implies — both permit the same tests. The nominal-vs-ordinal-vs-interval boundary matters enormously. An IQ of 130 is not twice as intelligent as 65, because IQ has no true zero; that is an interval-scale statement and it is simply meaningless. But the mean of IQ scores is fine, because intervals are treated as equal.

Two further distinctions live alongside the scales and are examined with them. Discrete variables take separate values with nothing in between (number of children, errors committed); continuous variables can in principle take any value in a range (time, weight). And a variable's role in a design — independent, dependent, confounding, mediating, moderating — is a separate question from its scale. Reaction time is ratio whether it is your outcome or your predictor.

Frequency distributions: looking before you compute

A frequency distribution is just a count of how often each value occurs. It is also the last point at which your data are still fully visible. Every statistic after this is a compression — the mean throws away everything except the centre — and the compression can only be trusted if you have seen what is being compressed.

For a small number of categories, a simple frequency table does it: value, frequency f, relative frequency f/N, cumulative percentage. For continuous data you group into class intervals, and the number of intervals is a real decision: too few and you hide structure, too many and you show noise. Ten to twenty bins is the usual working range, and the honest move is to try three bin widths and report whether your conclusion changes.

This is not a preliminary. It is the analysis step where the most consequential decisions are made, and it is the step most often skipped in favour of running the test that was planned before the data existed.

Central tendency, and when each measure lies

Three ways to say where a distribution sits, each answering a subtly different question. The mean is the balance point: the value that makes the deviations sum to exactly zero. The median is the positional middle: half above, half below. The mode is the most frequent value, and the only centre available for nominal data.

The mean uses every value, which is its strength and its weakness: it is the most efficient estimator when the distribution is roughly symmetric, and it is dragged bodily toward any extreme score. Add one participant earning ten million to a sample of students and the mean income becomes a number no student has. The median would not move at all.

Dispersion, and why you divide by n − 1

Two groups can have identical means and be completely different data. Centre without spread is a half-description, and spread is the more consequential half: every inferential test you will ever run is essentially a signal-to-noise ratio, where the noise term is a measure of dispersion. Improving measurement precision is a statistical intervention, not just a methodological nicety.

The range (max − min) uses two values and is hostage to both. The interquartile range (Q3 − Q1) covers the middle fifty per cent and is resistant to outliers, which is why it partners the median. The variance and its square root the standard deviation use every value and are the basis of nearly all parametric statistics.

Why n − 1, asked properly: your deviations are taken from M, not from μ, and M is the value that makes the sum of squared deviations as small as it can possibly be for this sample. So SS computed around M systematically underestimates SS around the true μ. Dividing by the smaller number n − 1 inflates the estimate by exactly the right amount to remove that bias. The same logic — subtract one degree of freedom per parameter estimated from the data — reappears in every df you will ever compute.

Shape: skewness, kurtosis, and what to do about outliers

Centre and spread describe a distribution only if you already know its shape. Two more numbers finish the job. Skewness measures asymmetry: zero for symmetric, positive when the long tail runs right, negative when it runs left. Kurtosis measures tail weight relative to a normal curve: positive (leptokurtic) means heavy tails and a sharp peak, negative (platykurtic) means light tails and a flat middle.

Rules of thumb that get you through both exams and reviewers: skewness and kurtosis within ±1 is unremarkable, ±2 is worth mentioning, beyond that you address it explicitly. A more useful check is the ratio of the statistic to its standard error — beyond about ±3 in samples under 300 is a real departure. But in samples above a few hundred, tiny meaningless departures become 'significant', which is why the Kolmogorov-Smirnov and Shapiro-Wilk normality tests are nearly useless at the sample sizes where normality matters least.

The deeper point: outliers are information. A participant whose reaction times are four SDs slow may be the only person in your sample who was actually doing the task the way you designed it. In clinical samples, the extreme cases are frequently the ones the research is for.

z-scores: putting everything on one scale

A raw score means nothing on its own. Is 47 good? It depends entirely on the mean and the spread of the distribution it came from. A z-score answers that question by re-expressing the score as a distance from the mean, measured in standard deviations — which makes scores from completely different tests directly comparable.

Standardising a whole variable — converting every score to z — leaves the shape of the distribution completely unchanged. This is the most commonly missed point on the topic. Standardising does not normalise. It is a linear transformation: it slides the distribution so the mean is 0 and stretches it so the SD is 1. A skewed distribution of z-scores is exactly as skewed as it was before.

The normal curve, and what it actually promises

The normal distribution is a mathematical idealisation — symmetric, unimodal, defined entirely by μ and σ, with tails that approach but never touch zero. No real psychological variable is exactly normal. Its importance is not descriptive. It matters because of what happens to sample means when you draw repeatedly from any population, which is the central limit theorem and the business of volume 2.

What the curve gives you immediately is the mapping from a z-score to a proportion of the distribution. The empirical rule — 68 per cent of cases within ±1 SD, 95 per cent within ±1.96, 99.7 per cent within ±3 — is worth committing to memory because it turns any mean and SD into a statement about where people fall.

Drag the observed statistic above to z = 1.96 with a two-tailed α of .05 and watch the blue area exactly fill the terracotta one: that is the definition of the critical value, and the whole of null hypothesis testing is this picture with different distributions. Switch to the t tab and shrink df to 5 — the tails thicken, and the critical value moves out past 2.5. That is the price of estimating σ from a small sample, made visible.

Standard scores: T, stanine, IQ and the rest

z-scores are perfect mathematically and awkward in practice: half of them are negative, and most are small decimals. So psychometrics rescales them. Every standard score you have ever seen on a test report is a z-score with a new mean and SD bolted on, chosen to make the numbers friendly.

Transformations: legitimate repair versus cosmetics

When a distribution violates an assumption badly, one option is to re-express the variable on a different scale. A transformation applies the same monotone function to every score, so it preserves the order of participants while changing the spacing between them — which is precisely how it pulls in a long tail.

The modern alternative is usually better: fit a model that assumes the distribution you actually have. Generalized linear models handle counts and binary outcomes directly (days 62–63); robust methods and bootstrapping (day 110) handle heavy tails without touching the scale; mixed models handle reaction-time distributions honestly. Transformation is a 1960s solution that remains examinable and occasionally still the neatest fix.

Reporting descriptives so a reviewer trusts the rest

Your results section opens with descriptives, and experienced readers use them to decide how much of the rest to believe. A clean Table 1 that reconciles with the text buys you goodwill; a table where the ns do not add up costs you all of it, before anyone reaches your hypothesis test.

One structural habit worth building now. Write your Table 1 and its accompanying paragraph before you run a single inferential test. It forces you to look at your data, it makes any impossible value obvious while there is still time, and it means the most-read part of your results was written by someone who was not yet hoping for a particular answer.

The volume 1 formula sheet

Everything in this volume that you may have to compute under exam conditions, with its trap noted. Learn the structure, not the string of symbols — every one of these reappears inside a larger formula later.

Notation you will meet for the next 101 days

Latin letters for sample statistics, Greek for population parameters. Once that convention is automatic, half of statistical notation reads itself.

Ten questions before you move on

Answer from memory, then reveal. Below seven, reread the day named in the feedback — volume 2 assumes every idea here without reintroducing it.