statistics

Two classes at 80% and 90% doesn’t mean 85% overall unless the classes are the same size. Percentages are summaries with hidden denominators… bring the denominators back and compute a weighted mean. Your story changes the moment you respect the weights… and so do your decisions.

The Central Limit Theorem isn’t a spell… it’s a deal. If observations are roughly independent and from the same process, the distribution of the sample mean drifts toward normal as n grows. Break the deal (copy-paste data, time-series dependence, duplicated records) and your “bell” learns to lie. Audit independence first, celebrate normality second.

Sample size solves lots of problems… until it solves the wrong one. If your sampling is biased or your measure is sloppy, n just makes you confident in the wrong direction. First fix how you sample and what you measure. Then grow n for stability. Sequence matters: method, measure, then magnitude.

Standard deviation measures spread in people. Standard error measures spread in statistics. It shrinks with √n, which means doubling your sample is progress… but not magic. Treat standard error like the scale on your ruler… if it’s wide, you can’t draw fine lines. Before you chase more decimals, check whether your standard error can justify

You never hold a population mean in your hand… you hold a sample mean. That mean isn’t a replica; it’s a messenger. Its message carries error, and that error has a shape. When your sample is honest and big enough, the shape becomes comfortably bell-ish even when the population isn’t. That’s the quiet magic of

1) Pictures are Prophesy  Draw what you’re looking for and then find it. The picture is a prophesy of the promise you made yourself. 2) The Complement Saves Time  “P(at least)” problems are usually “1 – P(none).” Do the complement and move on. 3) Standardize, Then Think  Convert x → z before reading any table. Standardization clears the

Percentages don’t help unless they become headcounts. After you find an area, multiply by N and speak human: “P (IQ ≥ 120)  ≈ 0.091 → about 9 in 100 students.” Decisions don’t live in curves; they live in classrooms, budgets, and schedules. Always translate the area into how many (and for whom) you’ll do something

The tails tell you who needs an exception or an intervention (“top 10%,” “bottom 5%”). The middle tells you how to run Tuesday (“most students finish between 12–16 minutes”). If you’re writing policy, pick a tail and be explicit about the cutoff. If you’re improving practice, describe the middle and make the common case better.

A z-score is a distance in standard deviations… a unit of surprise. “z = +2” means “two σ above typical,” not “smart” or “good.” Use z to compare across scales (minutes, points, dollars) and to speak percentile without the drama. The trick isn’t memorizing every table entry; it’s knowing where the tails start to matter

Here’s the repeatable sequence: draw → set μ & σ → pick region → choose area or value → compute → translate. If you need a probability, compute an area. If you need a cutoff, get a value. The workflow isn’t fancy; it’s protective. It keeps you from solving the same problem six different ways