Percentages donât help unless they become headcounts. After you find an area, multiply by N and speak human: âP (IQ â„ 120)Â â 0.091 â about 9 in 100 students.â Decisions donât live in curves; they live in classrooms, budgets, and schedules. Always translate the area into how many (and for whom) youâll do something […]
Turn Percent into People Read More »
The tails tell you who needs an exception or an intervention (âtop 10%,â âbottom 5%â). The middle tells you how to run Tuesday (âmost students finish between 12â16 minutesâ). If youâre writing policy, pick a tail and be explicit about the cutoff. If youâre improving practice, describe the middle and make the common case better.
Tails Make Policies. Middles Make Plans. Read More »
A z-score is a distance in standard deviations… a unit of surprise. âz = +2â means âtwo Ï above typical,â not âsmartâ or âgood.â Use z to compare across scales (minutes, points, dollars) and to speak percentile without the drama. The trick isnât memorizing every table entry; itâs knowing where the tails start to matter
Z-Scores Are Units of Surprise Read More »
Hereâs the repeatable sequence: draw â set ÎŒ & Ï â pick region â choose area or value â compute â translate. If you need a probability, compute an area. If you need a cutoff, get a value. The workflow isnât fancy; itâs protective. It keeps you from solving the same problem six different ways
One Workflow to Rule Your Bell Curve Read More »
Numbers are shy until you sketch them. With the normal curve, the picture is the plan: mark ÎŒ and Ï, shade the region, then choose the move… area or cutoff. Most errors arenât math mistakes; theyâre region mistakes. We solve for a left tail when we meant the middle. We read the wrong column. Drawing
Draw the Curve Before You Decide Read More »
1) Draw Before You Measure  If you havenât drawn the scatterplot, you havenât earned r. Shape first, number second. 2) Check the Residuals  If the residuals curve, your relationship did too. Fit ideas, not just lines. 3) Try Spearman When Life Bends  Monotonic but curved? Spearman respects order without demanding straightness. 4) Simulate Your Intuition  When a probability feels
Eight Tiny Experiments for Patterns First, Probabilities Second Read More »
âIndependentâ does not mean âunrelated in spirit.â It means: knowing one outcome tells you nothing about the other. Dice rolls? Independent. Midterms and sleep? Probably not. Donât confuse mutually exclusive (canât both happen) with independent (donât inform each other). If events are mutually exclusive and both have positive probability, they are not independent. Start with
Independence Isnât Friendship Read More »
A probability isnât a prediction; itâs a contract about the long run. âThirty percent rainâ means: if you could replay today a thousand times, it would rain in about 300 of them. On any single day, reality is lumpy. Act like a portfolio manager: define the sample space, list outcomes, assign fair odds, and make
Probability Is a Contract, Not a Crystal Ball Read More »
Two very different clouds can share the same correlation. One may be tight with a rebel; the other, loose but clean. Numbers rhyme; pictures speak. Thatâs why Anscombeâs lesson endures: always sketch the data. Write one sentence for each plot: direction, form, strength, outliers… and only then compute r. If your sentence and your statistic
Same r, Different Story Read More »
An outlier with long arms (extreme x) can drag your regression line across the room. Thatâs leverage… and it talks louder than the rest. Before you trust r or a slope, label the loud points and ask three questions: (1) Is it real? (2) Does it represent a scenario we care about? (3) What happens
The One Point With a Megaphone Read More »