Academic Blog

Same test name, completely different setups. One question decides it: is each value in group A naturally linked to a specific value in group B? Why the paired test cancels between-person noise and is far more powerful when it applies — and how picking wrong quietly throws away half your statistical power.

A z-score is how many standard deviations a value sits from the mean — a universal ruler that lets you compare across any scale. The z = (x−μ)/σ formula, the 68–95–99.7 rule, the three-step method for “what percentage scored above X?”, and the direction error that catches everyone.

Set the hypotheses up wrong and the whole test answers the wrong question. The three rules that never change: hypotheses are about the population not the sample, the null always carries the equals sign, and direction is chosen before you see the data — plus why choosing one-sided after peeking is cheating.

Average enough independent things and the average follows a normal distribution — even when the individuals don’t. The three distributions students blur together, why the normal keeps appearing, how large n really needs to be, and why the CLT is the engine under every confidence interval and t-test.

Everyone recites the slogan; almost nobody can list the actual reasons. If X and Y correlate, there are exactly five possibilities — X causes Y, Y causes X, a confounder causes both, coincidence, or selection bias — and only one is causation. With the HRT case that cost real lives.

One question decides it: do you know the population standard deviation? If yes, z-test; if you’re estimating it from your sample, t-test — which is almost always. Why the t-distribution has fatter tails, the Guinness brewery origin story, and why “n > 30 use z” is a shortcut, not the rule.

Treatment A beats Treatment B in every single subgroup of patients — and yet, combine all the subgroups together, and Treatment B wins overall. This isn't a contradiction or a calculation error. It's a real, well-documented phenomenon called Simpson's Paradox, and the Statistics Made Simple Complete Bundle works through the exact case study that made it famous. In 1973, the University of...

A lottery ticket, an insurance policy, and a casino game all share the same underlying question: on average, across many repetitions, what should you expect to happen? The Statistics Made Simple Complete Bundle builds this idea up from first principles, with the same worked datasets used throughout the book. A lottery ticket costs $5. There's a 1-in-1,000,000 chance of winning $2,000,000, and...