Every z table you'll ever use, as one living picture. Set the distribution, slide the cutoff, read the area. Then meet the t distribution and watch its fat tails deflate as df grows.
The normal curve
μ 50
σ 10
Area
x 60
x₂ 70
—
z-score
—
Shaded area
—
Percentile of x
Normal vs. t — why small samples need wider cutoffs
Degrees of freedom 4
Standard normal zt with your dfExtra tail weight t carries beyond |2|
—
P(|t| > 2)
4.6%
P(|z| > 2)
Try this:
IQ: set μ = 100? It's capped at 100 here, so use μ = 50, σ = 10 and ask: what percent of people score above 65? (Find x = 65, choose "Above x," check the z.) Now answer the same question for any scale by thinking in z.
Set "Between" and find the values that trap the middle 68%, then 95%. Compare to the empirical rule you memorized — now you've SEEN it.
On the t panel, set df = 3 and look at the extra tail area beyond |2|. This is why a t test at n = 4 demands a bigger statistic than a z test. Slide df to 30 — where did the difference go?
Sigma says: A z-score is just a score with its unit swapped to "standard deviations from the mean." Once you think in z, every normal question — heights, reaction times, exam scores — is the same question.