Feel the Standard Normal Distribution
If z = 1.96 rings a bell, you've already met statistics' universal language — one curve sits beneath every test, every interval, every t-distribution.
Standard Normal — The Origin of Everything
Honestly — without this single curve, none of what follows (tests, confidence intervals, the t-distribution, regression) would work.
The standard normal N(0, 1) is a bell curve with mean 0 and standard deviation 1.
The one-line trick "z = (x − μ) / σ" lets every normal distribution collapse onto this same curve — and that's how a single paper table can compute probabilities for the entire world.
In other words, it's not the final boss of statistics; it's the origin. Once you own this, the rest of the page reads as "applications of the standard normal".
- Step 1: Set k to 1.0 → area is ~68%. That's "±1σ covers about 70%."
- Step 2: Set k to 1.96 → area is ~95%. You'll see this number everywhere in testing and CIs.
- Step 3: Stretch k to 3.0 → nearly 100%. Almost nothing lies outside.
▶ "68 - 95 - 99.7" — no memorization, just see it
Slide the width k; the blue-filled area IS the probability.
± 1σ already covers ~68%, ± 2σ is ~95%, ± 3σ is nearly everything.
That famous number z = 1.96? It's the two-tail 5% critical value — hypothesis tests and confidence intervals all start there.
// Formula used here
Left: z = (x − μ) / σ
• x − μ: distance from the mean
• ÷ σ: convert to "how many σ away" — a universal ruler
• z: a standardized score comparable across any normal distribution
Right: φ(z)
• e−z²/2: maximum at z=0, drops rapidly as |z| grows → the bell shape
• 1/√(2π): normalizing constant ensuring total area = 1 (probabilities sum to 100%)
• φ(z) gives the curve's height (density) at that point; area under the curve = probability
- Step 1: Press ▶ Standardize → the μ=2, σ=1.5 curve morphs smoothly into N(0,1).
- Step 2: Change μ to −2, σ to 2.5, then ▶ again → a totally different curve snaps onto the same pink one.
- Step 3: Pause the progress slider midway → watch μ approach 0 and σ approach 1 in real time.
▶ Watch every normal collapse onto "that one curve"
Height, IQ, blood pressure readings, factory part errors — real-world normal-ish things all have different means and spreads.
Yet apply z = (x − μ) / σ and they all snap onto that pink curve.
It auto-plays on scroll (▶ to replay). That's why every statistical formula needs only one standard-normal table.
// Common misconceptions
The exact z-value where the two-sided probability equals 5% is 1.959964…; 1.96 is that number rounded to two decimal places. It wasn't chosen for convenience — it came straight out of the math. P(−1.96 ≤ Z ≤ 1.96) = 0.95000…, virtually identical to 0.95. For hand calculation, 1.96 is precise enough.
z = (x−μ)/σ is reversible: x = μ + zσ recovers the original value. Standardization rescales; it doesn't discard anything.
Any normal distribution transforms to the standard normal via z = (x−μ)/σ. Know this one distribution and you can compute probabilities for all of them. That's why it's "where everything begins."
// Shapes you'll meet again
Around the standard normal, the "standardize then look up" flow and its variations keep returning.
- Two table styles: the same z value reads differently depending on whether the table reports "upper tail" or "cumulative" probability — that distinction lives here
- Interval probability as a subtraction: P(0.5 ≤ Z ≤ 1.5) = Φ(1.5) − Φ(0.5). The "difference of two cumulatives" picture recurs across many problems
- The standardize-then-look-up flow: for X ~ N(170, 6²) with X ≥ 180, the form lines up as z = (180−170)/6 ≈ 1.67 before reaching the table. General normals always pass through this route
- The reverse-lookup shape: "what is the upper 5% point?" returns z = 1.645. The same motion — using the table in reverse — appears as a shape
See the home of the N(μ,σ²) → z translation at Normal Distribution.