Normal Distribution Visualization | z-Scores and Confidence Intervals

The idea: a bell set by two numbers

Heights, measurement errors and test scores pile up around an average and thin out on both sides in the same smooth, symmetric shape: the normal distribution, or bell curve. It is fixed by two numbers. The mean μ is the centre of the bell. The standard deviation σ is its width: the distance from μ to the point where the curve turns from bending down to bending up. We write N(μ, σ²).

The bell curve of test scores N(70, 10 squared), centred on mu = 70 with a bar from 70 to 80 marking sigma = 10. The area from 60 to 80 is dark blue, 50 to 90 medium blue and 40 to 100 light blue. Below the axis, three bars: 68.27% within mu plus or minus 1 sigma, 95.45% within 2 sigma, 99.73% within 3 sigma. — Test scores N(70, 10²): μ is the centre, σ the width; the shades are μ ± 1σ, 2σ and 3σ.

The page models test scores with μ = 70 and σ = 10. The bell on the canvas is built from thin slices so that any part of the area can be shaded. The total area under the curve is 1 (100% of the scores), and the area over a range is the fraction of scores in that range.

Changing μ and σ

The curve is f(x) = e^(−(x − μ)²/(2σ²)) / (σ√(2π)). Changing μ slides the whole bell left or right without changing its shape. Changing σ stretches it: because the total area must stay 1, a wider bell is also lower. Its peak is 1/(σ√(2π)), so σ = 15 gives a peak only 2/3 as tall as σ = 10 (Demo: changing μ and σ).

The 68–95–99.7 rule

For every normal distribution, the same fraction of the area lies within a given number of standard deviations of the mean (Demo: the 68–95–99.7 rule):

Range	Scores, μ = 70, σ = 10	Area inside	Outside
μ ± 1σ	60–80	68.27%	about 1 in 3
μ ± 2σ	50–90	95.45%	about 1 in 22
μ ± 3σ	40–100	99.73%	about 3 in 1000

This is the quick sanity check behind "three-sigma" limits in quality control: a value more than 3σ from the mean is rare enough to deserve a second look.

z-scores and the z table

To find the fraction below a score you would need the area under this bell. Instead of a table for every μ and σ, every score is moved onto one bell, the standard normal N(0, 1), with the z-score

z = (x − μ) / σ

z counts how many standard deviations x lies above (z > 0) or below (z < 0) the mean. The area left of z, written Φ(z), is listed in a z table; as a percentage it is the percentile. A score of 85 when μ = 70 and σ = 10 gives z = (85 − 70)/10 = 1.5 and Φ(1.5) = 0.9332: the student did better than 93.3% of all students, the 93rd percentile (Demo: score 85 → z = 1.5 → 93rd percentile).

Standardising keeps the area: left of 85 on N(70, 10²) is the same 93.32% as left of 1.5 on N(0, 1).

A table usually lists only z ≥ 0. For a negative z use the symmetry of the bell: Φ(−z) = 1 − Φ(z). A score of 55 gives z = −1.5 and Φ = 1 − 0.9332 = 0.0668, the 6th percentile.

z	Φ(z)	z	Φ(z)
0.0	0.5000	1.96	0.9750
0.5	0.6915	2.0	0.9772
1.0	0.8413	2.5	0.9938
1.5	0.9332	3.0	0.9987

Comparing two scores

Raw scores from different tests are not comparable, but z-scores are. Ann scores 85 on test A (μ = 70, σ = 10); Ben scores 88 on test B (μ = 80, σ = 8). Ben has more points, but his z = (88 − 80)/8 = 1.0 (84th percentile) is below Ann's z = 1.5 (93rd percentile): relative to the others who took the same test, Ann did better (Demo: compare scores from two tests).

Confidence intervals

In practice μ is unknown and we only see a sample. The page draws n = 25 scores and takes their mean x̄. By the central limit theorem x̄ is itself normal, with mean μ and standard deviation σ/√n, the standard error (10/5 = 2 points). Since 95% of a normal lies within 1.96 standard deviations, x̄ lands within 1.96·σ/√n of μ in 95% of samples, so the 95% confidence interval

x̄ ± 1.96 · σ / √n   =   x̄ ± 3.92   (σ = 10, n = 25)

contains μ in 95% of samples. The page draws 20 seeded samples: 19 intervals cover μ = 70 and sample #12 (x̄ = 66.05) misses. Drawing 80 more brings the total to 95 of 100 (Demo: 95% confidence intervals cover μ).

Twenty vertical intervals, one per sample of 25 scores, each centred on its sample mean with half-width 3.92, around a horizontal line at mu = 70. Nineteen are green and cross the line; interval 12, centred at 66.05, is red and lies entirely below it. — The 95% is about the method: 19 of these 20 intervals catch μ; #12 is the unlucky one.

"95% confidence" does not mean a given interval has a 95% chance of holding μ: μ is a fixed number, so each interval either holds it or not. It means the recipe catches μ 95% of the time. A wider interval (99%, z = 2.58) misses less often; a bigger n makes every interval narrower, like 1/√n. The same intervals decide A/B tests: if the interval for the difference between two variants excludes 0, the difference is significant at the 5% level. Bayes' theorem gives the other view, a probability for μ itself after seeing the data.

See also Mean vs Median vs Mode: on a symmetric bell all three coincide at μ, but an outlier or a skewed tail pulls them apart.

For counts of rare events the better approximation is Poisson, not normal: Binomial vs Poisson.

See also hypothesis testing with a coin: the p-value as a tail area, α, type I/II errors and power; a 95% interval excludes a value exactly when the two-sided test rejects it at α = 0.05.

See also Confidence Intervals: hundreds of repeated samples, the effect of n and of the 80–99% level, and why exactly the samples whose x̄ falls outside μ ± E miss.

Generative diffusion models add normal noise to data step by step until it becomes N(0, I), then learn to remove it.

What the page leaves out

Unknown σ: with the sample standard deviation s in place of σ, the interval uses Student's t with n − 1 degrees of freedom instead of 1.96 (2.06 for n = 25).
How Φ is computed: it has no closed form; the page uses a polynomial approximation of the error function.
Checking normality (Q–Q plots), skewed and heavy-tailed data, and transformations such as logs.
One-sided intervals, hypothesis tests and p-values, and intervals for proportions.
The multivariate normal and correlated variables.