Normal Distribution

Adjust the mean and standard deviation of the bell curve and measure the area under it

Beginner

Mean μ0.0Standard deviation σ2.0Bound a−2.0Bound b2.0

Tip: drag on the chart to move the nearest bound.

P(−2.0 < X < 2.0)

0.6827

68.27% of the area under the curve

= Φ(+1.00) − Φ(−1.00) = 0.8413 − 0.1587

z-scores (z = (x − μ) / σ)

a = −2.0: z = (−2.0 − 0.0) / 2.0 = −1.00
b = 2.0: z = (2.0 − 0.0) / 2.0 = +1.00

What is the normal distribution?

The normal (or Gaussian) distribution is the symmetric, bell-shaped distribution described by two numbers: the mean μ, which sets where the peak is, and the standard deviation σ, which sets how spread out it is. Changing μ slides the curve left or right; increasing σ makes it lower and wider.

The curve is a probability density, so the height at a point is not a probability. Probabilities are areas: the chance that X lands between a and b is the area under the curve between them, and the total area is always 1. That is why the curve gets shorter when it gets wider.

Every normal distribution is a stretched and shifted copy of the standard normal Z, which has μ = 0 and σ = 1. The z-score converts a value x into "how many standard deviations from the mean", so one table of Φ(z) answers questions about any normal distribution.

Key formulas

Probability density

f(x) = (1 / (σ√(2π))) · e^{−(x − μ)² / (2σ²)}

μ: mean (center of the curve)
σ: standard deviation (spread), σ > 0

z-score

z = (x − μ) / σ

Probabilities from the CDF

P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)

Φ(z): standard normal CDF, P(Z ≤ z) = ½[1 + erf(z/√2)]

Useful identities: P(X > a) = 1 − Φ(z_a) and, by symmetry, Φ(−z) = 1 − Φ(z).

Empirical rule (68-95-99.7)

P(|X − μ| < σ) ≈ 68.27%, P(|X − μ| < 2σ) ≈ 95.45%, P(|X − μ| < 3σ) ≈ 99.73%

Standard normal reference values

z	Φ(z) = P(Z < z)	P(−z < Z < z)
0.00	0.5000	0.0000
0.50	0.6915	0.3829
1.00	0.8413	0.6827
1.645	0.9500	0.9000
1.96	0.9750	0.9500
2.00	0.9772	0.9545
2.576	0.9950	0.9900
3.00	0.9987	0.9973

How to use this visualizer

1
Move the μ slider and watch the whole curve slide without changing shape.
2
Move the σ slider. The curve flattens as it widens because the total area must stay 1. The vertical scale is fixed so you can see this.
3
Choose Between, Below or Above, then drag on the chart (or use the sliders) to move the bounds a and b. The shaded area is the probability.
4
Read the z-score of each bound and the calculation Φ(z_b) − Φ(z_a) in the panels below the chart.
5
Turn on the 68-95-99.7 bands. Set a = μ − σ and b = μ + σ and check that the probability is about 0.6827 for any μ and σ.

Where it shows up

Measurement error

Repeated measurements of the same quantity scatter around the true value in a bell shape.

Test scores

Standardized tests are often scaled so scores are roughly normal, which makes percentiles easy to compute from z-scores.

Quality control

Factories flag parts more than 3σ from the target. Under a normal model only 0.27% of good parts fall that far out.

Biology

Traits influenced by many small independent factors, like adult height, are close to normal.

Why so common?

The central limit theorem says that the average of many independent random effects is approximately normal, almost whatever the individual effects look like. That is why the bell curve appears in so many unrelated places.

Practice problems

Problem 1: Area between two values

Adult heights in a population are normal with μ = 170 cm and σ = 8 cm. What fraction of people are between 162 cm and 186 cm tall?

Show solution

z for 162: (162 − 170) / 8 = −1

z for 186: (186 − 170) / 8 = 2

P = Φ(2) − Φ(−1) = 0.9772 − 0.1587

P ≈ 0.8186, about 81.9% of people

Problem 2: Working backwards from a probability

Exam scores are normal with μ = 70 and σ = 10. What score do you need to be in the top 5%?

Show solution

We need x with P(X > x) = 0.05, i.e. Φ(z) = 0.95.

From the table, Φ(1.645) = 0.95, so z = 1.645.

x = μ + zσ = 70 + 1.645 × 10

x ≈ 86.45, so a score of about 86.5 or higher

Discussion

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