Normal Distribution (Bell Curve): what it means

The normal distribution (also called the Gaussian distribution) is the most famous probability distribution in statistics. It’s the classic bell curve: values cluster near the average (the mean) and thin out symmetrically as you move away from the center.

You’ll see the normal distribution everywhere because many real-world processes are the result of lots of small, independent effects adding together. Heights, measurement noise, standardized test scores, manufacturing variation, and many “errors” in science can often be approximated by a normal curve. Even when a dataset is not perfectly normal, the bell curve is still a powerful approximation tool.

• Mean (μ): the center of the curve. Half of the probability mass is on each side of μ.
• Standard deviation (σ): how wide the curve is. Larger σ = wider, flatter curve; smaller σ = tighter curve.

• PDF (probability density function) tells you the height of the curve at x. It’s not a probability by itself. Think “density at a point.”
• CDF (cumulative distribution function) tells you the area under the curve up to x. This is a probability: P(X ≤ x).

If a random variable X is normally distributed with mean μ and standard deviation σ, we write: X ~ N(μ, σ²). (Notice it’s σ² in the notation because variance is σ².)

PDF:
f(x) = 1 / (σ √(2π)) · exp( − (x − μ)² / (2σ²) )

CDF: There isn’t a simple elementary closed form for the normal CDF, so calculators use a reliable numerical approximation. A common way to express it is via the error function erf:
Φ(x) = 1/2 · [ 1 + erf( (x − μ) / (σ √2) ) ]

The z-score converts your x into “how many standard deviations from the mean”:
z = (x − μ) / σ

When you compute z, you can compare values across totally different distributions. For example, a test score of 88 might be “above average” in one class and “average” in another. Z-scores put them on the same scale.

For a normal distribution:

• About 68% of values lie within μ ± 1σ.
• About 95% of values lie within μ ± 2σ.
• About 99.7% of values lie within μ ± 3σ.

This is why z-scores matter: z = 1 is “one sigma away,” which is common; z = 3 is “three sigma away,” which is rare.

Worked examples (with the exact steps)

Examples are the fastest way to feel confident with the bell curve. Below are three classic normal distribution questions you’ll see in homework, exams, and real life.

Suppose adult heights (in a simplified model) follow N(170, 7²) centimeters. What is the probability someone is ≤ 180 cm?

• Step 1: Compute the z-score: z = (180 − 170) / 7 = 10/7 ≈ 1.4286.
• Step 2: Convert z to a probability using the normal CDF: P(X ≤ 180) = Φ(1.4286).
• Step 3: Interpret: this is the percentile. A value around 0.92 means “about the 92nd percentile.”

With this calculator, you’d enter μ = 170, σ = 7, x = 180, and it returns the CDF and percentile instantly.

A machine fills bottles with volume modeled as N(500, 8²) ml. What fraction of bottles are ≥ 515 ml?

• Step 1: z = (515 − 500) / 8 = 15/8 = 1.875.
• Step 2: Left tail: P(X ≤ 515) = Φ(1.875).
• Step 3: Right tail: P(X ≥ 515) = 1 − Φ(1.875).

The “1 − CDF” step is the most common place people make mistakes. This calculator shows both P(X ≤ x) and P(X ≥ x) so you can sanity-check yourself.

Exam scores are modeled as N(72, 10²). What is the probability a student scores between 60 and 85?

• Step 1: z_a = (60 − 72) / 10 = −1.2
• Step 2: z_b = (85 − 72) / 10 = 1.3
• Step 3: P(60 ≤ X ≤ 85) = Φ(1.3) − Φ(−1.2)

That “subtract CDFs” pattern is the key identity for interval probabilities. In the calculator, switch mode to “Between (a to b)” and enter a = 60, b = 85.

• PDF is not probability: f(100) is a density (height), not “the probability of being exactly 100.”
• σ must be positive: if σ = 0, the distribution collapses and the normal formulas break.
• Right tail needs 1 − CDF: P(X ≥ x) = 1 − P(X ≤ x).
• Order your bounds: if a > b, swap them or you’ll get a negative interval width.

What this calculator computes (and how)

Under the hood, the calculator does three things:

• Standardize your input into a z-score: z = (x − μ) / σ.
• Compute the CDF using an error-function approximation: Φ(z) = 1/2 [1 + erf(z/√2)].
• Compute the PDF directly using the exponential bell curve formula.

For interval mode, it calculates the left CDF at b and the left CDF at a, then subtracts: P(a ≤ X ≤ b) = Φ(b) − Φ(a) (with μ and σ applied inside the z-scores).

• Main probability meter: shows P(X ≤ x) for single mode, or P(a ≤ X ≤ b) for between mode.
• PDF: density at x (single mode) or midpoint density (between mode).
• CDF: Φ(x) (single mode) or both Φ(a) and Φ(b) (between mode).
• Z-score(s): z for x, or z_a and z_b.
• Percentiles: CDF × 100.

• Post “I’m in the top 5% for ___” by converting a value to a percentile.
• Share a screenshot of an interval probability (“What are the odds we finish between 2–3pm?”).
• Turn real-life questions into z-scores (how unusual is this week’s sales number?).
• Use it in group chats for quick probability checks before meetings or exams.

Frequently Asked Questions

• What’s the difference between PDF and CDF?

  The PDF is the height of the bell curve at x (a density). The CDF is the area to the left of x (a probability). If you want “what percent are below this value,” you want the CDF.

• Why can’t I get a “probability at exactly x”?

  For continuous distributions like the normal, the probability of one exact point is effectively zero. Probabilities come from intervals (areas under the curve), not single points. That’s why PDF is a density.

• How accurate is this calculator?

  It uses standard normal formulas and a well-known approximation for the error function (erf), which is very accurate for typical calculator use. For scientific computing, cross-check with software like R, Python (SciPy), or a statistics table.

• What if my data isn’t really normal?

  The normal distribution is often an approximation. If your data is skewed, has heavy tails, or is bounded (like 0–100%), the normal model might misestimate extreme probabilities. Still, it’s a useful first pass, especially near the center.

• What does “σ” mean in plain English?

  σ is “typical spread.” If σ is small, values are tightly clustered around μ. If σ is large, values vary widely. The 68–95–99.7 rule tells you how much of the data sits within 1, 2, and 3 standard deviations of the mean.

• Can I use this for z-tables?

  Yes. If you set μ = 0 and σ = 1, you’re using the standard normal. Then x becomes z and the CDF output matches a z-table value.

• How do I compute P(X > x) instead of P(X ≤ x)?

  Use the complement rule: P(X > x) = 1 − P(X ≤ x). The results box shows both left-tail and right-tail probabilities.

• Does this tool store my inputs?

  The calculator runs in your browser. If you click “Save Result,” it stores only the result summary in localStorage on this device so you can compare later. Nothing is sent anywhere.