March 28, 20264 min read

Normal Distribution Calculator — Bell Curve, Z-Score & Probabilities

Calculate normal distribution probabilities with mean μ and standard deviation σ. Covers the 68-95-99.7 rule, Z-scores, and how to read a Z-table with worked examples.

normal distribution bell curve Z-score statistics calchub

The bell curve shows up constantly in nature and statistics — IQ scores, heights, measurement errors, financial returns. Understanding it means knowing what's "normal" vs. genuinely unusual. Use the CalcHub Normal Distribution Calculator to find areas and probabilities under the curve.

The Normal Distribution Formula

The probability density function (PDF) for a normal distribution:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Where μ (mu) is the mean and σ (sigma) is the standard deviation.

The shape is completely defined by just these two parameters. Change μ and the curve slides left or right. Change σ and it stretches wider or compresses taller.

The 68-95-99.7 Rule (Empirical Rule)

For any normal distribution:

Range	Percentage of Data
μ ± 1σ	68.27%
μ ± 2σ	95.45%
μ ± 3σ	99.73%
μ ± 4σ	99.994%

This means only 0.27% of values fall more than 3 standard deviations from the mean — events beyond 3σ are genuinely rare.

Converting to Z-Scores

To use standard normal tables, convert any value to a Z-score:

$$Z = \frac{x - \mu}{\sigma}$$

Z tells you how many standard deviations a value sits from the mean.

Example: Adult male heights are normally distributed with μ = 175 cm, σ = 7 cm. What's the probability of a man being shorter than 165 cm?

Z = (165 − 175) / 7 = −1.43

From the Z-table, P(Z < −1.43) ≈ 0.0764, so about 7.6% of men are shorter than 165 cm.

Z-Table Excerpt (Cumulative Probabilities for Z ≥ 0)

Z	.00	.01	.02	.03	.04	.05
0.0	.5000	.5040	.5080	.5120	.5160	.5199
0.5	.6915	.6950	.6985	.7019	.7054	.7088
1.0	.8413	.8438	.8461	.8485	.8508	.8531
1.5	.9332	.9345	.9357	.9370	.9382	.9394
2.0	.9772	.9778	.9783	.9788	.9793	.9798
2.5	.9938	.9940	.9941	.9943	.9945	.9946
3.0	.9987	.9987	.9987	.9988	.9988	.9989

For negative Z values, use the symmetry: P(Z < −a) = 1 − P(Z < a)

Finding Probabilities Between Two Values

Example: IQ scores: μ = 100, σ = 15. What fraction of people score between 85 and 115?

Z₁ = (85 − 100) / 15 = −1.00 → P = 0.1587
Z₂ = (115 − 100) / 15 = +1.00 → P = 0.8413

P(85 < X < 115) = 0.8413 − 0.1587 = 0.6827 (the 68% rule confirmed)

Standard Normal vs. General Normal

The standard normal distribution has μ = 0 and σ = 1 — it's just the reference form. Any normal distribution can be standardized using the Z-score formula. Statistical tables only need to cover this one standard form.

What makes the normal distribution so common?

The Central Limit Theorem: the average of many independent random variables tends toward a normal distribution, regardless of the original distribution. This is why errors average out to bell-shaped distributions in measurement, and why sample means are normally distributed even when individual data isn't.

What's the difference between a PDF and a CDF?

The PDF (probability density function) gives the height of the bell curve at any point — it doesn't directly give probabilities for a specific value (since that probability is technically zero for any single point in a continuous distribution). The CDF (cumulative distribution function) gives the area to the left of a value — that's the actual probability you're after.

Can data be "approximately" normal?

Yes, and this is how it usually works in practice. You can test normality with a histogram, Q-Q plot, or formal tests like Shapiro-Wilk or Kolmogorov-Smirnov. Many statistical methods are robust to moderate departures from normality, especially with larger sample sizes.