Visual Normal Distribution & Z-Score Bell Curve Percentile Finder
A normal distribution $\mathcal{N}(\mu, \sigma^2)$ has probability density $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}((x-\mu)/\sigma)^2}$, where standardizing via $Z = \frac{x - \mu}{\sigma}$ measures how many standard deviations $x$ lies from the mean.
Z = (x − μ) / σ, Φ(Z) = ∫_{−∞}^Z (1/√2π)·e^(−t²/2) dtInputs & Parameters (3)
20-Row Comparison Table
| # | Population Mean (μ) | Left-Tail Percentile Φ(Z) | Raw Value x | Central ±|Z| Area |
|---|---|---|---|---|
| 01 | 0 | 89.435 % | 18.75 | 78.87% |
| 02 | 11 | 89.435 % | 29.75 | 78.87% |
| 03 | 21 | 89.435 % | 39.75 | 78.87% |
| 04 | 32 | 89.435 % | 50.75 | 78.87% |
| 05 | 42 | 89.435 % | 60.75 | 78.87% |
| 06 | 53 | 89.435 % | 71.75 | 78.87% |
| 07 | 63 | 89.435 % | 81.75 | 78.87% |
| 08 | 74 | 89.435 % | 92.75 | 78.87% |
| 09 | 84 | 89.435 % | 102.75 | 78.87% |
| 10 | 95 | 89.435 % | 113.75 | 78.87% |
| 11 | 105 | 89.435 % | 123.75 | 78.87% |
| 12 | 116 | 89.435 % | 134.75 | 78.87% |
| 13 | 126 | 89.435 % | 144.75 | 78.87% |
| 14 | 137 | 89.435 % | 155.75 | 78.87% |
| 15 | 147 | 89.435 % | 165.75 | 78.87% |
| 16 | 158 | 89.435 % | 176.75 | 78.87% |
| 17 | 168 | 89.435 % | 186.75 | 78.87% |
| 18 | 179 | 89.435 % | 197.75 | 78.87% |
| 19 | 189 | 89.435 % | 207.75 | 78.87% |
| 20 | 200 | 89.435 % | 218.75 | 78.87% |
Key Analytical Takeaways
- 01At $Z = 0$ ($x = \mu$), the bell curve splits into two symmetric $50.0\%$ halves ($\Phi(0) = 0.5000$).
- 02With default IQ parameters ($\mu = 100, \sigma = 15$), a score of $x = 130$ corresponds to $Z = +2.00$, placing it in the $97.72\text{th}$ percentile.
- 03Both the cumulative left-tail percentile $P(X \le x)$ and symmetric central area $P(|Z| \le |z|)$ update in real time.
Frequently Asked Questions
What does a negative Z-score mean?+
A negative Z-score simply means the raw observation x is below the population mean μ, placing it in the lower 50% (left half) of the bell curve.
Why doesn't the normal distribution have a simple closed-form integral?+
The Gaussian integral ∫ e^(−t²/2) dt cannot be expressed in elementary functions; probabilities are computed using the error function erf(z/√2) or high-precision rational approximations.
How do I convert a two-tailed probability to a one-tailed Z-score?+
Because the Gaussian bell curve is symmetric around Z = 0, a central confidence area (1 − α) splits missing tail probability equally into α/2 on each side, so look up Φ⁻¹(1 − α/2).