Normal Distribution Calculator
Calculate normal bell curve cumulative probabilities ($P(X < x)$), probability density ($f(x)$), Z-scores, and percentiles for $N(\mu, \sigma^2)$.
Calculate Normal Bell Curve Probabilities
Enter observed value (X), mean (μ), and standard deviation (σ).
Calculation Results
Calculated using normal distribution density & cumulative equations: f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}
Quick Summary
The Normal Distribution Calculator evaluates Gaussian bell curve probabilities $P(X < x)$, density $f(x)$, standardized Z-score, and percentile ranks.
Formula Explanation
f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)
P(X < x) = \Phi\left( \frac{x - \mu}{\sigma} \right)How It Works
The normal distribution (Gaussian curve) models natural phenomena where data clusters symmetrically around the mean. The Normal Distribution Calculator standardizes $X$ into $Z = \frac{X - \mu}{\sigma}$ and calculates cumulative bell curve area probabilities.
Step-by-Step Worked Example
Practical Problem: For an IQ distribution with mean $\mu = 100$ and SD $\sigma = 15$, find the probability $P(X < 115)$.
- Step 1: Calculate Z-score: $Z = (115 - 100) / 15 = 15 / 15 = \mathbf{+1.0000}$.
- Step 2: Evaluate cumulative left-tail area $P(Z < 1.0000)$: $P(Z < 1.00) = \mathbf{0.8413}$.
- Step 3: Calculate right-tail probability $P(X > 115)$: $1 - 0.8413 = \mathbf{0.1587}$.
- Step 4: Calculate probability density $f(115)$: $f(115) = \mathbf{0.016209}$.
- Step 5: Interpretation: 84.13% of the population has an IQ score of 115 or below.
Real-World Calculation Examples
Scenario 1: IQ Score 115 (μ = 100, σ = 15)
Parameters: X = 115
Result: P(X < 115) = 0.8413 (84.13th percentile).
Scenario 2: Adult Male Height (X = 73 in, μ = 70 in, σ = 3 in)
Parameters: X = 73
Result: P(X < 73) = 0.8413 (Z = +1.00).
Scenario 3: Factory Bottle Filling (X = 500ml, μ = 505ml, σ = 2.5ml)
Parameters: X = 500
Result: P(X < 500) = 0.0228 (2.28% underfill risk).
Scenario 4: High SAT Score (X = 1400, μ = 1050, σ = 200)
Parameters: X = 1400
Result: P(X < 1400) = 0.9599 (Top 4% score).
Key Benefits of Using This Calculator
General Normal Distribution $N(\mu, \sigma^2)$
Handles any custom mean $\mu$ and standard deviation $\sigma$ directly.
Density & Cumulative Integration
Outputs both point probability density $f(x)$ and cumulative area $P(X < x)$.
Z-Score & Percentile Sync
Outputs standardized Z-score and percentile rank %.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a normal distribution?
A normal distribution (Gaussian distribution) is a symmetric bell-shaped continuous probability distribution defined by mean $\mu$ and variance $\sigma^2$.
What is the Central Limit Theorem (CLT)?
CLT states that the distribution of sample means approaches a normal distribution as sample size $n$ increases ($n \ge 30$), regardless of the population shape!
What is the 68-95-99.7 Empirical Rule?
In any normal curve: 68.27% of data lies within $\mu \pm 1\sigma$, 95.45% within $\mu \pm 2\sigma$, and 99.73% within $\mu \pm 3\sigma$.
How do I calculate normal probabilities in Excel?
Use formula =NORM.DIST(x, mean, standard_dev, TRUE).
What is the probability of an exact single point in a continuous distribution?
The probability of an exact single point $P(X = x)$ in any continuous distribution is technically 0; density $f(x)$ gives the height of the curve.
What is difference between $N(0,1)$ and $N(\mu, \sigma^2)$?
$N(0,1)$ is the Standard Normal Distribution ($\mu=0, \sigma=1$); $N(\mu, \sigma^2)$ is a general normal distribution.
What is skewness of a normal distribution?
The skewness of a perfectly normal distribution is 0 (perfect symmetry).
What is kurtosis of a normal distribution?
The kurtosis of a normal distribution is 3 (excess kurtosis = 0, mesokurtic).
How are normal distributions used in Quality Control (Six Sigma)?
Six Sigma uses normal curves to set tolerance limits at $\pm 6\sigma$, aiming for 3.4 defects per million.
What is normal probability plot / Q-Q plot?
A Q-Q plot compares sample quantiles against theoretical normal quantiles to assess normality.