Normal Distribution Probability Calculator
Calculate normal distribution interval probability ($P(a < X < b)$), lower and upper Z-scores, and population area percentages.
Calculate Interval Probability Between Two Bounds
Enter mean (μ), standard deviation (σ), lower bound (a), and upper bound (b).
Calculation Results
Calculated using normal distribution cumulative interval integration: P(a < X < b) = \Phi\left(\frac{b - \mu}{\sigma}\right) - \Phi\left(\frac{a - \mu}{\sigma}\right)
Quick Summary
The Normal Distribution Probability Calculator evaluates the probability $P(a < X < b)$ that a random variable falls between lower bound $a$ and upper bound $b$.
Formula Explanation
P(a < X < b) = P(X < b) - P(X < a) = \Phi(Z_b) - \Phi(Z_a)
Z_a = \frac{a - \mu}{\sigma}, \quad Z_b = \frac{b - \mu}{\sigma}How It Works
To find the probability between two limits $a$ and $b$, convert both limits into Z-scores ($Z_a$ and $Z_b$) and subtract cumulative left-tail area $\Phi(Z_a)$ from $\Phi(Z_b)$. The Normal Distribution Probability Calculator computes exact bell curve interval probabilities instantly.
Step-by-Step Worked Example
Practical Problem: For an IQ distribution with mean $\mu = 100$ and SD $\sigma = 15$, find the probability of scoring between $a = 85$ and $b = 115$.
- Step 1: Calculate lower Z-score ($Z_a$): $Z_a = (85 - 100) / 15 = -15 / 15 = \mathbf{-1.0000}$.
- Step 2: Calculate upper Z-score ($Z_b$): $Z_b = (115 - 100) / 15 = +15 / 15 = \mathbf{+1.0000}$.
- Step 3: Look up cumulative areas: $\Phi(-1.00) = 0.1587$; $\Phi(+1.00) = 0.8413$.
- Step 4: Subtract cumulative probabilities: $P(85 < X < 115) = 0.8413 - 0.1587 = \mathbf{0.6826\text{ (68.26\%)}}.$
- Step 5: Interpretation: 68.26% of the population scores between 85 and 115 on the IQ test ($\pm 1\sigma$).
Real-World Calculation Examples
Scenario 1: IQ Score Between 85 & 115 (±1σ)
Parameters: a = 85, b = 115, μ = 100, σ = 15
Result: P(85 < X < 115) = 0.6827 (68.27% of population).
Scenario 2: IQ Score Between 70 & 130 (±2σ)
Parameters: a = 70, b = 130, μ = 100, σ = 15
Result: P(70 < X < 130) = 0.9545 (95.45% of population).
Scenario 3: Factory Parts Within Spec (495mm to 505mm)
Parameters: a = 495, b = 505, μ = 500, σ = 2.5
Result: 95.45% Yield.
Scenario 4: Exam Scores Between 70 & 85
Parameters: a = 70, b = 85, μ = 75, σ = 8
Result: P(70 < X < 85) = 0.6284.
Key Benefits of Using This Calculator
Dual Bound Integration ($a$ to $b$)
Calculates exact area probability under the bell curve between any lower bound $a$ and upper bound $b$.
Lower & Upper Z-Score Output
Displays $Z_a$ and $Z_b$ standardized scores simultaneously.
Tail Probability Breakdown
Outputs lower tail $P(X < a)$ and upper tail $P(X > b)$ areas.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
How do I calculate normal distribution probability between two numbers?
Calculate Z-scores for both numbers ($Z_a$ and $Z_b$) and subtract cumulative left-tail area $\Phi(Z_a)$ from $\Phi(Z_b)$.
What is 68-95-99.7 Empirical Rule for normal intervals?
$\mu \pm 1\sigma \rightarrow 68.27\%$, $\mu \pm 2\sigma \rightarrow 95.45\%$, and $\mu \pm 3\sigma \rightarrow 99.73\%$.
How do I calculate normal interval probability in Excel?
Use formula =NORM.DIST(b, mean, sd, TRUE) - NORM.DIST(a, mean, sd, TRUE).
What if lower bound a is negative infinity?
If $a = -\infty$, the interval probability equals the left cumulative probability $P(X < b)$.
What if upper bound b is positive infinity?
If $b = +\infty$, the interval probability equals the right tail probability $P(X > a)$.
Can normal interval probability be negative?
No — since $b > a$, $\Phi(Z_b) \ge \Phi(Z_a)$, so probability is always between $0.0$ and $1.0$.
What is tolerance interval in quality engineering?
A tolerance interval calculates bounds $[a, b]$ expected to contain a specified proportion (e.g. 99%) of future production units.
How are normal interval probabilities used in finance (VaR)?
Value at Risk (VaR) calculates the maximum expected financial loss within a specified confidence interval.
What is central 90% normal interval boundary?
Central 90% area falls between $Z = -1.645$ and $Z = +1.645$ ($\mu \pm 1.645\sigma$).
What is central 95% normal interval boundary?
Central 95% area falls between $Z = -1.96$ and $Z = +1.96$ ($\mu \pm 1.96\sigma$).