How to Use the Normal Distribution Calculator
Use this calculator to find areas under a normal curve, convert raw values to z-scores, or work backward from a z-score or percentile to an x-value.
- Choose the calculation type you need.
- Enter the mean (μ) and standard deviation (σ).
- Enter the required x-value, bounds, z-score, or percentile.
- Press Calculate to see the probability, shaded bell curve, and steps.
For related statistics tasks, you can also use the Z-Score Calculator, Standard Deviation Calculator, and Probability Calculator.
Normal Distribution Formula
Z-score:
z = (x − μ) ÷ σ
Cumulative probability:
P(X ≤ x) = Φ(z)
Probability between two values:
P(a ≤ X ≤ b) = Φ(zb) − Φ(za)
Here, μ is the mean, σ is the standard deviation, and Φ(z) is the standard normal cumulative distribution function.
A normal distribution, also called a Gaussian distribution, is a continuous and symmetric probability distribution described by its mean and standard deviation. Its bell shape places the highest density near the mean and progressively less density farther from the center.
Worked Examples
Example 1: Probability Below a Value
Suppose exam scores follow a normal distribution with mean 75 and standard deviation 10. What proportion of scores are 85 or lower?
Step 1: z = (85 − 75) ÷ 10 = 1
Step 2: Φ(1) ≈ 0.841345
Answer: About 84.13% of scores are expected to be 85 or lower.
Example 2: Probability Between Two Values
Using the same distribution, find the probability that a score lies between 70 and 90.
z1 = (70 − 75) ÷ 10 = −0.5
z2 = (90 − 75) ÷ 10 = 1.5
P(70 ≤ X ≤ 90) = Φ(1.5) − Φ(−0.5)
P ≈ 0.933193 − 0.308538 = 0.624655
Answer: Approximately 62.47%.
Understanding the Bell Curve
The mean controls the center of a normal distribution and the standard deviation controls its spread. A smaller standard deviation creates a narrower curve, while a larger standard deviation spreads probability across a wider range. The total area under the curve equals 1, or 100% probability.
| Property | Meaning |
|---|---|
| Symmetry | The curve is symmetric around the mean. |
| Mean = Median = Mode | For a normal distribution, all three measures coincide at the center. |
| Total area = 1 | All possible values together represent 100% probability. |
| Mean (μ) | Determines the center of the distribution. |
| Standard deviation (σ) | Determines how widely values are spread around the mean. |
Z-Score and Probability Reference
| Z-Score | Left-Tail Φ(z) | Right-Tail | Central Area from −z to +z |
|---|---|---|---|
| 0.00 | 50.00% | 50.00% | 0.00% |
| 1.00 | 84.13% | 15.87% | 68.27% |
| 1.64 | 94.95% | 5.05% | 89.90% |
| 1.96 | 97.50% | 2.50% | 95.00% |
| 2.58 | 99.51% | 0.49% | 99.01% |
| 3.00 | 99.87% | 0.13% | 99.73% |
Common Normal Distribution Mistakes
- Using σ ≤ 0: A normal distribution requires a standard deviation greater than zero.
- Confusing left and right tails: P(X ≤ x) and P(X ≥ x) measure different areas of the curve.
- Treating density as probability: The height of a continuous density curve at one point is not the probability of that exact value.
- Assuming every dataset is normal: This calculator gives results under the assumption that the specified normal model is appropriate.
How to Check Your Result
Check the z-score first. A positive z-score places the x-value above the mean, while a negative z-score places it below the mean. Left-tail and right-tail probabilities for the same boundary should add to 1, and any probability between two valid bounds must remain between 0 and 1. The shaded region on the graph should also match the probability type you selected.
Manufacturing Tolerance Example
Suppose a process has a mean of 10.05 mm and a standard deviation of 0.15 mm, with acceptable limits from 9.8 mm to 10.2 mm. The modeled probability of falling inside the limits is about 79.4%.
If the process is also re-centered at 10.00 mm and its standard deviation is reduced to 0.10 mm, the probability inside 9.8–10.2 mm becomes about 95.45%, corresponding to a modeled defect rate of about 4.55%.
Calculation Methodology
Utilixea converts raw values to standard z-scores using z = (x − μ) ÷ σ and evaluates the standard normal cumulative distribution function. Interval probabilities are calculated from the difference between cumulative probabilities at the specified boundaries.
Last reviewed: July 22, 2026
Frequently Asked Questions
It calculates probabilities, z-scores and percentiles for a normally distributed variable using its mean and standard deviation.
A general normal distribution may have any mean and any positive standard deviation. The standard normal distribution has mean 0 and standard deviation 1.
Convert x to a z-score using z = (x − μ) ÷ σ, then evaluate Φ(z). This gives the area to the left of x.
Calculate the left-tail probability first, then subtract it from 1: P(X ≥ x) = 1 − Φ(z).
Convert both boundaries to z-scores, then subtract their cumulative probabilities: P(a ≤ X ≤ b) = Φ(zb) − Φ(za).
A negative z-score means the x-value lies below the mean. For example, z = −1 means the value is one standard deviation below the mean.
