Normal Distribution Calculator
Calculate probabilities, z-scores, and density values for normal distributions. Essential tool for statistical analysis, hypothesis testing, and Six Sigma quality control.
Normal Distribution Calculator Guide
This calculator finds probabilities and z-scores for a normally distributed variable X with mean μ and standard deviation σ. It answers questions like the share of values below a cutoff, above a cutoff, or inside an interval.
- Enter the mean (μ) and standard deviation (σ).
- Choose P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b).
- Enter x, or enter a and b for the interval case, then press Calculate.
Formulas Used
Φ(z) is the standard normal CDF. This tool uses a standard error function approximation, so small rounding differences versus z-tables are expected.
Worked Examples
Example: Exam scores
If exam scores are normal with μ = 70 and σ = 10, the share scoring 85 or lower is found by choosing P(X ≤ x) and setting x = 85. The probability is Φ((85 - 70)/10).
Example: Quality control interval
If a part length has μ = 15 cm and σ = 0.2 cm, the probability of landing between 14.7 and 15.3 is found by choosing the interval option and entering a = 14.7 and b = 15.3.
Example: Right tail probability
To find the chance of exceeding a threshold, select P(X ≥ x). This is a right tail probability computed as 1 - Φ(z).
Interpreting Z-Scores
A z-score tells you how far a value is from the mean, measured in standard deviations. This makes it easy to compare values across different scales. For example, z = 1 means the value is one standard deviation above the mean, and z = -2 means it is two standard deviations below the mean.
- z = 0: the value equals the mean.
- z > 0: the value is above the mean.
- z < 0: the value is below the mean.
- |z| grows as the value becomes more unusual under the model.
In many applications, z-scores around 2 or more in absolute value are treated as relatively rare outcomes (tail events), although the right threshold depends on your domain.
When a Normal Model Makes Sense
Normal distributions are a good fit for many measurement processes and for averages of many small, independent effects. You will often see the normal model used for:
- Measurement error and instrument noise.
- Manufacturing variation around a target dimension.
- Standardized test scores and scaled metrics.
- Normal approximations of other distributions when sample sizes are large.
If your data is strongly skewed, has hard bounds (like percentages near 0 or 100), or has heavy tails, a normal model can underestimate extreme outcomes.
PDF vs CDF
The density f(x) is the height of the bell curve at x. The probability P(X ≤ x) is the CDF, which is the area under the curve to the left of x. Probabilities come from areas, not from the height at a single point.
Common Pitfalls
- σ must be positive. A standard deviation of 0 does not define a valid normal curve.
- For interval probabilities, make sure a is less than or equal to b.
- Probabilities are areas under the curve, not heights. A density value f(x) can be greater than 1 when σ is small, but the total area is still 1.