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.

Related guides: the empirical rule (68-95-99.7) and how to read a z-table.

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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.

  1. Enter the mean (μ) and standard deviation (σ).
  2. Choose P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b).
  3. Enter x, or enter a and b for the interval case, then press Calculate.

Formulas Used

z = (x - μ) / σ
P(X ≤ x) = Φ(z)
P(X ≥ x) = 1 - Φ(z)
P(a ≤ X ≤ b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)

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.

Worked Example: Adult Heights

Suppose adult heights in a population follow a normal distribution with μ = 170 cm and σ = 10 cm. Three questions the calculator answers directly:

  1. P(X < 180): z = (180 − 170) ÷ 10 = 1.0, and Φ(1.0) ≈ 0.8413, about 84% of people are shorter than 180 cm.
  2. P(160 < X < 180): the bounds give z = −1 and z = +1, so the probability is Φ(1) − Φ(−1) ≈ 0.8413 − 0.1587 = 0.6827: the familiar 68% of the empirical rule.
  3. P(X ≥ 185): z = 1.5 and 1 − Φ(1.5) ≈ 1 − 0.9332 = 0.0668, roughly a 6.7% chance of being 185 cm or taller.

Each result is an area under the bell curve: left tail, central band, and right tail respectively. Sketching which region you need before calculating is the most reliable way to avoid picking the wrong probability type.

Related Resources

Frequently Asked Questions

What is the 68-95-99.7 (empirical) rule?

For any normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a fast sanity check: if your calculated probability for the band from mean minus sigma to mean plus sigma is far from 0.68, an input is probably wrong.

Why is the probability of X equal to an exact value zero?

For continuous distributions, probability is area under the curve, and the area above a single point is zero. That is why the calculator works with ranges - less than x, greater than x, or between a and b - rather than exact equality.

How do I go backwards from a percentile to a value?

That is the inverse problem: find z for your target percentile (for example, z of about 1.645 for the 95th percentile) from a standard normal table, then convert with x equals mu plus z times sigma. With mean 170 and sigma 10, the 95th percentile is about 186.4.

What is the difference between the standard normal and a general normal distribution?

The standard normal has mean 0 and standard deviation 1. Any normal variable converts to it through the z-transform z equals (x minus mu) over sigma, which is exactly what this calculator does internally before looking up probabilities.

When is a normal model a poor choice?

When data is strongly skewed (incomes), bounded near an edge (percentages close to 0 or 100), count-based with small counts, or heavy-tailed (financial returns). In those cases normal probabilities can badly underestimate extreme outcomes; consider lognormal, binomial, Poisson, or t-distributions instead.

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