Live model
Locate an observation and interval inside a normal model
Calculate the observation's standard score, density, percentile, upper tail, interval probability, and symmetric extremeness tail. Density is kept separate from probability so a curve height is not misreported as the chance of one exact continuous value.
| Landmark | Raw value | z-score | CDF | Interpretive role |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
z=(x-mu)/sigma; f(x)=phi(z)/sigma; P(X<=x)=Phi(z); P(a<=X<=b)=Phi(zb)-Phi(za)
The CDF uses a numerical error-function approximation after standardization. The PDF divides standard-normal height by sigma, preserving density units. The two-sided tail doubles the one-sided standard-normal tail beyond |z|.
Use the calculator
Five steps for interpreting a normal observation
- Verify a normal model is defensible. Review data shape, process knowledge, and sampling design first.
- Enter mu and a positive sigma. Keep all raw values in the same measurement unit.
- Enter the observation x. Its z-score states standardized distance, not probability.
- Define an ordered interval. Use it for a tolerance, central range, or decision window.
- Match the result to the question. Use density for local curve height, CDF for rank, upper tail for exceedance, and CDF difference for interval mass.
Five foundations
Normal density and probability
1. The model is continuous
A single exact value has probability zero. Probability belongs to intervals and tails; density measures local concentration per unit.
2. mu sets location
Moving mu shifts every raw-value landmark without changing standardized shape when sigma remains fixed.
3. sigma sets scale
A larger sigma spreads mass and lowers peak density. sigma must be strictly positive.
4. z is dimensionless
Standardization allows values in dollars, millimeters, or seconds to use the same standard normal CDF.
5. CDF and upper tail reconcile
At the same x, P(X<=x)+P(X>x)=1. For a continuous variable, endpoint inclusion has no numerical effect.
Calculation anatomy
Symbols, units, and default calculation
| Symbol | Meaning | Unit |
|---|---|---|
| mu | Distribution mean | measurement unit |
| sigma | Distribution SD | same unit, positive |
| x | Observation | same unit |
| z | Standardized observation | dimensionless |
| phi(z) | Standard-normal density | dimensionless |
| f(x) | Raw-unit density | per measurement unit |
| Phi(z) | Standard-normal CDF | 0-1 |
Defaults give z=(118-100)/15=1.2. The raw density equals phi(1.2)/15, the percentile is Phi(1.2), and interval mass is Phi((115-100)/15)-Phi((85-100)/15).
Deep analysis
Three interpretations that should not be collapsed
Percentile position
The CDF says what proportion lies at or below x. It is a rank-like model statement, not the probability that x itself occurs.
Interval coverage
Subtract endpoint CDFs for a tolerance or central band. Using density times width is only a local approximation for very narrow intervals.
Symmetric extremeness
The two-sided tail compares absolute distance from mu in both directions. It does not automatically constitute a hypothesis-test p-value without a testing design.
Decision cases
Measurement and scale-boundary examples
Inspection measurement
A dimension averages 100 mm with SD 15 mm. An observation of 118 mm is located by z and percentile, while the 85-115 mm interval is evaluated separately as a coverage requirement.
Near-zero spread entry
As sigma becomes very small, z and density can become extreme and parameter error dominates. sigma=0 is rejected because it defines a point mass rather than a normal density.
Terms
Normal-distribution vocabulary
- Probability density
- Curve height per unit, not point probability.
- Cumulative distribution
- Probability at or below a value.
- Upper tail
- Probability above an observation.
- z-score
- Signed standard deviations from the mean.
- Interval mass
- CDF difference between ordered endpoints.
- Two-sided tail
- Mass at least as distant from mu in either direction.
FAQ
Questions about normal density and tails
Is density at x a probability?
No. It is probability per unit near x.
What does percentile report?
The modeled proportion at or below x.
Why must sigma be positive?
Zero spread is not a normal density.
How is interval probability found?
Subtract the lower endpoint CDF from the upper endpoint CDF.
What is the symmetric two-sided tail?
Combined mass beyond +/-|z| from the mean.
Can I use it for skewed measurements?
Only if a normal approximation is justified for the decision, especially in tails.
Limits and evidence
Distribution-model boundaries
- The normal model and fixed mu and sigma are assumed, not fitted from entered raw data.
- Density values carry inverse measurement units and are not probabilities.
- The numerical CDF approximation is designed for practical calculation, not symbolic proof.
- Tail decisions can be sensitive to outliers, skew, mixtures, and estimated parameters.
- The symmetric tail is descriptive unless embedded in a pre-specified test.
Sources and related tools