Standard normal CDF · one-tail and two-tail map
Normal Tail Probability Calculator
Enter a z-score directly or standardize an observation from a normal mean and standard deviation. Compare left, right, and symmetric two-tail areas without losing track of which event each probability represents.
Set the cutoff
Continuity note: For a continuous normal variable, using < versus ≤ or > versus ≥ does not change probability at one exact cutoff.
Tail-area map
About 2.4998% of a standard normal distribution lies at or above z = 1.96. The symmetric two-tail area is about 4.9996%.
Normal tails and the standard score
A z-score states how many standard deviations an observation lies from a normal mean. Standardization uses z = (x−μ)/σ. A positive z is above the mean and a negative z is below it. Once standardized, probabilities use the same standard normal curve with mean zero and standard deviation one.
The cumulative distribution Φ(z) is the area to the left of z. The right tail is 1−Φ(z). A symmetric two-tail probability beyond |z| adds the lower area below −|z| and upper area above +|z|.
P(Z ≤ z) = Φ(z); P(Z ≥ z) = 1 − Φ(z)
P(|Z| ≥ |z|) = 2[1 − Φ(|z|)]
P(−|z| ≤ Z ≤ |z|) = 1 − two-tail probability.
Worked cutoff z = 1.96
At z = 1.96, the left-tail area is approximately 0.975002 and the right-tail area is approximately 0.024998. Reflecting the cutoff gives Φ(−1.96) ≈ 0.024998 by normal symmetry.
The symmetric two-tail area is twice the right tail beyond +1.96, approximately 0.049996. The central area between −1.96 and +1.96 is about 0.950004. Rounded critical-value tables often describe these as 2.5%, 5%, and 95%.
The density at 1.96 is about 0.058441. Density is curve height, not a probability at the exact point. A continuous variable has zero probability of equaling one exact value; probabilities come from areas over intervals.
Left, right, and two-tail events
Left tail
Use when the event asks for values below a cutoff, such as P(X ≤ x). Negative z-values often make this tail small.
Right tail
Use for values above a cutoff, such as P(X ≥ x). Positive z-values often make this tail small.
Symmetric two tail
Use for values at least |z| standard deviations from the mean in either direction. It is not every arbitrary two-sided interval.
Tail choice should follow the event or prespecified alternative, not whichever probability is smaller. A two-sided hypothesis p-value doubles the tail at an equally extreme magnitude under a symmetric null model; a directional one-sided p-value uses one tail.
Raw observations and units
In raw mode, x, μ, and σ must share units. If x is inches and μ is centimeters, standardization is invalid until conversion. The standard deviation must be positive; a zero spread does not define a continuous normal distribution.
Suppose test scores are modeled normal with mean 100 and SD 10. A score of 119.6 gives z = (119.6−100)/10 = 1.96 and therefore the same right-tail probability. Converting back uses x = μ + zσ.
The z-score is unitless, while the density of a raw normal variable has inverse units. This calculator displays standard normal density after transformation.
Symmetry and complements as audit checks
The standard normal is symmetric: Φ(−z)=1−Φ(z). Left and right areas at one cutoff must sum to one. The central area and symmetric two-tail area must also sum to one. The dashboard exposes all four values to catch a tail-selection or subtraction error.
At z=0, left and right tails are both 0.5, two-tail beyond |0| is 1, and the central interval from −0 to +0 has probability zero. At very large positive z, left approaches one and right approaches zero without reaching either for finite z.
Tail odds shown as 1 : k divide the complement probability by the selected right-tail probability. They are a descriptive ratio, not gambling odds or a likelihood ratio.
Approximation and numerical accuracy
The browser uses a standard high-accuracy error-function approximation. Ordinary displayed probabilities agree closely with statistical tables, but extreme tails can underflow or lose relative precision in floating-point subtraction from one.
A display of 0.000000 can mean a positive probability smaller than half the final shown unit. Increase decimals or use specialist survival-function software for z magnitudes above about eight, where direct 1−CDF calculations are numerically delicate.
Published tables may round z or probabilities. A table entry for 1.96 often shows 0.9750, while this calculator retains more digits. Do not infer measurement precision from extra probability digits.
Normal model assumptions
The calculation is exact for a variable that truly follows the specified normal distribution. Real measurements may be skewed, heavy-tailed, bounded, discrete, mixed, or affected by outliers. A normal approximation should be justified by design, theory, diagnostics, or a limit theorem.
A z test for a sample statistic needs its sampling distribution and standard error, not merely the raw-data standard deviation. Estimated standard deviations and small samples often lead to a t distribution.
For binomial counts, Poisson events, and other discrete distributions, continuity and distribution-specific probabilities matter. Converting everything to z without checking approximation conditions can misstate tails.
Percentiles and critical values
The percentile is 100Φ(z). A z of 1.96 is near the 97.5002nd percentile, meaning about 97.5% lies at or below it under the model. Percentile does not mean the observation scored 97.5% on its original scale.
Critical values reverse the CDF question: find z for a chosen tail area. Common approximate values include 1.644854 for an upper 5% tail, 1.959964 for symmetric 5% total tails, and 2.575829 for symmetric 1% total tails.
Match one-sided or two-sided confidence conventions carefully.
Decision boundary: A small normal tail is a probability under a specified model. It is not by itself evidence of causality, fraud, clinical abnormality, or practical importance.
Sampling distributions and standard errors
A raw observation has standard deviation σ under a normal population model. A sample mean of n independent observations has standard error σ/√n. These denominators answer different questions. Standardizing a sample mean with the raw σ instead of its SE makes the z magnitude too small.
For example, if individual values have mean 100 and σ = 10, the mean of 25 independent values has SE = 2. A sample mean of 104 is therefore z = (104−100)/2 = 2, not 0.4. The right-tail probability for the mean is approximately 0.0228 under the known-σ normal model.
If σ is estimated from the same small sample, a t statistic normally replaces z. Dependence, finite populations, weights, or clusters change the standard error further. Identify the random quantity before selecting the denominator.
Finding a cutoff from a target tail
An inverse-normal calculation starts with probability and returns z. For an upper tail α, find z satisfying Φ(z)=1−α. A 5% upper tail gives z≈1.644854. For a symmetric two-tail total α, use upper tail α/2; total 5% gives z≈1.959964.
One-sided and two-sided critical values differ because two-sided probability is divided across both tails. Using 1.645 for a 95% two-sided interval leaves about 10% outside, not 5%. Always state the tail allocation.
Percentile ranks reverse the same relationship. The 90th percentile has left area 0.90 and z≈1.281552. A lower 10th percentile is its reflection, −1.281552.
Extreme-tail computation
For moderate z, computing the right tail as 1−Φ(z) is accurate enough. When Φ(z) is extremely close to one, subtraction removes leading significant digits, a problem called catastrophic cancellation. Statistical libraries therefore provide a survival function calculated directly.
Log survival probabilities are useful when a tail is smaller than the smallest representable ordinary number. Scientific software may return log P rather than zero. Asymptotic Mills-ratio approximations provide another check for large positive z.
This calculator limits |z| to 12 and is designed for educational probabilities, not genome-scale thresholds or reliability calculations requiring accurately resolved probabilities near 10⁻³⁰. Cross-check such work with a specialist library.
Three event-translation examples
For “below 85” with μ=100 and σ=10, z=−1.5 and the requested event is the left tail. For “above 115,” z=1.5 and the requested event is the right tail. Symmetry makes those two probabilities equal under the same normal model.
For “more than 15 units away from 100,” the cutoffs are 85 and 115, so use the symmetric two-tail event beyond |z|=1.5. For “between 85 and 115,” use the central probability within ±1.5. These are complements.
An interval such as 90 through 120 is not symmetric about 100. Calculate Φ(2)−Φ(−1), rather than using the central ±z output. Event wording should be converted into explicit inequalities before arithmetic.
Frequently asked questions
Is the right tail at 1.96 equal to 5%?
No. It is about 2.5%; the symmetric area beyond ±1.96 is about 5%.
Does ≤ differ from < for a normal variable?
No. One exact point has probability zero in a continuous distribution.
What does a negative z-score mean?
The observation lies below the mean by that many standard deviations.
Is density the same as probability?
No. Density is curve height; integrate over an interval to obtain probability.
Can I use sample mean and sample SD as μ and σ?
You can standardize descriptively, but inferential sampling uncertainty may require a t distribution or another model.
Why does an extreme tail display zero?
It may be below display or floating-point precision rather than mathematically zero.
When the probability is known but the corresponding cutoff is not, solve the inverse problem with the inverse normal distribution calculator.
References
NIST Digital Library of Mathematical Functions — error functions and normal probabilities