Normal Distribution Tail Calculator

Calculator Input

Enter raw values or z-scores, choose a tail type, and calculate probabilities, percentiles, and critical cutoffs instantly.

Mean (μ)

Standard Deviation (σ)

Input Mode

Tail Type

Primary Value / z₁

Upper Value / z₂

Confidence Level (%)

Decimal Places

Distribution Plot

The curve shows the normal distribution. The shaded region marks the selected tail probability.

Example Data Table

Case	Mean	SD	Tail Type	x₁	x₂	Use Case
Exam scores	70	10	Right tail	85	-	Chance of scoring at least 85
Manufacturing	50	4	Left tail	44	-	Probability of falling below tolerance
Service times	18	3	Between	15	21	Middle process coverage band
z-test review	0	1	Two-tailed	2.1	-	Two-sided extremity probability

Formula Used

The standardized score is computed with: z = (x - μ) / σ

The standard normal cumulative distribution function is: Φ(z) = 0.5 × [1 + erf(z / √2)]

Tail probabilities are then computed as:

Left tail: P(X ≤ x) = Φ(z)
Right tail: P(X ≥ x) = 1 - Φ(z)
Between: P(x₁ ≤ X ≤ x₂) = Φ(z₂) - Φ(z₁)
Two-tailed: P(|Z| ≥ |z|) = 2 × [1 - Φ(|z|)]

The density at a z-score is: φ(z) = (1 / √(2π)) e^-z²/2

How to Use This Calculator

Enter the distribution mean and standard deviation.
Select whether your inputs are raw x values or z-scores.
Choose left, right, between, or two-tailed probability.
Enter the primary cutoff and upper cutoff if needed.
Set the confidence level for critical value references.
Pick the number of decimal places to display.
Press the calculate button to show probability results above the form.
Review the graph, metrics, and downloadable output files.

FAQs

1. What does a tail probability represent?

A tail probability is the area under the normal curve beyond a cutoff. It measures how likely values are in an extreme region, either below, above, or outside selected thresholds.

2. When should I use raw values instead of z-scores?

Use raw values when you know the original measurement scale, such as test scores or production weights. Use z-scores when data has already been standardized or when comparing across different scales.

3. What is the difference between right tail and two-tailed?

Right tail measures probability above one cutoff only. Two-tailed measures combined probability in both extremes, symmetrically away from the mean, and is common in two-sided hypothesis testing.

4. Why must standard deviation be positive?

Standard deviation measures spread. A zero or negative value would not define a valid normal distribution shape, so the calculator requires a positive number for accurate probability and density results.

5. What does the percentile output mean?

The percentile shows the percentage of observations expected at or below a selected value. For example, the 90th percentile means about 90% of values fall below that cutoff.

6. Are the interval boundaries inclusive?

Yes, the displayed labels use inclusive notation. For a continuous normal distribution, including or excluding exact boundary points does not change the probability because single points have zero area.

7. What are critical values used for?

Critical values help identify rejection regions, confidence interval cutoffs, and significance thresholds. They are useful in hypothesis testing, quality limits, and decision rules based on tail areas.

8. Can I use this for real business or research data?

Yes, as long as the normal model is appropriate. It works well for many standardized metrics, measurement errors, sampling distributions, and approximate process data under normality assumptions.