Z Score Probability Calculator

Calculator Inputs

Calculation Mode

Probability Type

Decimal Precision

Observed Value (x)

Mean (μ)

Standard Deviation (σ)

Lower Bound

Upper Bound

Use raw mode when you know the original value, mean, and standard deviation. Use direct z mode when the score is already standardized.

How to Use This Calculator

Select Raw Value to Z Score or Direct Z Score.
Choose the probability region: left tail, right tail, between bounds, or outside bounds.
Enter the required values. Raw mode needs mean and standard deviation.
Set the decimal precision for detailed reporting.
Press Calculate Probability to show results above the form.
Review the z score, probability, percentile, density, and interpretation.
Use the CSV and PDF buttons to export your results.

Formula Used

1) Standardization formula

z = (x - μ) / σ

2) Standard normal density

f(z) = (1 / √(2π)) × e^(-z² / 2)

3) Left-tail probability

P(Z ≤ z) = Φ(z)

4) Right-tail probability

P(Z ≥ z) = 1 - Φ(z)

5) Probability between two z values

P(z₁ ≤ Z ≤ z₂) = Φ(z₂) - Φ(z₁)

6) Probability outside two z values

P(Z ≤ z₁ or Z ≥ z₂) = Φ(z₁) + [1 - Φ(z₂)]

Here, Φ(z) is the cumulative distribution function of the standard normal distribution.

Example Data Table

Observed Value	Mean	Std. Deviation	Z Score	Left-Tail Probability	Percentile Rank
50	50	10	0.0000	0.5000	50.00%
72	65	8	0.8750	0.8092	80.92%
80	70	10	1.0000	0.8413	84.13%
40	55	12	-1.2500	0.1056	10.56%
90	70	15	1.3333	0.9088	90.88%

What This Calculator Reports

Z score conversion Left and right tails Between and outside intervals Percentile rank Normal density Two-tailed p value Sigma band interpretation Chart shading CSV export PDF export

FAQs

1) What does a z score represent?

A z score shows how far a value sits from the mean in standard deviation units. Positive scores are above the mean, and negative scores are below it.

2) When should I use raw mode?

Use raw mode when you know the observed value, the distribution mean, and the standard deviation. The calculator converts that value into a z score first.

3) When should I use direct z mode?

Use direct z mode when your data is already standardized. This is common in statistics textbooks, probability tables, and model validation workflows.

4) What is left-tail probability?

Left-tail probability is the area under the normal curve from negative infinity up to the selected z score or value. It gives cumulative probability.

5) What is right-tail probability?

Right-tail probability is the area from the selected z score to positive infinity. It is useful for exceedance analysis, risk thresholds, and one-sided tests.

6) What does percentile rank mean here?

Percentile rank gives the percentage of observations expected to fall at or below the selected value under a normal distribution assumption.

7) Why must standard deviation be positive?

Standard deviation measures spread, so it cannot be zero or negative in this calculator. A nonpositive value would make standardization impossible.

8) Can I export the calculated results?

Yes. After a successful calculation, use the CSV or PDF buttons above the form results to save a structured summary for reporting or review.