Compute left, right, and interval normal probabilities instantly. Adjust means, deviations, and bounds with confidence. Download tables and visualize cumulative areas with interactive graphs.
| Scenario | Mean | Standard Deviation | Lower or Target | Upper | Probability |
|---|---|---|---|---|---|
| Left Tail | 100.00 | 15.00 | 85.00 | — | 0.158655 |
| Right Tail | 50.00 | 10.00 | 62.00 | — | 0.115070 |
| Between | 0.00 | 1.00 | -1.00 | 1.00 | 0.682689 |
| Outside | 500.00 | 80.00 | 420.00 | 620.00 | 0.225462 |
The cumulative distribution function for a normal variable is:
F(x) = 0.5 × [1 + erf((x − μ) / (σ × √2))]
The standard score is:
z = (x − μ) / σ
Common probability forms are:
This file uses a standard numerical approximation for the normal CDF, which is accurate for practical probability work, quality control, finance, forecasting, and statistical learning.
A normal CDF probability calculator helps you measure the chance that a value falls below, above, inside, or outside selected limits for a normal distribution. This is useful in statistics, manufacturing, finance, education, operations, and data analysis.
The calculator works with any mean and any positive standard deviation. That makes it suitable for both standard normal questions and custom distributions based on real business or research data. The included z-score outputs help you interpret how far each value is from the mean in standard deviation units.
Use left tail mode for values below a target. Use right tail mode for exceedance probability. Use between mode for range coverage. Use outside mode when you need the combined probability in both tails. The table, downloadable files, and graph make the result easier to verify and explain.
CDF means cumulative distribution function. It gives the probability that a normally distributed variable is less than or equal to a chosen value.
PDF shows density at a point. CDF shows accumulated probability up to a point. For actual probability questions, the CDF is usually the direct tool.
Yes. Enter any mean and any positive standard deviation. The calculator converts values into z-scores and then evaluates the normal cumulative probability.
Use left tail for values at or below a cutoff. Use right tail for values at or above a cutoff, such as exceedance or defect thresholds.
You need both when calculating probability between two limits or outside two limits. Single-tail calculations only require one value.
The result uses a reliable numerical approximation of the normal CDF. It is highly accurate for standard practical work, reporting, and educational use.
The z-score tells you how many standard deviations a value is above or below the mean. It helps compare values across normal distributions.
CSV is useful for spreadsheets and audits. PDF is useful for sharing, reporting, and keeping a fixed record of the calculated probability and inputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.