Normal CDF Calculator

Find cumulative probabilities for the normal distribution of measurements, real-world data, and any continuous variable. This assumes you already know the mean and standard deviation.

X Value Value to evaluate P(X ≤ x)

Mean (μ) Center of distribution

Std Deviation (σ) Spread (must be > 0)

Try an Example:

Cumulative Probability P(X ≤ x)

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Probability Visualization

P(X ≤ x) = —

P(X > x) = —

Probability Summary

Left Tail P(X ≤ x)

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Right Tail P(X > x)

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Within ±x from μ

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Outside ±x from μ (Two-Tailed)

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Z-Score Conversion

Formula

z = (x − μ) / σ

Z-Score

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z = (— − —) / — = —

Enter values above to see interpretation

Inverse CDF (Percentile Lookup)

Desired Probability

Common percentiles:

Standard Normal Critical Values

Confidence Level	α (Two-Tailed)	Z Critical (±)	Interval Coverage
90%	0.10	±1.645	μ ± 1.645σ
95%	0.05	±1.960	μ ± 1.960σ
99%	0.01	±2.576	μ ± 2.576σ
99.9%	0.001	±3.291	μ ± 3.291σ

Empirical Rule: For any normal distribution, approximately 68% falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.

Calculation Methodology

Standardize the value

Convert x to z-score: z = (x − μ) / σ

Apply the CDF formula

Φ(z) = ½[1 + erf(z / √2)]

Interpret the probability

P(X ≤ x) gives the proportion of values at or below x

Understanding the Normal CDF

📊 What is the CDF?

CDF: Cumulative Distribution Function gives P(X ≤ x)
Returns the probability that a random variable is less than or equal to x
Always between 0 and 1, monotonically increasing
At x = μ, CDF = 0.5 (50% below the mean)

💡 Practical Applications

Quality control: finding defect rates
Finance: Value at Risk (VaR) calculations
Testing: determining p-values and critical regions
Grading: setting percentile-based cutoffs

The normal (Gaussian) distribution is fundamental in statistics, describing phenomena from measurement errors to IQ scores. The CDF tells you "what percentage falls below this value."

The Normal CDF… Explained:

The cumulative distribution function (CDF) of the normal distribution answers a fundamental question in statistics:

What is the probability that a randomly selected value falls at or below a specific point?

This probability accumulates from negative infinity up to your chosen value, giving you the total area under the bell curve to the left of that point.

For any normal distribution defined by its mean (μ) and standard deviation (σ), the CDF transforms raw values into probabilities between 0 and 1. At the mean, the CDF equals exactly 0.5, since half of all values fall below the center of a symmetric distribution.

Interpreting Your Result

When the calculator returns a value like 0.8413, this means 84.13% of all values in the distribution fall at or below your X value. Here’s how to translate that number into actionable insight:

Result close to 0 (e.g., 0.05): Your X value is in the lower tail. Only 5% of values fall at or below this point—it’s unusually low.
Result around 0.5: Your X value is near the mean. Roughly half the distribution falls below, half above.
Result close to 1 (e.g., 0.95): Your X value is in the upper tail. 95% of values fall below this point—it’s unusually high.

To find the probability of being above your X value, simply calculate 1 minus the CDF result. If P(X ≤ 72) = 0.81, then P(X > 72) = 0.19, meaning 19% of values exceed 72.

Common Normal CDF Lookup Scenarios

Test Scores and Percentiles

A class exam has a mean of 75 and standard deviation of 10. You scored 88. Entering X=88, μ=75, σ=10 gives a CDF of approximately 0.9032. This means you scored higher than about 90% of the class—you’re in the 90th percentile.

Manufacturing Quality Control

A machine produces bolts with a target diameter of 10mm and standard deviation of 0.15mm. Bolts smaller than 9.7mm are rejected. With X=9.7, μ=10, σ=0.15, the CDF equals roughly 0.0228. This tells you about 2.3% of bolts will be rejected for being too small.

Delivery Time Estimates

A shipping company’s delivery times average 5 days with a standard deviation of 1.2 days. A customer needs their package within 7 days. Setting X=7, μ=5, σ=1.2 returns approximately 0.9525—there’s a 95% chance the package arrives on time.

Investment Returns

A stock’s monthly returns have a mean of 1.5% and standard deviation of 4%. What’s the probability of a negative month? With X=0, μ=1.5, σ=4, the CDF is about 0.3538. There’s roughly a 35% chance of losing money in any given month.

How to Use This Calculator

Enter your X value along with the distribution’s mean and standard deviation. The calculator instantly computes P(X ≤ x) and displays both the decimal probability and percentage. You’ll also see the corresponding z-score, which standardizes your value relative to the distribution’s center and spread.

You Should Understand Z-Scores:

The z-score tells you how many standard deviations your value sits from the mean. A z-score of 1.5 means the value is 1.5 standard deviations above average. Negative z-scores indicate values below the mean. This standardization allows comparison across different normal distributions.

0.05

Lower Tail

Only 5% fall at or below this value

0.50

At the Mean

Exactly half below, half above

0.95

Upper Tail

95% fall at or below this value

Z-Score Quick Reference

-3σ

0.13%

-2σ

2.28%

-1σ

15.87%

50%

+1σ

84.13%

+2σ

97.72%

+3σ

99.87%

The Empirical Rule

For any normal distribution, approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This 68-95-99.7 rule provides quick mental estimates before calculating exact probabilities. Values beyond three standard deviations occur less than 0.3% of the time, making them statistical outliers worthy of investigation.