Normalcdf Calculator
Calculate probabilities using the normal distribution cumulative distribution function (CDF). Find the probability that a value falls between two bounds, compute Z-scores, or find percentiles with inverse normal.
Normal Distribution CDF Calculator
Calculate P(a < X < b) for a normal distribution with mean μ and standard deviation σ
Leave empty or check -∞ for negative infinity
Leave empty or check +∞ for positive infinity
Center of the distribution (default: 0 for standard normal)
Spread of the distribution (must be positive, default: 1 for standard normal)
Result
Probability
Visual Representation
Interpretation
Z-Score Calculator
Convert a value to its Z-score and find the probability: Z = (X - μ) / σ
Z-Score
P(Z < z) - Left Tail
Inverse Normal Calculator (invNorm)
Find the value (X) at a given percentile or probability in the normal distribution
Enter 0.5 for median, 0.95 for 95th percentile, etc.
X Value at Percentile
Common Z-Score Reference Table
| Z-Score | Probability (Left Tail) | Percentile |
|---|---|---|
| -3.00 | 0.0013 | 0.13% |
| -2.00 | 0.0228 | 2.28% |
| -1.96 | 0.0250 | 2.50% (95% CI lower bound) |
| -1.00 | 0.1587 | 15.87% |
| 0.00 | 0.5000 | 50% (Median) |
| 1.00 | 0.8413 | 84.13% |
| 1.96 | 0.9750 | 97.50% (95% CI upper bound) |
| 2.00 | 0.9772 | 97.72% |
| 3.00 | 0.9987 | 99.87% |
Understanding Normal Distribution
What is Normal Distribution?
The normal distribution, also known as the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric around its mean. It's characterized by two parameters: mean (μ) and standard deviation (σ). Many natural phenomena follow this distribution, including heights, test scores, and measurement errors.
The 68-95-99.7 Rule (Empirical Rule)
- 68% of data falls within 1 standard deviation of the mean (μ ± σ)
- 95% of data falls within 2 standard deviations of the mean (μ ± 2σ)
- 99.7% of data falls within 3 standard deviations of the mean (μ ± 3σ)
When to Use Each Calculator
normalcdf
Use when you know the range (a, b) and want to find the probability
Z-Score
Use to standardize a value and compare across different distributions
invNorm
Use when you know the probability and want to find the corresponding value
Standard Normal Distribution
When μ = 0 and σ = 1, the distribution is called the standard normal distribution or Z-distribution. Any normal distribution can be converted to standard normal using the Z-score formula: Z = (X - μ) / σ
Key Formulas
- normalcdf(a, b, μ, σ): P(a < X < b) for normal distribution
- Z-Score: Z = (X - μ) / σ standardizes any value
- invNorm(p, μ, σ): Finds X such that P(X < x) = p
- Probability Density Function: f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
- Total Area under curve: Always equals 1 (100%)
Frequently Asked Questions
What is normalcdf and how do I use it?
Normalcdf (normal cumulative distribution function) calculates the probability that a value falls within a specified range in a normal distribution. You input the lower bound (a), upper bound (b), mean (μ), and standard deviation (σ) to find P(a < X < b). This is essential for determining probabilities in bell curve distributions and is commonly used in statistics for hypothesis testing and confidence intervals.
What is a Z-score and how is it calculated?
A Z-score measures how many standard deviations a data point is from the mean. It's calculated using the formula Z = (X - μ) / σ, where X is your value, μ is the mean, and σ is the standard deviation. Z-scores standardize values, allowing you to compare data from different distributions. For example, a Z-score of 2.0 means the value is 2 standard deviations above the mean.
What is the 68-95-99.7 rule (Empirical Rule)?
The Empirical Rule states that in a normal distribution: approximately 68% of data falls within 1 standard deviation of the mean (μ ± σ), 95% falls within 2 standard deviations (μ ± 2σ), and 99.7% falls within 3 standard deviations (μ ± 3σ). This rule helps quickly estimate probabilities and identify outliers in normally distributed data without complex calculations.
How do I calculate percentages using normalcdf?
To convert normalcdf results to percentages, multiply the probability by 100. For example, if normalcdf returns 0.8413, this equals 84.13%. This percentage represents the proportion of data falling within your specified range. For more general percentage calculations, check out our percentage calculator.
What is the difference between normalcdf and invNorm?
Normalcdf and invNorm are inverse operations. Normalcdf takes a range of values and returns the probability (area under the curve), while invNorm takes a probability and returns the corresponding value (X). Use normalcdf when you know the value range and want to find probability; use invNorm when you know the probability/percentile and want to find the value.
What is a confidence interval and how does it relate to normal distribution?
A confidence interval is a range of values within which a population parameter is likely to fall with a certain probability. Common confidence intervals include 95% (Z = ±1.96) and 99% (Z = ±2.58). You can use normalcdf to find the probability within any confidence interval, or invNorm to find the Z-scores that define a specific confidence level.
How do I interpret standard deviation in a normal distribution?
Standard deviation (σ) measures the spread or variability of data in a normal distribution. A larger standard deviation means data is more spread out from the mean, creating a wider, flatter bell curve. A smaller standard deviation means data clusters tightly around the mean, creating a taller, narrower curve. In the standard normal distribution, σ = 1.
What is the standard normal distribution (Z-distribution)?
The standard normal distribution is a special case of the normal distribution with mean μ = 0 and standard deviation σ = 1. It's also called the Z-distribution. Any normal distribution can be converted to standard normal using Z-scores. This standardization allows statisticians to use Z-tables and makes comparing different distributions easier.
How is normalcdf used in calculating GPA probabilities?
If GPAs in a class follow a normal distribution, you can use normalcdf to find the probability of achieving a certain GPA range. For example, if the class average is 3.0 with a standard deviation of 0.5, normalcdf can tell you what percentage of students score between 3.5 and 4.0. Use our GPA calculator for detailed GPA calculations.
What are the practical applications of normal distribution in real life?
Normal distribution appears throughout nature and science: human heights and weights, test scores (SAT, IQ), measurement errors, blood pressure readings, product dimensions in manufacturing, and financial returns. Understanding normalcdf helps analyze these phenomena, make predictions, set quality control limits, and determine probabilities for decision-making in fields like medicine, engineering, finance, and social sciences.