Normal Distribution Calculator — Step-by-Step Solutions

Q: What is the empirical rule (68-95-99.7 rule)?

In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This provides a quick way to estimate probabilities.

Q: How do I know if my data is normally distributed?

You can use visual methods (histogram, Q-Q plot) or formal tests (Shapiro-Wilk, Anderson-Darling). Many natural phenomena approximate normality due to the Central Limit Theorem.

Q: What is the difference between the normal and standard normal distribution?

The standard normal distribution is a special case with mean 0 and standard deviation 1. Any normal distribution can be converted to it using z-scores: z = (x - mu) / sigma.

Q: Can the normal distribution produce negative values?

Yes. The normal distribution extends from negative infinity to positive infinity. If you are modelling a quantity that cannot be negative (like height), it works as an approximation when the mean is several standard deviations above zero.

Q: What numerical method does this calculator use?

This calculator uses the Abramowitz and Stegun rational polynomial approximation for the CDF and the Acklam algorithm for the inverse CDF, both providing accuracy to at least 8 significant digits.

The Normal Distribution Calculator computes probabilities for any normal distribution. It finds the probability between two values, calculates percentiles, and performs inverse lookups to find the value corresponding to a given probability.

Enter the mean and standard deviation of your distribution along with your target values. The calculator uses numerical integration to produce accurate results, displayed alongside an interactive bell curve that shades the region of interest.

Mean (mu)

Std Dev (sigma)

Lower bound

Upper bound

Your calculations will appear here

How to Use the Normal Distribution Calculator

Select a mode. "Probability Between" finds the probability that a value falls between two bounds. "Percentile" finds what percentage of values fall below a given value. "Inverse" finds the value corresponding to a given cumulative probability.
Enter the mean (mu) and standard deviation (sigma) of your normal distribution.
Enter the additional inputs required by your chosen mode: lower and upper bounds, a value, or a probability.
Review the results, step-by-step working, and the bell curve with shaded area showing the region of interest.

Formula and Theory

The normal distribution (also called the Gaussian distribution) is the most important probability distribution in statistics. It is defined by two parameters: the mean (mu), which determines the centre, and the standard deviation (sigma), which determines the spread. The probability density function forms the familiar bell-shaped curve.

The probability of a value falling within a specific range is found by computing the area under the curve between those bounds. This requires evaluating the cumulative distribution function (CDF). Since the normal CDF has no closed-form expression, it is computed using numerical methods such as rational polynomial approximations.

Key properties of the normal distribution include: it is symmetric about the mean, the mean equals the median and mode, and approximately 68% of values fall within one standard deviation, 95% within two, and 99.7% within three (the empirical rule). The standard normal distribution is the special case with mean 0 and standard deviation 1.

Worked Examples

Example 1: Probability between two values

Problem: IQ scores follow a normal distribution with mean 100 and standard deviation 15. What percentage of people have IQ between 85 and 115?

Solution: Convert to z-scores: z1 = (85 - 100) / 15 = -1, z2 = (115 - 100) / 15 = 1. P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6827.

Answer: About 68.27% of people have IQ between 85 and 115

Example 2: Percentile calculation

Problem: In a normally distributed dataset with mean 50 and standard deviation 8, what percentile is a score of 62?

Solution: z = (62 - 50) / 8 = 1.5. P(Z < 1.5) = 0.9332.

Answer: A score of 62 is at the 93.32nd percentile

Example 3: Inverse normal

Problem: Heights of adult men are normally distributed with mean 175 cm and standard deviation 7 cm. What height marks the top 10%?

Solution: The 90th percentile: z = 1.2816. Height = 175 + 1.2816 * 7 = 175 + 8.97 = 183.97.

Answer: The top 10% are taller than approximately 184 cm

Frequently Asked Questions

What is the empirical rule (68-95-99.7 rule)?

In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This provides a quick way to estimate probabilities.

How do I know if my data is normally distributed?

You can use visual methods (histogram, Q-Q plot) or formal tests (Shapiro-Wilk, Anderson-Darling). Many natural phenomena approximate normality due to the Central Limit Theorem.

What is the difference between the normal and standard normal distribution?

The standard normal distribution is a special case with mean 0 and standard deviation 1. Any normal distribution can be converted to it using z-scores: z = (x - mu) / sigma.

Can the normal distribution produce negative values?

Yes. The normal distribution extends from negative infinity to positive infinity. If you are modelling a quantity that cannot be negative (like height), it works as an approximation when the mean is several standard deviations above zero.

What numerical method does this calculator use?

This calculator uses the Abramowitz and Stegun rational polynomial approximation for the CDF and the Acklam algorithm for the inverse CDF, both providing accuracy to at least 8 significant digits.