Normal Distribution Calculator
This normal distribution calculator (also a bell curve calculator) calculates the area under a bell curve and establishes the probability of a value being higher or lower than any arbitrary value X. You can also use this probability distribution calculator to find the probability that your variable is in any arbitrary range, X to X₂, just by using the normal distribution mean and standard deviation values.
Normal Distribution Definition
The normal distribution (also known as the Gaussian) is a continuous probability distribution. Most data is close to a central value, with no bias to left or right. Many observations in nature, such as the height of people or blood pressure, follow this distribution.
In a normal distribution, the mean value is also the median (the "middle" number of a sorted list of data) and the mode (the value with the highest frequency of occurrence). As this distribution is symmetric about the center, 50% of values are lower than the mean, and 50% of values are higher than the mean.
Another parameter characterizing the normal distribution is the standard deviation. It describes how widespread the numbers are. Generally, 68% of values should be within 1 standard deviation from the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. The number of standard deviations from the mean is called the z-score.
What Is the Standard Normal Distribution?
You can standardize any normal distribution through a process known as the standard score. A standard normal distribution has the following properties:
- Mean value is equal to 0
- Standard deviation is equal to 1
- Total area under the curve is equal to 1
- Every value of variable x is converted into the corresponding z-score
The total area under the standard normal distribution curve equals 1, which corresponds to probability. You can calculate the probability that your value is lower than any arbitrary X — written as P(x < X) — as the area under the graph to the left of the z-score of X.
The Normal CDF Formula
The cumulative distribution function (CDF) of the normal distribution is:
P(x < X) = 0.5 × [1 + erf((X − μ) / (σ × √2))]
Where:
- μ — mean of the distribution
- σ — standard deviation
- X — the value you are evaluating
- erf — the error function
Formula for Normal Distribution: An Example
Suppose test scores are normally distributed with a mean of 70 and a standard deviation of 10. What is the probability that a randomly selected student scores less than 80?
- Calculate the z-score: Z = (80 − 70) / 10 = 1.0
- Apply the CDF: P(x < 80) = 0.5 × [1 + erf(1.0 / √2)] ≈ 84.13%
This means about 84% of students score below 80 points.
How to Use the Normal Distribution Calculator: An Example
- Enter the mean (μ) — the average value of your dataset
- Enter the standard deviation (σ) — a positive number indicating data spread
- Enter the X value — the point at which you want to evaluate the probability
- Optionally enter X₂ to compute the probability within a range [X, X₂]
- Select a measurement system (pure number, metric, US/imperial, or currency) to display units
- Click Calculate to see z-scores, P(x < X), P(x > X), and range probability
Units of Measurement
The normal distribution applies to any measurable quantity. This calculator supports labelling values in:
- Pure numbers — statistical scores, IQ, test results
- Metric units — cm, kg, m, °C and more
- US / Imperial units — ft, lb, °F, gal and more
- Currencies — USD ($), RUB (₽), EUR (€), GBP (£), JPY (¥), CNY (¥), and 16 other world currencies — useful for financial data analysis
The Bell Curve: Key Properties
- 68–95–99.7 rule — approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ
- Symmetry — the curve is perfectly symmetric around the mean
- Unimodal — it has exactly one peak at the mean
- Asymptotic — the tails approach but never touch zero
The Central Limit Theorem
The central limit theorem states that the sum (or average) of a large number of independent, identically distributed random variables tends toward a normal distribution, regardless of the original distribution. This is why the normal distribution appears so frequently in statistics and data science — it describes the behavior of sample means, measurement errors, and many natural phenomena.
Normal Distribution and Statistical Testing
The normal distribution is the foundation of many statistical tests:
- Z-test — compares a sample mean to a known population mean
- T-test — used when the population standard deviation is unknown
- ANOVA — compares means across multiple groups, assuming normality
- Confidence intervals — built using z-scores from the normal distribution
- P-values — derived from the area under the normal curve
Financial Applications
In finance, the normal distribution is commonly used to model:
- Stock price returns (USD, RUB, EUR, etc.)
- Risk assessment and Value at Risk (VaR)
- Option pricing models (e.g., Black-Scholes)
- Portfolio performance analysis
FAQs
What is a z-score?
A z-score (standard score) tells you how many standard deviations a value X is from the mean: Z = (X − μ) / σ. A z-score of 0 means the value equals the mean; a z-score of 1 means it is one standard deviation above the mean.
What does P(x < X) mean?
It is the probability that a randomly chosen value from the distribution is less than X. It equals the area under the bell curve to the left of X.
How is P(x > X) calculated?
P(x > X) = 1 − P(x < X). Because all probabilities sum to 1, the right-tail probability is simply the complement of the left-tail.
What is the difference between normal distribution and standard normal distribution?
Any normal distribution can be standardized by converting to z-scores (subtracting the mean and dividing by the standard deviation). The standard normal distribution has μ = 0 and σ = 1.
Can this calculator be used for currencies?
Yes. Select "Currency" as the measurement system and choose your preferred currency — USD ($), Russian Ruble (₽), Euro (€), or any of the 22 supported currencies. The results will display with the chosen currency symbol, making it easy to analyze financial data distributions.