Normal Distribution Calculator
Compute left-tail, right-tail, or two-tail probabilities for any normal distribution given its mean and standard deviation. Widely used in statistics, finance, and scientific research.
About this calculator
The normal distribution is a symmetric, bell-shaped continuous distribution fully described by its mean (μ) and standard deviation (σ). Any value x is first converted to a standard z-score: z = (x − μ) / σ. The cumulative distribution function (CDF) gives the probability that a random variable falls below x. This calculator uses a Taylor-series approximation of the error function: CDF ≈ 0.5 · (1 + erf(z/√2)). A left-tail result returns CDF directly; a right-tail returns 1 − CDF; a two-tail returns 2 · min(CDF, 1 − CDF), capturing both extremes. About 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ.
How to use
Suppose exam scores are normally distributed with mean μ = 70 and standard deviation σ = 10, and you want the probability that a student scores below 85. Set mean = 70, std_dev = 10, x_value = 85, tail_type = 'left'. The calculator computes z = (85 − 70) / 10 = 1.5, then evaluates the CDF at z = 1.5 ≈ 0.9332. There is approximately a 93.3% probability that a randomly chosen student scores below 85.
Frequently asked questions
What does the z-score mean in a normal distribution calculation?
The z-score measures how many standard deviations a value x sits above or below the mean: z = (x − μ) / σ. A z-score of 0 means x equals the mean; z = 1 means x is one standard deviation above average. Z-scores allow different normal distributions to be compared on a single standard scale. Once you have z, you look it up in (or compute) the standard normal CDF to obtain the probability.
How do I interpret left-tail versus right-tail probability in a normal distribution?
The left-tail probability P(X < x) is the area under the curve to the left of your chosen value — it tells you what fraction of the population scores below x. The right-tail P(X > x) = 1 − P(X < x) gives the fraction above x. The two-tail result 2·min(left, right) captures the probability of being at least as extreme as x in either direction, which is directly used as the p-value in two-sided hypothesis tests.
When is it appropriate to use a normal distribution instead of other distributions?
The normal distribution is appropriate when your data is continuous, roughly symmetric, and clusters around a central value — heights, measurement errors, and standardized test scores are classic examples. The Central Limit Theorem guarantees that the mean of a large enough sample will be approximately normal regardless of the underlying distribution, making it foundational in inferential statistics. Avoid it for skewed data (use log-normal), count data (use Poisson), or bounded proportions (use beta distribution).