Normal Distribution Calculator

The normal distribution is a continuous probability model shaped like a symmetric bell curve. Its mean determines the center, and its standard deviation determines how widely values spread around that center. Many measurement errors, biological traits, test-score summaries, and sampling statistics are approximately normal, which makes the distribution a central tool in statistics and data analysis. This calculator accepts any finite mean and any positive standard deviation. The probability density function, or PDF, reports the height of the curve at the lower value x. Density is not itself a probability because a continuous variable has zero probability of equaling one exact point. Instead, probabilities are areas under the curve. The cumulative distribution function, or CDF, reports the area to the left of x, P(X <= x). The upper-tail result is the complementary area to the right, P(X > x), and therefore equals one minus the CDF. To calculate a range, enter x as the lower endpoint and enter an upper endpoint greater than or equal to x. The calculator subtracts the lower cumulative probability from the upper cumulative probability. For example, the standard normal area from zero to one is about 0.3413. Symmetry gives the same area from minus one to zero, so about 68.27 percent of a normal population lies within one standard deviation of its mean. The z-score standardizes x by subtracting the mean and dividing by the standard deviation. A z-score of zero is exactly at the mean, a score of one is one standard deviation above it, and a score of minus two is two standard deviations below it. Standardization lets one common normal CDF evaluate distributions expressed in different units. The calculator uses a well-established numerical approximation to the error function to evaluate these areas. Normal models are powerful but should not be applied automatically. Strong skew, heavy tails, multiple peaks, hard boundaries, or influential outliers can make a normal curve misleading. A normal probability also describes a model, not proof about a real population. Check the data-generating context and use diagnostic plots when observations are available. For discrete counts, a continuity correction may be appropriate, while binomial, Poisson, t, or other distributions may better match the actual process.