Normal distribution

A continuous probability distribution with a bell-shaped density symmetric about its mean. Its mean sets the center, and its variance sets the spread.

Probability density function: A function whose values describe the relative density of a continuous random variable across its possible values. The normal curve is a specific density function; probabilities come from areas beneath it.
Expected value: The probability-weighted average of a random variable's possible values. For a normal distribution, the expected value is its center of symmetry.
Central limit theorem: A theorem stating that suitably standardized sums of many independent random variables often approach a normal distribution. It explains why normal approximations arise for sample sums and averages.
Abraham de Moivre: An eighteenth-century mathematician whose work included an early normal approximation to the binomial distribution. His 1733 approximation anticipated the normal curve's role in probability.
Student's t-distribution: A symmetric distribution with heavier tails than the standard normal, indexed by degrees of freedom. It accounts for extra uncertainty when estimating variance from small samples.
Standard normal distribution: The normal distribution with mean zero and variance one. Standardizing any normal variable maps it to this reference distribution.
Variance: The expected squared deviation of a random variable from its mean. It is the normal distribution's second parameter and controls its spread.
Confidence interval: A data-based interval procedure designed to cover an unknown parameter at a stated long-run rate. Normal quantiles set interval widths when estimates have approximately normal sampling distributions.
Pierre-Simon Laplace: A French mathematician who developed major results in probability, celestial mechanics, and statistics. His work extended normal approximations and helped establish their mathematical foundations.
Uniform distribution: A probability distribution assigning equal density across a specified interval. Its flat density contrasts with the normal distribution's concentration around its mean.

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