Normal Distribution

The normal distribution, also called the Gaussian distribution or the bell curve, is a probability distribution in which values cluster symmetrically around an average, with fewer and fewer observations further away in either direction. It is fully described by two numbers: the mean, which sets the centre, and the standard deviation, which sets the spread. Human heights, measurement errors and many averages of random quantities are close to normal, which is why it is the most used distribution in statistics.

In economics and finance it underpins much of econometrics, the classic theory of portfolio diversification, option pricing and risk measures such as value at risk. It is also often criticised in finance, because market returns show more extreme moves than a normal curve predicts, a property known as fat tails. Distributions such as the Pareto distribution and other power laws are used where extremes dominate.

Picture the queue of people at a bakery sorted by height. Most are near the average, a fair number are a bit taller or shorter, and only a handful are very tall or very short. If you drew a chart of how many people fall at each height, it would rise to a smooth hump in the middle and fall away evenly on both sides, like a bell.

That shape appears whenever an outcome is the sum of many small, independent nudges, each pushing a little up or down. The nudges mostly cancel out, so results bunch near the middle, and getting far from the middle requires many nudges to go the same way, which is rare.

The probability density of a normal distribution is:

• $x$ is a possible value of the variable.
• $\mu$ is the mean, the centre of the distribution.
• $\sigma$ is the standard deviation, the typical distance from the mean.
• $\pi$ and $e$ are the mathematical constants (about 3.1416 and 2.7183).

A useful rule of thumb is that about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. For example, if a fund's annual return is assumed normal with a mean of 7% and a standard deviation of 10%, then in roughly two years out of three the return would fall between $7 - 10 = -3\%$ and $7 + 10 = 17\%$, and in about 19 years out of 20 between $-13\%$ and $27\%$. A loss of more than 23%, three standard deviations below the mean, would be expected only about once in 700 years under that assumption.

The French mathematician Abraham de Moivre derived the curve in 1733 as an approximation to the probabilities of coin tosses. Carl Friedrich Gauss used it in the early 19th century to describe errors in astronomical measurements, which is why it bears his name, and Pierre-Simon Laplace developed the central limit theorem, which explains why it appears so often: the average of many independent random variables tends towards a normal distribution, whatever their individual shapes. In finance, Louis Bachelier modelled price changes as normal in 1900, and the idea later fed into the random-walk view of markets and the efficient market hypothesis.

• Statistical inference. Standard errors, confidence intervals and hypothesis tests in randomized controlled trials and regressions often rely on normal approximations.
• Portfolio theory. Harry Markowitz's mean-variance framework treats volatility as the measure of risk, an approach that fits normal returns best, and ratios such as the Sharpe ratio build on it.
• Option pricing. The Black-Scholes model assumes that the logarithm of a share's price follows a normal distribution, so prices are lognormal.
• Risk management. Parametric value-at-risk models used by banks estimate potential losses from the mean and standard deviation of returns.
• Economic data. Errors in forecasts, measurement noise and many survey averages are often modelled as normal.

Real financial returns are usually not normal. They have fat tails, meaning extreme gains and losses happen far more often than the bell curve suggests, and they can be skewed, with large falls more common than large rises. On 19 October 1987 the Dow Jones Industrial Average fell by about 22.6% in a single day, a move that would be essentially impossible under a normal model calibrated to earlier data. Models assuming normal returns also underestimated losses in the 2008 financial crisis. Writers such as Nassim Nicholas Taleb, who popularised the idea of the black swan event, and Benoit Mandelbrot argued that relying on the bell curve breeds false confidence.

In response, regulators moved away from pure value at risk: the Basel Committee's market risk framework now uses expected shortfall, which looks at the average loss in the worst cases, and banks run stress tests on extreme scenarios. Many variables in economics, such as income and wealth, are also strongly skewed, so averages computed as if they were normal can mislead; studies of inequality use other distributions instead. The normal distribution remains a powerful default, but analysts are expected to check whether the data fit it before relying on it.

• NIST/SEMATECH e-Handbook of Statistical Methods: Normal Distribution