Normal Distribution (Gaussian Bell Curve)

The normal distribution is a model for continuous quantities whose spread arises from many small, independent influences, such as body heights or measurement errors. Its density function $f(x) =\frac{ 1}{(\sigma{}\sqrt{(2\pi{})})}\cdot{}e^{(-\frac{(x-\mu{})^{2}}{(2\sigma{}^{2})})}$ describes a symmetric bell curve around the mean μ. The parameter μ sets the location of the maximum, the standard deviation σ the width. The prefactor $\frac{1}{(\sigma{}\sqrt{(2\pi{})})}$ ensures that the total area under the curve is exactly one, as it must be for a probability density. The exponential term penalizes deviations from the mean quadratically, which is why far-away values become very unlikely. Probabilities correspond to areas under the curve.

The rule describes the share of a normal distribution within specified multiples of its standard deviation. About 68.27 percent lies within μ ± σ, 95.45 percent within μ ± 2σ, and 99.73 percent within μ ± 3σ. Example height model with $\mu{} = 178 cm$ and $\sigma{} = 7 cm$: the central 2σ interval runs from 164 to 192 cm and contains 95.45 percent of modeled values. The rule of thumb gives quick estimates without evaluating integrals. It applies to a normal distribution; for real measurements, first check whether that model is appropriate. Values beyond three standard deviations are rare in the model, but not impossible.

The z-score states how many standard deviations a value is away from the mean. You compute it with $z =\frac{ (x - \mu{})}{\sigma{}}$, so you subtract the mean and divide by the standard deviation. A z-score of 2 means the value lies two standard deviations above the mean, a negative z-score correspondingly below it. Through standardization you convert any normal distribution into the standard normal distribution with mean zero and standard deviation one. Only then can you use tabulated values or the function Φ to read off probabilities. Standardization is the key to making different normal distributions comparable with each other.

The variance $\sigma{}^{2}$ is the mean square of the deviations from the mean and measures the spread in squared units. The standard deviation σ is the square root of the variance and has the same unit as the original data, which makes it more intuitive. For body heights in centimetres, σ is also in centimetres, whereas $\sigma{}^{2}$ is in square centimetres. In the density function of the normal distribution σ appears directly as the width parameter, while $\sigma{}^{2}$ sits in the exponent. A common mistake is to confuse σ and $\sigma{}^{2}$, for example to compute the z-score with the variance instead of the standard deviation. Remember: σ is the typical deviation, $\sigma{}^{2}$ its square.

The reason is the central limit theorem of statistics. It states that the sum or the mean of many independent random influences is approximately normally distributed, almost regardless of how the individual influences are distributed. Many natural quantities arise exactly this way: body height depends on countless genetic and environmental factors, a measurement error on many small disturbances. As a result the influences add up to a bell-shaped distribution. That is why the normal distribution appears everywhere, from biology through physics to quality control. It is also the reason why many statistical methods assume the normal distribution as a standard model and build on it.