Properties of the normal distribution; when the Poisson tends to the normal

Properties of the normal distribution. A normal distribution N(μ, σ²) is a continuous distribution described completely by its two parameters, the mean μ and the standard deviation σ. Its main properties:

1. Bell-shaped and symmetrical about the mean, so its skewness is zero.
2. Mean = median = mode, all at the centre μ, where the curve is highest.
3. Asymptotic: the two tails come ever closer to the horizontal axis but never touch it; X can in theory take any real value.
4. Total area under the curve is 1, half on each side of the mean.
5. Points of inflexion at μ − σ and μ + σ, where the curve changes its bend: it is concave upward to the left of μ − σ, concave downward between μ − σ and μ + σ, and concave upward again to the right of μ + σ.
6. Fixed areas: about 68.27% of the area lies within μ ± σ, 95.45% within μ ± 2σ and 99.73% within μ ± 3σ.
7. Quartile deviation and mean deviation are about 0.675σ and 0.8σ.
8. Mesokurtic: β₂ = 3, the benchmark against which kurtosis is judged.
9. Standard form: z = (X − μ) / σ follows N(0, 1), so a single table of areas serves every normal distribution.

When the Poisson distribution tends to the normal. A Poisson distribution with mean m (its variance is also m) is skewed to the right when m is small. As m becomes large (in theory m → ∞; in practice, m above about 10 to 20) the distribution becomes more and more symmetrical and bell-shaped, and can be approximated by a normal distribution with mean m and standard deviation √m, z = (X − m) / √m. (A continuity correction of 0.5 is used, since the Poisson is discrete and the normal continuous.)

Example: if a call centre receives on average 60 calls an hour, the chance of more than 70 calls in an hour can be found from the normal curve with mean 60 and SD √60 = 7.75 instead of adding many Poisson terms.