Normal characteristics; relation to standard normal

Major characteristics of the normal distribution

A normal random variable X with mean μ and standard deviation σ has the pdf

f(x) = (1/(σ√(2π))) e^(−(x − μ)²/(2σ²)), −∞ < x < ∞.

1. The curve is bell shaped and symmetric about x = μ.
2. Mean = median = mode = μ.
3. The total area under the curve is 1, half on each side of μ.
4. The curve is asymptotic to the x-axis on both sides.
5. Points of inflection are at μ ± σ.
6. Skewness is 0 and kurtosis is 3.
7. Empirical rule: about 68.27 %, 95.45 % and 99.73 % of the area lies within μ ± σ, μ ± 2σ and μ ± 3σ.
8. It is fully described by two parameters, μ and σ.
9. A linear combination of independent normal variables is also normal.

Relation with the standard normal distribution

Every normal distribution N(μ, σ²) can be converted into the standard normal distribution N(0, 1) by the transformation

Z = (X − μ)/σ

• Z has mean 0 and standard deviation 1.
• P(a < X < b) = P((a − μ)/σ < Z < (b − μ)/σ), found from one standard table.
• The reverse is X = μ + σZ.

So the standard normal is simply the normal distribution measured in units of standard deviations from the mean, and all normal curves have the same shape after standardising.