Statistics
What two parameters define a normal distribution?
Its mean and its standard deviation.
Card 1 of 276. Answer: Its mean and its standard deviation.
Its mean and its standard deviation.
Card 1 of 276. Answer: Its mean and its standard deviation.
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Its mean μ and standard deviation σ, usually written N(μ, σ²) with the variance, which carries the same information. μ sets the location, σ the scale, and together they fix the whole curve: any normal variable is a shifted, stretched standard normal, X = μ + σZ with Z ~ N(0, 1).
Two numbers answer any probability question. If exam scores are normal with μ = 70 and σ = 8, 86 is two σ above the mean, so the share above it is P(Z > 2) ≈ 2.3%, and the 90th percentile is 70 + 1.2816 × 8 ≈ 80.3. The shape never changes: skewness is always 0 and excess kurtosis 0, so no third parameter is needed.
Their maximum likelihood estimates are the sample mean and the average squared deviation, dividing by n. A common trap: every dataset has a mean and standard deviation, but matching them does not make it normal, so check a Q-Q plot. A multivariate normal needs a mean vector and a covariance matrix.