Statistics
In a normal distribution which parameter indicates where the bell is centered?
The mean.
Card 1 of 259. Answer: The mean.
The mean.
Card 1 of 259. Answer: The mean.
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The mean, μ. A normal distribution N(μ, σ²) has two parameters: μ sets where the bell is centered and the standard deviation σ sets its width. Because the curve is symmetric, μ is also the median and the mode, the point where the density peaks.
The density is f(x) = (1 / (σ√(2π))) · exp(−(x − μ)² / (2σ²)). Its exponent is 0 only at x = μ, so the peak sits exactly at the mean. Changing μ slides the curve without changing its shape: many IQ tests are scaled to mean 100 and standard deviation 15, and raising the mean to 110 would shift every percentile up 10 points.
Interviewers may contrast μ, the population parameter, with the sample mean x̄, its maximum likelihood estimate. In a right-skewed distribution such as income, the mean is pulled above the median, so "center" stops being one number. And not every bell-shaped curve has a mean: the Cauchy distribution is centered on its median, but its mean is undefined.